mit_ocw_relativity_8_033.jsonl

{"unit_id": "mit-ocw:0001-course-study-guide-handout-5a9725ed-a1d0160e:page-0001", "text": "# MIT Course 8.033, Fall 2005, Study Guide\n\nProf. Max Tegmark\n(Last revised September 7, 2005.)\n\n# MATH\n\n- Be able to do basic calculations with matrices (add, multiply, transpose, etc.).\n- Be able to solve variational calculus problems with the Euler-Lagrange equation.\n\n# SPECIAL RELATIVITY BASICS & KINEMATICS\n\nBe able to solve problems that involve converting between different intertial frames. Key concepts:\n\n- Invariance/symmetry under a transformation (translation, rotation, Galilean, Lorentz, etc.)\n- Aether, Michaelson-Morley experiment\n- The two postulates of special relativity\n- Event, inertial frame\n- Lorentz transformation, transforming a 4-vector\n- c, β, γ\n- 4-vectors: spacetime 4-vector, velocity 4-vector, wave 4-vector\n- Time dilation, length contraction, relativity of simultaneity, spacetime\n- Invariants: rest length, proper time interval\n- Event separations: spacelike, timelike & null\n- Velocity addition, aberration, Doppler shift\n- Numerical value of speed of light.\n\n# Breadth\n\n- Key experimental evidence for special relativity\n- Galilieo, Newton, Maxwell, Michaelson & Morley, Lorentz, Emmy Noether, Einstein (if you can't remember their key achievement(s) discussed in class, Google them – same for the people listed below)\n\n# SPECIAL RELATIVITY DYNAMICS\n\nBe able to solve problems that involve acceleration and force. Examples: Rocket problems, particle accelerators. Key concepts:\n\n- Mass-energy unification\n- 4-vectors: momentum 4-vector, acceleration 4-vector, force 4-vector\n- Rest mass invariant\n\n# PARTICLE PHYSICS\n\n# Depth\n\nBe able to solve particle physics problems (collisions etc.) using energy-momentum conservation and these key concepts:\n\n- Conservation laws: energy-momentum, charge, lepton number, etc.\n- Rest mass, rest energy, kinetic energy, total energy, binding energy\n\n# Breadth\n\nBasic familiarity with this:\n\n- Particles: electron, muon, proton, neutron, pion, neutrino, photon, graviton, nucleus, atom, hydrogen, deuterium, helium, carbon, iron, ion, molecule, alpha particle, gamma ray, the six quarks, the six leptons, baryon, fermion, boson, antiparticles\n- Ballpark masses of electron, proton, neutron.\n- Terminology: atomic number, atomic weight, isotope, mass excess,\n- The four fundamental interactions, chemical reactions, nuclear reactions, elementary particle reactions\n- Beta decay, Compton scattering, recoil, Mössbauer effect, Pound Rebka experiment\n- Particle accelerators: linear accelerator, circular accelerator, fixed-target experiment vs. colliding beam experiment", "source": "mit-ocw", "source_doc_id": "0001-course-study-guide-handout-5a9725ed-a1d0160e", "source_title": "MIT Course 8.033, Fall 2005, Study Guide: MATH", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Study Guide: MATH", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0001-course-study-guide-handout-5a9725ed-a1d0160e:page-0002", "text": "- Fusion in stars, Eddington\n- Qualitative knowledge of some key unsolved problems in particle physics\n\n# ELECTROMAGNETISM\n\nBe able to solve problems involving the Lorentz force law and transforming the electric and magnetic fields between frames.\n\n- Lorentz force law\n- Lorentz transforming the electromagnetic field\n- Current 4-vector\n- Electromagnetic field from moving charge, retarded position\n- Concept: What's meant by “E implies B”\n\n# GENERAL RELATIVITY BASICS\n\nBe able to solve problems involving particle motion in various metrics (Minkowski, Newtonian, FRW and Schwarzschild).\n\n# Depth\n\n- General concept of a metric, how to work with it.\n- How a metric transforms when you change coordinates.\n- For a spatial metric (2D, 3D), how to compute length of a given curve.\n- For a spacetime metric (4D), definition of timelike, spacelike and null curves through spacetime.\n- How to compute ageing along a timelike curve.\n- How to compute proper length of a spacelike curve.\n- Definition of geodesic.\n- How do compute geodesics given a metric.\n- How to compute the trajectory of a massive particle.\n- How compute the trajectory of a photon.\n- How to compute gravitational redshift.\n\n# Breadth\n\n- Definition of weak and strong equivalence principle.\n- Effects of spatial curvature (on angles, “parallel lines”). Effects of spacetime curvature (can’t set up global inertial frame; no coordinate transformation gives Minkowski metric).\n- Experimental evidence for general relativity.\n\n# COSMOLOGY\n\n# Depth\n\n- The FRW metric\n- Interpretation of FRW metric (spatial curvature, expansion, comoving objects, geodesics, cosmological redshift)\n- The Friedmann equation, how its solution depends qualitatively on the cosmological parameters.\n- How  $\\rho$ depends on  $a$ for various de density components.\n- Cosmological parameters:  $\\Omega_{\\gamma}$,  $\\Omega_{\\mathrm{m}}$,  $\\Omega_{\\mathrm{k}}$,  $\\Omega_{\\Lambda}$,  $\\Omega_{b}$,  $\\Omega_{d}$,  $h$, their meaning\n- Age of the Universe, qualitative dependence on cosmological parameters.\n\n# Breadth\n\n- Evidence for Big Bang\n- Rough timeline for the history of the Universe (planck time, inflation, nucleosynthesis, recombination, first stars, typical galaxy formation, formation of Earth, now, death of Sun, ultimate cosmic fate) and rough qualitative understanding of these stages.\n- Rough values of key cosmic distance scales (size of Earth's orbit, size of solar system, distance to nearest star, size of Milky Way galaxy, distance to Andromeda galaxy, size of observable universe).\n- Rough measured values of the above-mentioned cosmological parameters.\n- Qualitative observational knowledge about the metric of our Universe from Science article (curvature, topology, expansion history, fluctuation growth, black holes).\n- Qualitative knowledge of some key unsolved problems in Cosmology\n- Friedmann, Gamow, Hubble, Penzias & Wilson", "source": "mit-ocw", "source_doc_id": "0001-course-study-guide-handout-5a9725ed-a1d0160e", "source_title": "MIT Course 8.033, Fall 2005, Study Guide: MATH", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Study Guide: MATH", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0001-course-study-guide-handout-5a9725ed-a1d0160e:page-0003", "text": "# SCHWARZSCHILD METRIC & BLACK HOLES\n\n# Depth\n\nBe able to solve problems using the Schwarzschild metric.\n\n- The Schwarzschild metric\n- Interpretation of the Schwarzschild metric (t-coordinate, r-coordinate, shell coordinates, gravitational redshift, event horizon, Schwarzschild radius, event horizon)\n- Geodesics of the Schwarzschild metric: radial and angular, stable and unstable circular orbits, computing general geodesics using the effective potential, computing weak light deflection and Mercury perihelion shift, tidal forces\n- Definition of a black hole\n- Gravitational lensing, Einstein rings\n\n# Breadth\n\n- Evidence that General Relativity is correct\n- Evidence that black holes exist (both stellar mass and supermassive)\n- How black holes probably form\n- No-hair theorem: the three properties of a black hole\n- Singularity\n- Hawking radiation\n- Falling into a black hole: what it feels like and what it looks like from afar\n- River model of black holes, sense in which nothing special happens at the event horizon\n- Time travel: possibilities for going forward and backward, wormholes\n- Taylor-Hulse binary neutron star system\n- Shapiro time delay\n- The GPS satellite system\n- Schwarzschild, Wheeler, Hawking\n- Qualitative knowledge of some key unsolved problems in general relativity", "source": "mit-ocw", "source_doc_id": "0001-course-study-guide-handout-5a9725ed-a1d0160e", "source_title": "MIT Course 8.033, Fall 2005, Study Guide: MATH", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Study Guide: MATH", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-symmetry-handout-c44cab00-ef8e3300:page-0001", "text": "# MIT Course 8.033, Fall 2005, Symmetry & Invariance\n\n# Max Tegmark\n\n# Last revised September 13 2005\n\n# 1 Topics\n\n- The notion of symmetry in physics. Key concepts:\n\n- Frame\n- Inertial frame\n- Transformation\n- Invariant\n- Invariance\n- Symmetry\n- Relativity\n\n• Examples of possible symmetry:\n\n- Translation\n- Rotation\n- Parity\n- Galilean\n- Lorentz\n- Diffeomorphism\n- Gauge\n\n• We will study the symmetry of:\n\n- Classical mechanics — both initial conditions (v, E, etc.) and laws\n- Electromagnetism — in particular\n- * The wave equation\n- * Observed properties of light — does speed depend on wavelength? On motion of source? On motion of observer (frame)?\n\n• Key people:\n\n- Galileo Galilei, 1564-1642\n- Emmy Noether, 1882-1935\n- Michaelson & Morley", "source": "mit-ocw", "source_doc_id": "0002-symmetry-handout-c44cab00-ef8e3300", "source_title": "MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-symmetry-handout-c44cab00-ef8e3300:page-0002", "text": "# 2 Formula summary: transformations\n\n- Translation:\n\n$$\n\\left\\{ \\begin{array}{l l} \\mathbf {r} ^ {\\prime} = \\mathbf {r} + \\Delta r \\\\ t ^ {\\prime} = t + \\Delta t \\end{array} \\right.\n$$\n\n- Rotation:\n\n$$\n\\left\\{ \\begin{array}{l} \\mathbf {r} ^ {\\prime} = \\mathbf {R r} \\\\ t ^ {\\prime} = t \\end{array} \\right.\n$$\n\n- Parity\n\n$$\n\\left\\{ \\begin{array}{l l} \\mathbf {r} ^ {\\prime} = - \\mathbf {r} \\\\ t ^ {\\prime} = t \\end{array} \\right.\n$$\n\n- Galilean:\n\n$$\n\\left\\{ \\begin{array}{l l} \\mathbf {r} ^ {\\prime} = \\mathbf {r} - \\mathbf {v} t \\\\ t ^ {\\prime} = t \\end{array} \\right.\n$$\n\n- Lorentz:\n\n$\\mathbf{x}^{\\prime} = \\boldsymbol {\\Lambda}\\mathbf{x},\\quad i.e.,$ for  $\\mathbf{v}$ in  $x$ -direction,\n\n$$\n\\left\\{ \\begin{array}{l l} x ^ {\\prime} = \\gamma (x - v t) \\\\ y ^ {\\prime} = y \\\\ z ^ {\\prime} = z \\\\ t ^ {\\prime} = \\gamma (t - v x / c ^ {2}) \\end{array} \\right.,\n$$\n\n$$\n\\gamma \\equiv \\frac {1}{\\sqrt {1 - \\frac {v ^ {2}}{c ^ {2}}}}.\n$$", "source": "mit-ocw", "source_doc_id": "0002-symmetry-handout-c44cab00-ef8e3300", "source_title": "MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-symmetry-handout-c44cab00-ef8e3300:page-0003", "text": "# 3 Symmetry in physics\n\n# 3.1 Glossary of key concepts\n\n- Frame: A prescription for measuring the physical quantities (e.g.,  $\\mathbf{r}, t$) that appear in our equations (an operational definition of them).\n- Inertial frame: A particular kind of frame in which Newton's 1st law holds.\n- Transformation: The mathematical operation converting the quantities measured in one frame into the quantities measured in another. To avoid confusion, we usually put primes on the quantities in one of the two frames.\n- Invariant: A quantity or equation that is left unchanged by a transformation. A quantity  $Q$ is invariant if  $Q' = Q$. An equation is invariant if it stays true in the new frame (i.e., when you put primes on all quantities).\n- Invariance: Being invariant. For example, the equations of classical mechanics are said to be Galilean invariant because they are invariant under Galilean transformations.\n- Symmetry: In physics, essentially a synonym for invariance.\n- Relativity: Essentially a synonym for invariance and symmetry, but usually restricted to the symmetries of spacetime.\n\n# 3.2 Invariance under translation\n\n- No experiment within your lab can determine whether it's been shifted sideways. In other words, the laws of nature appear to be translationally invariant.\n\n- Original frame: masses at  $\\mathbf{r}_1$ and  $\\mathbf{r}_2$.\n\n$$\nF = \\frac {G m M}{| \\mathbf {r} _ {2} - \\mathbf {r} _ {1} | ^ {2}}\n$$\n\n- Primed frame: masses at  $\\mathbf{r}_1' \\equiv \\mathbf{r}_1 + \\mathbf{a}$ and  $\\mathbf{r}_2' \\equiv \\mathbf{r}_2 + \\mathbf{a}$.\n\n$$\nF ^ {\\prime} = \\frac {G m M}{| \\mathbf {r} _ {2} ^ {\\prime} - \\mathbf {r} _ {1} ^ {\\prime} | ^ {2}} = \\frac {G m M}{| (\\mathbf {r} _ {2} + \\mathbf {a}) - (\\mathbf {r} _ {1} + \\mathbf {a}) | ^ {2}} = \\frac {G m M}{| \\mathbf {r} _ {2} - \\mathbf {r} _ {1} | ^ {2}} = F\n$$\n\n- We have time translationak invariance in time as well as in space.", "source": "mit-ocw", "source_doc_id": "0002-symmetry-handout-c44cab00-ef8e3300", "source_title": "MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-symmetry-handout-c44cab00-ef8e3300:page-0004", "text": "# 3.3 Invariance under rotation\n\n- No experiment within your spaceship can determine whether it's been rotated. In other words, the laws of nature appear to be rotationally invariant.\n- If you're not cool with  $3 \\times 3$ matrices, please read the matrix primer handout.\n- Primed frame: masses at  $\\mathbf{r}_1' \\equiv \\mathbf{R}\\mathbf{r}_1$ and  $\\mathbf{r}_2' \\equiv \\mathbf{R}\\mathbf{r}_2$\n\n$$\nF ^ {\\prime} = \\frac {G m M}{\\left| \\mathbf {R r} _ {2} - \\mathbf {R r} _ {1} \\right| ^ {2}} = \\frac {G m M}{\\left| \\mathbf {R} \\left(\\mathbf {r} _ {2} - \\mathbf {r} _ {1}\\right) \\right| ^ {2}} = \\frac {G m M}{\\left| \\mathbf {r} _ {2} - \\mathbf {r} _ {1} \\right| ^ {2}} = F\n$$\n\n- Another example: Maxwell's equations in vacuum imply\n\n$$\n\\nabla^ {2} \\mathbf {E} = \\frac {1}{c ^ {2}} \\ddot {\\mathbf {E}}.\n$$\n\nSince only differences in position and time enter, it's translationally invariant.\n\nHere it's infinitesimal differences (derivatives), above it was a finite difference  $(\\mathbf{r}_2 - \\mathbf{r}_1)$.\n\n- $\\nabla^2$ is invariant under rotation (remember Gauss' theorem)\n- At MIT:\n\n$$\n\\mathbf {E} = \\frac {1}{c ^ {2}} \\ddot {\\mathbf {E}} = \\left( \\begin{array}{c} 0 \\\\ 0 \\\\ 1 \\end{array} \\right).\n$$\n\n- Near Australia:\n\n$$\n\\mathbf {E} = \\frac {1}{c ^ {2}} \\ddot {\\mathbf {E}} = \\left( \\begin{array}{c} 0 \\\\ 0 \\\\ - 1 \\end{array} \\right).\n$$\n\n- So both observer's agree that Maxwell was right, i.e., the wave equation is translationally and rotationally invariant.", "source": "mit-ocw", "source_doc_id": "0002-symmetry-handout-c44cab00-ef8e3300", "source_title": "MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-symmetry-handout-c44cab00-ef8e3300:page-0005", "text": "# 3.4 Invariance under reflection (parity)?\n\n- Yes for all of classical physics\n- Considered self-evident and obvious\n- 1956: Chen Ning Yang & Tsung-Dao Lee propose that weak interactions violate parity; Chien-Shiung Wu demonstrates it with cobalt 60, Leon Lederman with accelerator. (Yang & Lee get 1957 Nobel prize.)\n\n# 3.5 Symmetry is at the heart of modern physics\n\n- Special relativity is all about so-called Lorentz symmetry.\n- General relativity is about so-called diffeomorphism symmetry.\n- Key topics in particle physics are C, P and T symmetry and combinations like CP and CPT symmetry.\n- A cornerstone of particle physics is gauge symmetry\n- In 2007, the Large Hadron Collider at CERN will search for supersymmetry.", "source": "mit-ocw", "source_doc_id": "0002-symmetry-handout-c44cab00-ef8e3300", "source_title": "MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-symmetry-handout-c44cab00-ef8e3300:page-0006", "text": "# 4 The symmetry properties of classical mechanics\n\nUnder what transformations are the laws of classical mechanics invariant?\n\nAnswer: Galilean transformations.\n\n# 4.1 Invariance under Galilean transformation\n\n- Demo with colliding carts, ball.\n- So Newtonian mechanics appears to be invariant - let's understand exactly what the transformation is, and why this is so.\n- Inertial frame definition (a = 0 if F = 0)\n- Are we in an inertial frame? (PS1)\n- Galilean transformation definition (between 2 inertial frames)\n- Definition of event: a 4D point  $(x, y, z, t)$. Examples?\n\n•   $r' = r - vt$\n\n- Lengths invariant:  $\\Delta \\mathbf{r}' \\equiv \\mathbf{r}_2' - \\mathbf{r}_1' = (\\mathbf{r}_2 - \\mathbf{v}t) - (\\mathbf{r}_1 - \\mathbf{v}t) = \\Delta \\mathbf{r}$\n\n- But we must measure  $\\mathbf{r}_1'$ and  $\\mathbf{r}_2'$ at the same time!\n\n- Which we can, since time is invariant and unambiguous:  $t' = t$\n\n# 4.2 Transforming velocity\n\n- How does  $\\mathbf{u}$ transform under a Galilean transformation?\n\n$$\n\\begin{array}{l} \\mathbf {u} \\equiv \\frac {d \\mathbf {r}}{d t} \\\\ \\mathbf {u} ^ {\\prime} \\equiv \\frac {d \\mathbf {r} ^ {\\prime}}{d t ^ {\\prime}} = \\frac {d}{d t} (\\mathbf {r} - \\mathbf {v} t) = \\frac {d \\mathbf {r}}{d t} - \\mathbf {v} = \\mathbf {u} - \\mathbf {v} \\\\ \\end{array}\n$$\n\nSo velocities add/subtract as you'd expect:  $\\mathbf{u}' = \\mathbf{u} - \\mathbf{v}$\n\n- But what about the flashlight on the train?\n\n# 4.3 Transforming acceleration\n\n$$\n\\begin{array}{l} \\mathbf {a} \\equiv \\frac {d \\mathbf {u}}{d t} \\\\ \\mathbf {a} ^ {\\prime} \\equiv \\frac {d \\mathbf {u} ^ {\\prime}}{d t ^ {\\prime}} = \\frac {d}{d t} (\\mathbf {u} - \\mathbf {v}) = \\frac {d \\mathbf {u}}{d t} = \\mathbf {a} \\\\ \\end{array}\n$$\n\nSo acceleration is invariant.", "source": "mit-ocw", "source_doc_id": "0002-symmetry-handout-c44cab00-ef8e3300", "source_title": "MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 5, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "Page 6"], "page_start": 6, "page_end": 6, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-symmetry-handout-c44cab00-ef8e3300:page-0007", "text": "# 4.4 Transforming F = ma\n\n- Consider forces that depend on separation:\n\n$$\n- \\text {   Spring:   } F = k (x _ {2} - x _ {1})\n$$\n\n$$\n- \\text {   Gravity:   } F = \\frac {G m M}{| \\mathbf {r} _ {2} - \\mathbf {r} _ {1} | ^ {2}}\n$$\n\nThey are invariant, since lengths are.\n\n- $m$ is invariant\n- Since  $\\mathbf{F}$,  $\\mathbf{a}$ and  $m$ are all invariant, so is the equation  $\\mathbf{F} = m\\mathbf{a}$.\n- So the physical law is invariant, but not the initial conditions!\n\n# 4.5 Transforming energy & momentum\n\n- Neither is invariant, since  $\\mathbf{v}$ isn't.\n- But the conservation laws are invariant:  $E$ and  $\\mathbf{p}$ are conserved in any frame (PS1).\n- Work-energy theorem:\n\n$$\nW = \\Delta K E,\n$$\n\nwhere work defined as\n\n$$\nW = \\int_ {x _ {1}} ^ {x _ {2}} F d x.\n$$\n\n- Proof:\n\n$$\n\\begin{array}{l} W = \\int_ {x _ {1}} ^ {x _ {2}} F d x = \\int_ {x _ {1}} ^ {x _ {2}} m a d x = m \\int_ {x _ {1}} ^ {x _ {2}} \\frac {d v}{d t} d x \\\\ = m \\int_ {v _ {1}} ^ {v _ {2}} \\frac {d x}{d t} d v = m \\int_ {v _ {1}} ^ {v _ {2}} v d v = \\frac {m v _ {2} ^ {2}}{2} - \\frac {m v _ {1} ^ {2}}{2} = \\Delta K E. \\\\ \\end{array}\n$$\n\nOnly assumption here was F = ma, which is invariant, so the work-energy theorem is also invariant.\n\n-  $W$ and  $KE$ alone are not invariant.\n\n# 4.6 Transforming trajectories\n\n- Is the 3D shape of a trajectory not invariant?\n- No! Basket ball example: line in frame A is parabola in frame B.\n\n# 4.7 Key Galilean non-invariants\n\n$$\n\\left\\{ \\begin{array}{l l} \\mathbf {r} ^ {\\prime} = \\mathbf {r} - \\mathbf {v} t \\\\ \\mathbf {u} ^ {\\prime} = \\mathbf {u} - \\mathbf {v} \\\\ \\mathbf {p} ^ {\\prime} = \\mathbf {p} - m \\mathbf {v} \\end{array} \\right.\n$$", "source": "mit-ocw", "source_doc_id": "0002-symmetry-handout-c44cab00-ef8e3300", "source_title": "MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 6, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "Page 7"], "page_start": 7, "page_end": 7, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-symmetry-handout-c44cab00-ef8e3300:page-0008", "text": "| Quantity | Invariance |  |  |\n| --- | --- | --- | --- |\n|  | Translational? | Rotational? | Gallilian? |\n| $t$ | N | Y | Y |\n| $\\mathbf{r}$ | N | N | N |\n| $\\Delta t$ | Y | Y | Y |\n| $\\Delta \\mathbf{r}$ | Y | N | Y |\n| $|\\Delta \\mathbf{r}|$ | Y | Y | Y |\n| $d/dt$ | Y | Y | Y |\n| $\\nabla$ | Y | N | Y |\n| $\\nabla^{2}$ | Y | Y | Y |\n| $\\mathbf{v}$ | Y | N | N |\n| $\\mathbf{p}$ | Y | N | N |\n| $\\mathbf{a}$ | Y | N | Y |\n| $\\mathbf{F}$ | Y | N | Y |\n| $m$ | Y | Y | Y |\n| $E_{\\text{kin}}$ | Y | Y | N |\n| $W$ | Y | Y | N |\n| $\\mathbf{F} = ma$ | Y | Y | Y |\n| Newt. mechanics | Y | Y | Y |\n| Electromagnetism | Y | Y | N |", "source": "mit-ocw", "source_doc_id": "0002-symmetry-handout-c44cab00-ef8e3300", "source_title": "MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 7, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "Page 8"], "page_start": 8, "page_end": 8, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-symmetry-handout-c44cab00-ef8e3300:page-0009", "text": "# 5 The symmetry properties of electromagnetism\n\nUnder what transformations are the laws of electromagnetism invariant? Let's focus on a simple special case: the laws that govern the propagation of light.\n\n# 5.1 The classical wave equation\n\n• Classical wave equation (8.03):\n\n$$\n\\nabla^ {2} \\mathbf {E} - \\frac {1}{c _ {w} ^ {2}} \\ddot {\\mathbf {E}} = 0.\n$$\n\nFor example,  $E$ could denote:\n\n- One of the three component of the electric field\n- One of the three component of the magnetic field\n- Air density\n- Height of water surface (2D)\n- Deflection of guitar string (1D)\n\n- 1D special case:\n\n$$\n\\frac {d ^ {2} E}{d x ^ {2}} - \\frac {1}{c _ {w} ^ {2}} \\frac {d ^ {2} E}{d t ^ {2}} = 0.\n$$\n\n- General solution (show on PS2):\n\n$$\ny = A f (x - c _ {w} t) + B f (x + c _ {w} t),\n$$\n\nfor arbitrary smooth function  $f$ and constants  $A \\& B$.\n\n- More complicated in 3D, but wavefronts still propagate with speed  $c_{w}$.\n\n# 5.2 Symmetry of the wave equation\n\n- We learned that classical mechanics was invariant under Galilean transformations.\n- The wave equation can be derived from classical mechanics.\n\nQuestion: is the classical wave equation invariant under Galilean transformations?\n\n- 1. Yes\n- 2. No\n- 3. Yes, but only if wave speed  $c_{w} \\ll c$", "source": "mit-ocw", "source_doc_id": "0002-symmetry-handout-c44cab00-ef8e3300", "source_title": "MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 8, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "Page 9"], "page_start": 9, "page_end": 9, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-symmetry-handout-c44cab00-ef8e3300:page-0010", "text": "# 5.3 Transforming the wave equation\n\n- Apply Galilean transformation to 1D wave equation:\n\n$$\n\\frac {d ^ {2} E}{d x ^ {2}} - \\frac {1}{c ^ {2}} \\frac {d ^ {2} E}{d t ^ {2}} = 0.\n$$\n\n- Do this on PS2 - hints:\n\n$$\n- x ^ {\\prime} = x + v t\n$$\n\n$$\n- t ^ {\\prime} = t\n$$\n\n- Use chain rule for derivatives:\n\n$$\n\\frac {\\partial}{\\partial x} = \\frac {\\partial x ^ {\\prime}}{\\partial x} \\frac {\\partial}{\\partial x ^ {\\prime}} + \\frac {\\partial t ^ {\\prime}}{\\partial x} \\frac {\\partial}{\\partial t ^ {\\prime}} = \\frac {\\partial}{\\partial x ^ {\\prime}}\n$$\n\n$$\n{\\frac {\\partial}{\\partial t}} {=} {\\frac {\\partial x ^ {\\prime}}{\\partial t} \\frac {\\partial}{\\partial x ^ {\\prime}} + \\frac {\\partial t ^ {\\prime}}{\\partial t} \\frac {\\partial}{\\partial t ^ {\\prime}} = v \\frac {\\partial}{\\partial x ^ {\\prime}} + \\frac {\\partial}{\\partial t ^ {\\prime}}}\n$$\n\n- Work out 2nd derivatives too\n\n- Result:\n\n$$\n\\left(1 - \\frac {v ^ {2}}{c ^ {2}}\\right) \\frac {d ^ {2} E}{d x ^ {\\prime 2}} - \\frac {1}{c ^ {2}} \\frac {d ^ {2} E}{d t ^ {\\prime 2}} - 2 \\frac {v}{c} \\frac {d ^ {2} E}{d x ^ {\\prime} d t ^ {\\prime}} = 0.\n$$\n\n- Wave equation not invariant under Galilean transformation\n\n- Show on PS2: the new equation has solution\n\n$$\ny = A f (x - [ c - v ] t) + B f (x + [ c + v ] t),\n$$\n\ni.e., waves travel slower forward than backward.\n\n- Just what you'd expect for waves in a substance, “aether” (velocities add).\n- How can this be consistent with the wave equation being derived from classical mechanics, which is Galilean invariant?", "source": "mit-ocw", "source_doc_id": "0002-symmetry-handout-c44cab00-ef8e3300", "source_title": "MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 9, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "Page 10"], "page_start": 10, "page_end": 10, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-symmetry-handout-c44cab00-ef8e3300:page-0011", "text": "# 5.4 Observed properties of speed of light\n\n- Does speed depend on wavelength? No!\n- Does speed depend on motion of source? No!\n- Does speed depend on motion of observer (frame)? No!\n\nFor all three cases, let's now look at the evidence.\n\n# 5.4.1 Does  $c$ depend on wavelength?\n\n- Does light speed through glass depend on wavelength?\n- But what about light speed through vacuum?\n- Gamma-ray bursts provide great test\n- Gamma-ray bursts last a few seconds to minutes\n- Old speculations: nefarious nukes, civilization annihilation, nearby neutron stars\n- Recently shown to originate at cosmological distances (few billion light years  $\\sim 10^{17}$ light-seconds).\n- Flash seen also at x-rays and optical wavelengths, all within of order a minute  $\\sim 10^{2}$ seconds, so\n\n$$\n\\frac {\\Delta t}{t} \\lesssim \\frac {1 0 ^ {2} \\mathrm{s}}{1 0 ^ {1 7} \\mathrm{s}} = 1 0 ^ {- 1 5}.\n$$\n\n-  $c = d / t$, so relative speed variation with wavelength is\n\n$$\n\\frac {\\Delta c}{c} \\approx \\frac {\\Delta t}{t} \\lesssim 1 0 ^ {- 1 5}.\n$$\n\nAnswer: No, at least not more than about   $10^{-15}c \\approx 300$  nm/s.", "source": "mit-ocw", "source_doc_id": "0002-symmetry-handout-c44cab00-ef8e3300", "source_title": "MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 10, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "Page 11"], "page_start": 11, "page_end": 11, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-symmetry-handout-c44cab00-ef8e3300:page-0012", "text": "# 5.4.2 Does c depend on source motion?\n\n- Does speed of a bullet depend on speed of rifle?\n- Does sound speed of a gun shot depend on speed of rifle?\n- Binary stars provide great test\n- If velocities add, then\n\n$$\nt _ {1} = \\frac {d}{c - v}\n$$\n\n$$\nt _ {2} = \\frac {d}{c + v}\n$$\n\n$$\n\\Delta t \\equiv t _ {1} - t _ {2} = \\frac {2 d v}{c ^ {2} - v ^ {2}} \\approx 2 \\frac {d}{c} \\frac {v}{c} = 2 t \\frac {v}{c} \\approx 2 0 0 \\mathrm{years,say}\n$$\n\n(for a pulsar in the Large Magellanic Cloud with  $v = 300 \\, \\mathrm{km/s}$,  $d = 100000$ lightyears)\n\n- But half an orbit takes only 2 days, say\n- You'd see new “Doppler effect”  $\\propto a$ rather than  $v$\n- You'd see things moving backward in time whenever  $a > \\frac{c^2}{d}$ towards you\n\nAnswer: No dependence on source motion observed (and should be dramatic).\n\n# 5.4.3 Does  $c$ depend on observer motion (frame)?\n\n- No 1st order effect had been seen\n- Michelson-Morley experiment hammered it - let's see how\n- Consider interferometer moving with velocity  $\\mathbf{v}$ w.r.t. aether and compute round trip flight times parallel  $(t_{\\parallel})$ and perpendicular  $(t_{\\perp})$ to  $\\mathbf{v}$.\n- For light traveling in direction  $\\pm \\mathbf{v}$,\n\n$$\n\\begin{array}{l} c t _ {\\pm} = L _ {\\parallel} \\pm v t _ {\\pm} \\\\ t _ {\\pm} = \\frac {L _ {\\parallel}}{c m _ {\\mathrm{p}} v} \\\\ t _ {\\parallel} = t _ {+} + t _ {-} = \\frac {L _ {\\parallel}}{c - v} + \\frac {L _ {\\parallel}}{c + v} = \\frac {2 L _ {\\parallel}}{c} \\gamma^ {2}, s \\\\ \\end{array}\n$$\n\nwhere we have defined the quantity\n\n$$\n\\gamma \\equiv \\frac {1}{\\sqrt {1 - \\frac {v ^ {2}}{c ^ {2}}}}.\n$$\n\n- For light traveling perpendicularly to  $\\mathbf{v}$,\n\n$$\n(c t _ {\\perp} / 2) ^ {2} = \\sqrt {L _ {\\perp} ^ {2} + (v t _ {\\perp} / 2) ^ {2}}\n$$\n\n$$\nt _ {\\perp} = \\frac {2 L _ {\\perp}}{c} \\gamma\n$$", "source": "mit-ocw", "source_doc_id": "0002-symmetry-handout-c44cab00-ef8e3300", "source_title": "MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 11, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "Page 12"], "page_start": 12, "page_end": 12, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-symmetry-handout-c44cab00-ef8e3300:page-0013", "text": "• The difference is\n\n$$\n\\Delta t \\equiv t _ {\\perp} - t _ {\\parallel} = \\frac {2 L _ {\\perp}}{c} \\gamma - \\frac {2 L _ {\\parallel}}{c} \\gamma^ {2}\n$$\n\n- Rotating the interferometer by  $90^{\\circ}$ changes this to\n\n$$\n\\Delta t ^ {\\prime} = \\frac {2 L _ {\\perp}}{c} \\gamma^ {2} - \\frac {2 L _ {\\parallel}}{c} \\gamma ,\n$$\n\n- i.e., changes it by an amount\n\n$$\n\\Delta t ^ {\\prime} - \\Delta t = \\frac {2 L _ {\\perp}}{c} \\gamma^ {2} - \\frac {2 L _ {\\parallel}}{c} \\gamma - \\frac {2 L _ {\\perp}}{c} \\gamma + \\frac {2 L _ {\\parallel}}{c} \\gamma^ {2} = 2 \\gamma (\\gamma - 1) \\frac {L _ {\\parallel} + L _ {\\perp}}{c}.\n$$\n\n- To lowest order in  $v/c$, we have\n\n$$\n\\gamma \\approx 1 + \\frac {1}{2} \\left(\\frac {v}{c}\\right) ^ {2}\n$$\n\n$$\n\\Delta t ^ {\\prime} - \\Delta t \\approx \\frac {L _ {\\parallel} + L _ {\\perp}}{c} \\left(\\frac {v}{c}\\right) ^ {2},\n$$\n\n$$\n\\frac {\\Delta t ^ {\\prime} - \\Delta t}{t} \\approx \\left(\\frac {v}{c}\\right) ^ {2}.\n$$\n\n-  $v \\approx 30 \\mathrm{~km} / \\mathrm{s}$, so  $(v / c)^2 \\sim 10^{-8}$ — tough to measure!\n\n- But their  $L_{\\parallel} + L_{\\perp} = 11\\mathrm{m}$ was about  $2 \\times 10^{7}$ wavelengths  $\\lambda \\sim 500\\mathrm{nm}$, and they could see fringe shifts as small as  $0.01\\lambda$.\n\n- But they saw no fringe shift at all! So  $c$ appears not to depend on frame.\n\n# 5.5 Aether rescue attempts (see Resnick Table 1-2)\n\n- Lorentz-Fitzgerald contraction:  $L_{\\parallel}$ contracts to  $L_{\\parallel}/\\gamma$.\n\n- Ruled out by Kennedy & Thorndike (1932) using interferometer with  $L_{\\parallel} \\neq L_{\\perp}$\n\n- Aether drag hypothesis\n\n- Ruled out by stellar aberration\n\n- Also by light propagation in moving water (Fizeau 1851)\n\n- Emission theories ( $v$ depends on source speed)\n\n- Ruled out by binary stars (above)\n\n- Also ruled out by Michelson-Morley with extraterrestrial light\n\n- Also ruled out by measuring speed of  $\\gamma$-rays from CERN particle decays", "source": "mit-ocw", "source_doc_id": "0002-symmetry-handout-c44cab00-ef8e3300", "source_title": "MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 12, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "Page 13"], "page_start": 13, "page_end": 13, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-symmetry-handout-c44cab00-ef8e3300:page-0014", "text": "We've seen that classical mechanics is invariant under Galilean transformations but electromagnetism isn't.\n\n# Question: What is wrong?\n\n- 1. The idea that all inertial frames are equivalent\n- 2. Our theory of mechanics (8.01)\n- 3. Our theory electromagnetism (8.02)\n- 4. Nothing, because of Bohr's complementarity principle\n\n# 6 What are we to make of this?\n\n- Parity symmetry applied to some things, not others.\n- Is it the same with Galilean symmetry?\n- An experimental question: Is physics the same in all inertial frames?\n- A: Experiments suggest YES, both for mechanics and electromagnetism\n- A theoretical question: How describe this invariance mathematically, i.e., what is the transformation law that leaves physics invariant?\n- Galilean transformation? Works for mechanics but fails for  $E \\& M$\n- Lorentz transformation? Works for  $E \\& M$ (PS3) but fails for mechanics\n- No transformation works for both  $E \\& M$ and mechanics\n- So at least one of the two must be wrong!\n- Changing  $E \\& M$ to be have Galilean invariance is experimentally ruled out\n- So let's try changing mechanics to be Lorentz invariant!\n- BINGO! Not only OK with old experiments, but triumphed with new ones.", "source": "mit-ocw", "source_doc_id": "0002-symmetry-handout-c44cab00-ef8e3300", "source_title": "MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 13, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "Page 14"], "page_start": 14, "page_end": 14, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-symmetry-handout-c44cab00-ef8e3300:page-0015", "text": "# 7 Derivation of the Lorentz Transformation\n\n# 7.1 Battle Plan\n\nWe'll follow Einstein's approach and derive everything from two postulates:\n\n- 1. The laws of physics are the same in all inertial frames.\n- 2. The speed of light is same in all inertial frames.\n\nComments:\n\n- 2 follows from 1 if we consider the speed of light one of the laws of physics.\n- Einstein denoted inertial frame invariance “special relativity”\n- As opposed to “general relativity”, the generalization to non-inertial frames.\n\n# 7.2 Inertial frames done carefully: rods & clocks\n\n- Key concept: the event, a point in spacetime.\n- Define coordinate system with three perpendicular rigid measuring rods\n- Define time with local clocks\n- Synchronize clocks with light pulses\n- Minkowski diagram of synchronization procedure\n- Minkowski diagram basics\n- N.B. Don't confuse frame simultaneity with seeing things simultaneously: If you saw SN 1987A and a camera flash at the same time, did these two flashes go off simultaneously in your inertial frame?\n- The time  $t$ in an inertial frame is also called bookkeeper's time. Don't confuse with the time when you see something happen.\n- 1st shocker: Simultaneity is relative! Must abandon  $t' = t$.", "source": "mit-ocw", "source_doc_id": "0002-symmetry-handout-c44cab00-ef8e3300", "source_title": "MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 14, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "Page 15"], "page_start": 15, "page_end": 15, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-symmetry-handout-c44cab00-ef8e3300:page-0016", "text": "# 7.3 Transformation derivation, part I\n\n- Let's define 4-vectors that have units of length:\n\n$$\n\\mathbf {x} = \\left( \\begin{array}{c} x \\\\ y \\\\ z \\\\ c t \\end{array} \\right).\n$$\n\n- In this course, we'll often use units were  $c = 1$ (time measured in meters).\n- Given  $\\mathbf{v}$, the new 4-vector  $\\mathbf{x}'$ is some function of  $\\mathbf{x}$ - which function?\n- Translational invariance implies linearity:\n\n$$\n\\mathbf {x} ^ {\\prime} = \\boldsymbol {\\Lambda} (\\mathbf {v}) \\mathbf {x} + \\mathbf {x} _ {0},\n$$\n\nfor some offset   $x_{0}$  and some Lorentz matrix\n\n$$\n\\boldsymbol {\\Lambda} (\\mathbf {v}) = \\left( \\begin{array}{c c c c} \\Lambda_ {1 1} & \\Lambda_ {1 2} & \\Lambda_ {1 3} & \\Lambda_ {1 4} \\\\ \\Lambda_ {2 1} & \\Lambda_ {2 2} & \\Lambda_ {2 3} & \\Lambda_ {2 4} \\\\ \\Lambda_ {3 1} & \\Lambda_ {3 2} & \\Lambda_ {3 3} & \\Lambda_ {3 4} \\\\ \\Lambda_ {4 1} & \\Lambda_ {4 2} & \\Lambda_ {4 3} & \\Lambda_ {4 4} \\end{array} \\right).\n$$\n\n- Why? Because\n\n$$\n\\mathbf {x} _ {2} ^ {\\prime} - \\mathbf {x} _ {1} ^ {\\prime} = \\boldsymbol {\\Lambda} (\\mathbf {v}) (\\mathbf {x} _ {2} - \\mathbf {x} _ {1})\n$$\n\nfor linear relation — for any nonlinear (rigorously, non-affine) relation, the difference  $(\\mathbf{x}_2' - \\mathbf{x}_1')$ won't depend only on the difference  $(\\mathbf{x}_2 - \\mathbf{x}_1)$.\n\n- Notation warning: book uses notation where 4th coordinate is  $t$, not  $ct$, so there things get uglier and not all  $\\Lambda$-coefficients are dimensionless.\n- Notation warning: book uses  $\\mathbf{a}$, we use  $\\Lambda$ since it's more standard these days.\n- Velocity sign convention: velocity of primed frame in unprimed frame is  $\\mathbf{v}$, so velocity of unprimed frame in primed frame is  $-\\mathbf{v}$\n- WLOG no translation:  $\\mathbf{x}_0 = \\mathbf{0}$ (we can always translate later), so simply need to find the  $4 \\times 4$ matrix  $\\Lambda(\\mathbf{v})$\n- (WLOG=without loss of generality.)\n- WLOG no rotation (we can always rotate later), so  $\\mathbf{v} = \\mathbf{0}$ case gives identity matrix:\n\n$$\n\\boldsymbol {\\Lambda} (\\mathbf {0}) = \\left( \\begin{array}{c c c c} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{array} \\right).\n$$\n\n- WLOG v in  $x$-direction, since we can always rotate to make it so", "source": "mit-ocw", "source_doc_id": "0002-symmetry-handout-c44cab00-ef8e3300", "source_title": "MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 15, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "Page 16"], "page_start": 16, "page_end": 16, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-symmetry-handout-c44cab00-ef8e3300:page-0017", "text": "- Our transformation respects rotational symmetry around the  $x$-axis, so neither  $x'$ nor  $t'$ can depend on  $y$ or  $z$, i.e., we have  $\\Lambda_{12} = \\Lambda_{13} = \\Lambda_{42} = \\Lambda_{43} = 0$.\n\n- The  $x$-axis gets transformed into\n\n$$\n\\left( \\begin{array}{c c c c} \\Lambda_ {1 1} & 0 & 0 & \\Lambda_ {1 4} \\\\ \\Lambda_ {2 1} & \\Lambda_ {2 2} & \\Lambda_ {2 3} & \\Lambda_ {2 4} \\\\ \\Lambda_ {3 1} & \\Lambda_ {3 2} & \\Lambda_ {3 3} & \\Lambda_ {3 4} \\\\ \\Lambda_ {4 1} & 0 & 0 & \\Lambda_ {4 4} \\end{array} \\right) \\left( \\begin{array}{c} 1 \\\\ 0 \\\\ 0 \\\\ 0 \\end{array} \\right) = \\left( \\begin{array}{c} \\Lambda_ {1 1} \\\\ \\Lambda_ {2 1} \\\\ \\Lambda_ {3 1} \\\\ \\Lambda_ {4 1} \\end{array} \\right),\n$$\n\nso we have  $\\Lambda_{21} = \\Lambda_{31} = 0$ since, by construction, the spatial part of the  $x$-axis coincides continuously with the  $x'$-axis.\n\n- Consider events with  $x = t = 0$. They get transformed into\n\n$$\n\\left( \\begin{array}{c} x ^ {\\prime} \\\\ y ^ {\\prime} \\\\ z ^ {\\prime} \\\\ t ^ {\\prime} \\end{array} \\right) = \\left( \\begin{array}{c c c c} \\Lambda_ {1 1} & 0 & 0 & \\Lambda_ {1 4} \\\\ 0 & \\Lambda_ {2 2} & \\Lambda_ {2 3} & \\Lambda_ {2 4} \\\\ 0 & \\Lambda_ {3 2} & \\Lambda_ {3 3} & \\Lambda_ {3 4} \\\\ \\Lambda_ {4 1} & 0 & 0 & \\Lambda_ {4 4} \\end{array} \\right) \\left( \\begin{array}{c} 0 \\\\ y \\\\ z \\\\ 0 \\end{array} \\right) = \\left( \\begin{array}{c} 0 \\\\ \\Lambda_ {2 2} y + \\Lambda_ {2 3} z \\\\ \\Lambda_ {3 2} y + \\Lambda_ {3 3} z \\\\ 0 \\end{array} \\right).\n$$\n\nSo all events in this two-dimensional   $(y,z)$ -plane are simultaneous in both frames (with   $t' = t = 0$ ), making it trivial to compare measuring rods in the two frames since their two endpoints can coincide in space and time. This implies that the   $2 \\times 2$  transformation matrix in this plane must be the identity matrix, i.e.,   $\\Lambda_{23} = \\Lambda_{32} = 0$  and   $\\Lambda_{22} = \\Lambda_{33} = 1$ .\n\n- An object moving uniformly with  $x = vt$ in the unprimed frame remains at rest at the origin in the primed frame, so\n\n$$\n\\left( \\begin{array}{c c c c} \\Lambda_ {1 1} & 0 & 0 & \\Lambda_ {1 4} \\\\ 0 & 1 & 0 & \\Lambda_ {2 4} \\\\ 0 & 0 & 1 & \\Lambda_ {3 4} \\\\ \\Lambda_ {4 1} & 0 & 0 & \\Lambda_ {4 4} \\end{array} \\right) \\left( \\begin{array}{c} v t \\\\ 0 \\\\ 0 \\\\ c t \\end{array} \\right) = \\left( \\begin{array}{c} \\Lambda_ {1 1} v t + \\Lambda_ {1 4} c t \\\\ \\Lambda_ {2 4} c t \\\\ \\Lambda_ {3 4} c t \\\\ \\Lambda_ {4 1} v t + \\Lambda_ {4 4} c t \\end{array} \\right) = \\left( \\begin{array}{c} 0 \\\\ 0 \\\\ 0 \\\\ c t ^ {\\prime} \\end{array} \\right)\n$$\n\nso we have  $\\Lambda_{24} = \\Lambda_{34} = 0$ and\n\n$$\n\\Lambda_ {1 4} = - \\beta \\Lambda_ {1 1},\n$$\n\nwhere we've defined\n\n$$\n\\beta \\equiv \\frac {v}{c}.\n$$\n\n- (When we do research using relativity, we normally use units where  $c = 1$, so that we can write simply  $\\beta = v$.)\n\n- Progress update:\n\n$$\n\\boldsymbol {\\Lambda} (\\mathbf {v}) = \\left( \\begin{array}{c c c c} \\Lambda_ {1 1} & 0 & 0 & - \\beta \\Lambda_ {1 1} \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ \\Lambda_ {4 1} & 0 & 0 & \\Lambda_ {4 4} \\end{array} \\right), \\tag {1}\n$$\n\ni.e.\n\n$$\n\\left( \\begin{array}{c} x ^ {\\prime} \\\\ y ^ {\\prime} \\\\ z ^ {\\prime} \\\\ c t ^ {\\prime} \\end{array} \\right) = \\left( \\begin{array}{c c c c} \\Lambda_ {1 1} & 0 & 0 & - \\beta \\Lambda_ {1 1} \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ \\Lambda_ {4 1} & 0 & 0 & \\Lambda_ {4 4} \\end{array} \\right) \\left( \\begin{array}{c} x \\\\ y \\\\ z \\\\ c t \\end{array} \\right) = \\left( \\begin{array}{c} \\Lambda_ {1 1} (x - v t) \\\\ y \\\\ z \\\\ \\Lambda_ {4 1} x + \\Lambda_ {4 4} c t \\end{array} \\right). \\tag {2}\n$$", "source": "mit-ocw", "source_doc_id": "0002-symmetry-handout-c44cab00-ef8e3300", "source_title": "MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 16, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "Page 17"], "page_start": 17, "page_end": 17, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-symmetry-handout-c44cab00-ef8e3300:page-0018", "text": "# 7.4 Galileo and Einstein part ways\n\nSo far, we haven't assumed anything about the speed of light, so our results must still include both the Galilean transform and the Lorentz transform.\n\nLet's do the Galilean first:\n\n- Assuming that   $t' = t$  gives   $\\Lambda_{41} = 0$  and   $\\Lambda_{44} = 1$ .\n- Assuming that measuring rods have the same length in both frames implies  $\\Lambda_{11} = 1$.\n- This implies the Galilean transformation matrix:\n\n$$\n\\mathbf {G} (\\mathbf {v}) = \\left( \\begin{array}{c c c c} 1 & 0 & 0 & - \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{array} \\right).\n$$\n\n# 7.5 Transformation derivation, part II\n\nLet's revert to equation (1) and assume that light has same speed  $c$ in both frames.\n\n- Imagine a light flash created at  $\\mathbf{x} = (0,0,0,0)$ expanding with speed  $c$ in all directions, creating an expanding spherical wavefront of radii  $ct$ and  $ct'$ in the two frames. This light cone (a cone in 4D spacetime) is described by\n\n$$\nx ^ {2} + y ^ {2} + z ^ {2} - (c t) ^ {2} = 0 \\tag {3}\n$$\n\nand\n\n$$\nx ^ {\\prime 2} + y ^ {\\prime 2} + z ^ {\\prime 2} - \\left(c t ^ {\\prime}\\right) ^ {2} = 0 \\tag {4}\n$$\n\nin the two frames.\n\n- Substiting equation (2) into the last equation gives\n\n$$\n\\Lambda_ {1 1} ^ {2} (x - v t) ^ {2} + y ^ {2} + z ^ {2} - (\\Lambda_ {4 1} x + \\Lambda_ {4 4} c t) ^ {2} = 0\n$$\n\n- Rearranging terms:\n\n$$\n(\\Lambda_ {1 1} ^ {2} - \\Lambda_ {4 1} ^ {2}) x ^ {2} + y ^ {2} + z ^ {2} - (\\Lambda_ {4 4} - \\beta^ {2} \\Lambda_ {1 1} ^ {2}) (c t) ^ {2} - 2 (\\beta \\Lambda_ {1 1} ^ {2} + \\Lambda_ {4 1} \\Lambda_ {4 4}) c t x = 0.\n$$\n\n- So the light cone is where this quadratic polynomial in  $x, y, z$ and  $t$ vanishes.\n- This polynomial will vanish on the same cone as the polynomial of equation (3) only if the two polynomials are identical, i.e., if\n\n$$\n\\left\\{ \\begin{array}{l l} \\Lambda_ {1 1} ^ {2} - \\Lambda_ {4 1} ^ {2} = 1, \\\\ \\Lambda_ {4 4} ^ {2} - \\beta^ {2} \\Lambda_ {1 1} ^ {2} = 1, \\\\ \\beta \\Lambda_ {1 1} ^ {2} + \\Lambda_ {4 1} \\Lambda_ {4 4} = 0. \\end{array} \\right.\n$$", "source": "mit-ocw", "source_doc_id": "0002-symmetry-handout-c44cab00-ef8e3300", "source_title": "MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 17, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "Page 18"], "page_start": 18, "page_end": 18, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-symmetry-handout-c44cab00-ef8e3300:page-0019", "text": "- Solve:\n\n$$\n\\left\\{ \\begin{array}{l} \\Lambda_ {4 1} ^ {2} = \\Lambda_ {1 1} ^ {2} - 1, \\\\ \\Lambda_ {4 4} ^ {2} = \\beta^ {2} \\Lambda_ {1 1} ^ {2} + 1, \\\\ 0 = \\beta^ {2} \\Lambda_ {1 1} ^ {4} - \\Lambda_ {4 1} ^ {2} \\Lambda_ {4 4} ^ {2} = ... = 1 - (1 - \\beta^ {2}) \\Lambda_ {1 1} \\end{array} \\right.\n$$\n\nSolution:\n\n$$\n\\left\\{ \\begin{array}{l l} \\Lambda_ {1 1} = \\gamma , \\\\ \\Lambda_ {4 4} = \\gamma , \\\\ \\Lambda_ {4 1} = - \\beta \\gamma , \\\\ \\Lambda_ {1 4} = - \\beta \\gamma , \\end{array} \\right.,\n$$\n\nwhere we have defined\n\n$$\n\\gamma \\equiv \\frac {1}{\\sqrt {1 - \\beta^ {2}}} = \\frac {1}{\\sqrt {1 - \\frac {v ^ {2}}{c ^ {2}}}}.\n$$\n\n- (We don't care about the 2nd solution with  $\\Lambda_{11} = -\\gamma$, which corresponds to flipping the sign of  $t$ and  $x$, \"TP\".)\n\n- We're done! The Lorentz transformation is\n\n$$\n\\boldsymbol {\\Lambda} (\\mathbf {v}) = \\left( \\begin{array}{c c c c} \\gamma & 0 & 0 & - \\gamma \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ - \\gamma \\beta & 0 & 0 & \\gamma \\end{array} \\right),\n$$\n\ni.e.,\n\n$$\n\\left( \\begin{array}{c} x ^ {\\prime} \\\\ y ^ {\\prime} \\\\ z ^ {\\prime} \\\\ c t ^ {\\prime} \\end{array} \\right) = \\left( \\begin{array}{c c c c} \\gamma & 0 & 0 & - \\gamma \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ - \\gamma \\beta & 0 & 0 & \\gamma \\end{array} \\right) \\left( \\begin{array}{c} x \\\\ y \\\\ z \\\\ c t \\end{array} \\right) = \\left( \\begin{array}{c} \\gamma (x - \\beta c t) \\\\ y \\\\ z \\\\ \\gamma (c t - \\beta x) \\end{array} \\right).\n$$\n\n- Compare to Einstein's 1905 paper", "source": "mit-ocw", "source_doc_id": "0002-symmetry-handout-c44cab00-ef8e3300", "source_title": "MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 18, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "Page 19"], "page_start": 19, "page_end": 19, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-symmetry-handout-c44cab00-ef8e3300:page-0020", "text": "# 7.6 The inverse Lorentz transform\n\n- Since  $\\mathbf{x}' = \\boldsymbol{\\Lambda}(v)\\mathbf{x}$ and  $\\mathbf{x} = \\boldsymbol{\\Lambda}(-v)\\mathbf{x}'$, we get the consistency requirement\n\n$$\n\\mathbf {x} = \\boldsymbol {\\Lambda} (- v) \\mathbf {x} ^ {\\prime} = \\boldsymbol {\\Lambda} (- v) \\boldsymbol {\\Lambda} (v) \\mathbf {x}\n$$\n\nfor any event  $\\mathbf{x}$, so we must have  $\\Lambda(-v) = \\Lambda(v)^{-1}$, the matrix inverse of  $\\Lambda(v)$.\n\n- Is it?\n\n$$\n\\boldsymbol {\\Lambda} (- \\mathbf {v}) \\boldsymbol {\\Lambda} (\\mathbf {v}) = \\left( \\begin{array}{c c c c} \\gamma & 0 & 0 & \\gamma \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ \\gamma \\beta & 0 & 0 & \\gamma \\end{array} \\right) \\left( \\begin{array}{c c c c} \\gamma & 0 & 0 & - \\gamma \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ - \\gamma \\beta & 0 & 0 & \\gamma \\end{array} \\right) = \\left( \\begin{array}{c c c c} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{array} \\right),\n$$\n\ni.e., yes!\n\n# 7.7 Spacetime transformation summary\n\n- Translation:\n\n$$\n\\left\\{ \\begin{array}{l} \\mathbf {r} ^ {\\prime} = \\mathbf {r} + \\Delta r \\\\ t ^ {\\prime} = t + \\Delta t \\end{array} \\right.\n$$\n\n- Rotation:\n\n$$\n\\left\\{ \\begin{array}{l} \\mathbf {r} ^ {\\prime} = \\mathbf {R r} \\\\ t ^ {\\prime} = t \\end{array} \\right.\n$$\n\n- Parity\n\n$$\n\\left\\{ \\begin{array}{l l} \\mathbf {r} ^ {\\prime} = - \\mathbf {r} \\\\ t ^ {\\prime} = t \\end{array} \\right.\n$$\n\n- Galilean “boost”:\n\n$$\n\\left\\{ \\begin{array}{l} \\mathbf {r} ^ {\\prime} = \\mathbf {r} - \\mathbf {v} t \\\\ t ^ {\\prime} = t \\end{array} \\right.\n$$\n\n- Combined:\n\n$$\n\\left\\{ \\begin{array}{l} \\mathbf {r} ^ {\\prime} = \\pm \\mathbf {R r} + \\Delta \\mathbf {r} - \\mathbf {v} t \\\\ t ^ {\\prime} = t + \\Delta t \\end{array} \\right.\n$$\n\n- Lorentz “boost”:\n\n$$\n\\mathbf {x} ^ {\\prime} = \\boldsymbol {\\Lambda} \\mathbf {x}, \\quad i. e., \\text {   for   } \\mathbf {v} \\text {   in   } x \\text {-direction},\n$$\n\n$$\n\\left\\{ \\begin{array}{l l} x ^ {\\prime} = \\gamma (x - v t) \\\\ y ^ {\\prime} = y \\\\ z ^ {\\prime} = z \\\\ t ^ {\\prime} = \\gamma (t - v x / c ^ {2}) \\end{array} \\right.\n$$\n\n- Poincaré:\n\n$$\n\\mathbf {x} ^ {\\prime} = \\boldsymbol {\\Lambda} \\mathbf {x} + \\mathbf {x} _ {0}\n$$\n\nThis is the most general spacetime symmetry transformation of special relativity, with 10 parameters:   $x_{0}$  can give 3 independent translations in space and 1 translation in time),   $\\Lambda$  can give 3 independent rotations, 3 independent boosts and also reversal of space and/or time.", "source": "mit-ocw", "source_doc_id": "0002-symmetry-handout-c44cab00-ef8e3300", "source_title": "MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 19, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Symmetry & Invariance: Max Tegmark", "Page 20"], "page_start": 20, "page_end": 20, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0003-matrix-primer-handout-a08abcb1-3667296f:page-0001", "text": "# MIT Course 8.033, Fall 2005, Supplement\n\n# Max Tegmark\n\n# Matrix Primer\n\n- An  $m \\times n$ matrix is a rectangular array of numbers with  $m$ rows and  $n$ columns. Example of a  $2 \\times 3$ matrix:\n\n$$\n\\mathbf {A} = \\left( \\begin{array}{c c c} 3 & 1 & 4 \\\\ 1 & 5 & 9 \\end{array} \\right).\n$$\n\n-  $\\mathbf{A}_{ij}$ denotes the number on row  $i$ and column  $j$ — for example,  $\\mathbf{A}_{13} = 4$.\n\n- The transpose of a matrix, denoted by a superscripted  $t$, is a matrix with the rows and columns interchanged, i.e.,  $\\mathbf{A}_{ij}^t = \\mathbf{A}_{ji}$. For example,\n\n$$\n\\left( \\begin{array}{c c c} 3 & 1 & 4 \\\\ 1 & 5 & 9 \\end{array} \\right) ^ {t} = \\left( \\begin{array}{c c} 3 & 1 \\\\ 1 & 5 \\\\ 4 & 9 \\end{array} \\right),\n$$\n\n- Two matrices of identical shape can be added by adding their corresponding elements: If  $\\mathbf{C} = \\mathbf{A} + \\mathbf{B}$, then  $\\mathbf{C}_{ij} = \\mathbf{A}_{ij} + \\mathbf{B}_{ij}$. Example:\n\n$$\n\\left( \\begin{array}{c c} 1 & 2 \\\\ 3 & 4 \\end{array} \\right) + \\left( \\begin{array}{c c} 1 0 & 2 0 \\\\ 3 0 & 4 0 \\end{array} \\right) = \\left( \\begin{array}{c c} 1 1 & 2 2 \\\\ 3 3 & 4 4 \\end{array} \\right)\n$$\n\n- A matrix can be multiplied by a number by multiplying all of its elements by that number: If  $\\mathbf{B} = a\\mathbf{A}$, then  $\\mathbf{B}_{ij} = a\\mathbf{A}_{ij}$. Example:\n\n$$\n1 0 \\times \\left( \\begin{array}{c c} 1 & 2 \\\\ 3 & 4 \\end{array} \\right) = \\left( \\begin{array}{c c} 1 0 & 2 0 \\\\ 3 0 & 4 0 \\end{array} \\right)\n$$\n\n- The product  $\\mathbf{C} = \\mathbf{A}\\mathbf{B}$ of an  $l\\times m$ matrix  $\\mathbf{A}$ and an  $m\\times n$ matrix  $\\mathbf{B}$ is defined as\n\n$$\n\\mathbf {C} _ {i j} \\equiv \\sum_ {k = 1} ^ {m} \\mathbf {A} _ {i k} B _ {k j}.\n$$\n\nExample:\n\n$$\n\\left( \\begin{array}{c c} 1 & 2 \\\\ 3 & 4 \\end{array} \\right) \\left( \\begin{array}{c c} 1 0 & 2 0 \\\\ 3 0 & 4 0 \\end{array} \\right) = \\left( \\begin{array}{c c} 1 \\cdot 1 0 + 2 \\cdot 3 0 & 1 \\cdot 2 0 + 2 \\cdot 4 0 \\\\ 3 \\cdot 1 0 + 4 \\cdot 3 0 & 3 \\cdot 2 0 + 4 \\cdot 4 0 \\end{array} \\right) = \\left( \\begin{array}{c c} 7 0 & 9 0 \\\\ 1 5 0 & 2 2 0 \\end{array} \\right)\n$$\n\n- An identity matrix is a square matrix with 1 on the diagonal and 0 everywhere else. It is denoted I. Its acts like the number 1, since multiplying another matrix by it has no effect: IA = A and AI = A for any A. Example: the   $2 \\times 2$  identity matrix is\n\n$$\n\\left( \\begin{array}{c c} 1 & 0 \\\\ 0 & 1 \\end{array} \\right).\n$$", "source": "mit-ocw", "source_doc_id": "0003-matrix-primer-handout-a08abcb1-3667296f", "source_title": "MIT Course 8.033, Fall 2005, Supplement: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Supplement: Max Tegmark", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0003-matrix-primer-handout-a08abcb1-3667296f:page-0002", "text": "- A matrix  $\\mathbf{B}$ is said to be the inverse of a matrix  $\\mathbf{A}$ if  $\\mathbf{AB} = \\mathbf{I}$. Example:\n\n$$\n\\left( \\begin{array}{c c} 2 & 1 \\\\ 3 & 2 \\end{array} \\right) ^ {- 1} = \\left( \\begin{array}{c c} 2 & - 1 \\\\ - 3 & 2 \\end{array} \\right),\n$$\n\nsince\n\n$$\n\\left( \\begin{array}{c c} 2 & 1 \\\\ 3 & 2 \\end{array} \\right) \\left( \\begin{array}{c c} 2 & - 1 \\\\ - 3 & 2 \\end{array} \\right) = \\left( \\begin{array}{c c} 1 & 0 \\\\ 0 & 1 \\end{array} \\right).\n$$\n\n-  $1 \\times 1$ matrices are simply numbers, and it is easy to see that the above rules for addition, multiplication and inversion reduce to the familiar ones for this special case.\n\n- Vectors are special cases of matrices and therefore obey the above rules for addition and multiplication.\n\n- A matrix with only one column is called a column vector. All vectors in 8.033 are column vectors, usually referred to simply as vectors. Example:\n\n$$\n\\mathbf {a} = \\binom{2}{3}\n$$\n\n- A matrix with only one row is called a row vector. Example:\n\n$$\n\\mathbf {a} ^ {t} = \\left( \\begin{array}{c c} 2 & 3 \\end{array} \\right)\n$$\n\n- In a linear algebra class, you typically learn more advanced aspects of matrices, such as their determinant, eigenvalues and eigenvectors.", "source": "mit-ocw", "source_doc_id": "0003-matrix-primer-handout-a08abcb1-3667296f", "source_title": "MIT Course 8.033, Fall 2005, Supplement: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Supplement: Max Tegmark", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-kinematics-handout-42d52bee-882cb5f1:page-0001", "text": "# MIT Course 8.033, Fall 2006, Relativistic Kinematics\n\n# Max Tegmark\n\n# Last revised October 17 2006\n\n# Topics\n\n- Lorentz transformations toolbox\n\n- – formula summary\n- inverse\n- composition (v addition)\n- boosts as rotations\n- the invariant\n- wave 4-vector\n- velocity 4-vector\n- aberration\n- Doppler effect\n- – proper time under acceleration\n- calculus of variations\n- metrics, geodesics\n\n- Implications\n\n- Time dilation\n- Relativity of simultaneity, non-synchronization\n- Length contraction\n- $c$ as universal speed limit\n- Rest length, proper time", "source": "mit-ocw", "source_doc_id": "0004-kinematics-handout-42d52bee-882cb5f1", "source_title": "MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-kinematics-handout-42d52bee-882cb5f1:page-0002", "text": "# Formula summary: transformation toolbox\n\n- Lorentz transformation:\n\n$$\n\\boldsymbol {\\Lambda} (\\hat {\\mathbf {x}} v) = \\left( \\begin{array}{c c c c} \\gamma & 0 & 0 & - \\gamma \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ - \\gamma \\beta & 0 & 0 & \\gamma \\end{array} \\right),\n$$\n\ni.e.,\n\n$$\n\\left( \\begin{array}{c} x ^ {\\prime} \\\\ y ^ {\\prime} \\\\ z ^ {\\prime} \\\\ c t ^ {\\prime} \\end{array} \\right) = \\left( \\begin{array}{c} \\gamma (x - \\beta c t) \\\\ y \\\\ z \\\\ \\gamma (c t - \\beta x) \\end{array} \\right).\n$$\n\n- This implies all the equations below, derived on the following pages:\n\n- Inverse Lorentz transformation:\n\n$$\n\\boldsymbol {\\Lambda} (\\mathbf {v}) ^ {- 1} = \\boldsymbol {\\Lambda} (- \\mathbf {v})\n$$\n\n- Addition of parallel velocities:\n\n$$\n\\boldsymbol {\\Lambda} (v _ {1}) \\boldsymbol {\\Lambda} (v _ {2}) = \\boldsymbol {\\Lambda} \\left(\\frac {v _ {1} + v _ {2}}{1 + \\frac {v _ {1} v _ {2}}{c ^ {2}}}\\right)\n$$\n\n- Addition of arbitrary velocities:\n\n$$\n\\begin{array}{l} u _ {x} = \\frac {u _ {x} ^ {\\prime} + v}{1 + \\frac {u _ {x} ^ {\\prime} v}{c ^ {2}}} \\\\ u _ {y} = \\frac {u _ {y} ^ {\\prime} \\sqrt {1 - \\frac {v ^ {2}}{c ^ {2}}}}{1 + \\frac {u _ {x} ^ {\\prime} v}{c ^ {2}}} \\\\ u _ {z} = \\frac {u _ {z} ^ {\\prime} \\sqrt {1 - \\frac {v ^ {2}}{c ^ {2}}}}{1 + \\frac {u _ {x} ^ {\\prime} v}{c ^ {2}}} \\\\ \\end{array}\n$$\n\n- Boosts as generalized rotations:\n\n$$\n\\mathbf {\\Lambda} (- v) = \\left( \\begin{array}{c c c c} \\cosh \\eta & 0 & 0 & \\sinh \\eta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ \\sinh \\eta & 0 & 0 & \\cosh \\eta \\end{array} \\right),\n$$\n\nwhere  $\\eta \\equiv \\tanh^{-1}\\beta$\n\n- All Lorentz matrices  $\\Lambda$ satisfy\n\n$$\n\\boldsymbol {\\Lambda} ^ {t} \\boldsymbol {\\eta} \\boldsymbol {\\Lambda} = \\boldsymbol {\\eta},\n$$\n\nwhere the Minkowski metric is\n\n$$\n\\boldsymbol {\\eta} = \\left( \\begin{array}{c c c c} - 1 & 0 & 0 & 0 \\\\ 0 & - 1 & 0 & 0 \\\\ 0 & 0 & - 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{array} \\right),\n$$", "source": "mit-ocw", "source_doc_id": "0004-kinematics-handout-42d52bee-882cb5f1", "source_title": "MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-kinematics-handout-42d52bee-882cb5f1:page-0003", "text": "- All Lorentz transforms leave the interval\n\n$$\n\\Delta s ^ {2} \\equiv \\Delta \\mathbf {x} ^ {t} \\pmb {\\eta} \\Delta x = \\Delta x ^ {2} + \\Delta y ^ {2} + \\Delta z ^ {2} - (c \\Delta t) ^ {2}\n$$\n\ninvariant\n\n- Wave 4-vector\n\n$$\n\\mathbf {K} \\equiv \\gamma_ {u} \\left( \\begin{array}{c} k _ {x} \\\\ k _ {y} \\\\ k _ {z} \\\\ w / c \\end{array} \\right),\n$$\n\n- Velocity 4-vector\n\n$$\n\\mathbf {U} \\equiv \\gamma_ {u} \\left( \\begin{array}{c} u _ {x} \\\\ u _ {y} \\\\ u _ {z} \\\\ c \\end{array} \\right), \\quad \\gamma_ {u} \\equiv \\frac {1}{\\sqrt {1 - \\frac {u ^ {2}}{c ^ {2}}}}\n$$\n\n- Aberration:\n\n$$\n\\cos \\theta^ {\\prime} = \\frac {\\cos \\theta - \\beta}{1 - \\beta \\cos \\theta}\n$$\n\n- Doppler effect:\n\n$$\n\\omega^ {\\prime} = \\omega \\gamma (1 - \\beta \\cos \\theta)\n$$\n\n# Formula summary: other\n\n• Proper time interval:\n\n$$\n\\Delta \\tau = \\int_ {t _ {A}} ^ {t _ {B}} \\sqrt {1 - \\frac {| \\dot {\\mathbf {r}} (t) | ^ {2}}{c ^ {2}}} d t\n$$\n\n- Euler-Lagrange equation:\n\n$$\n\\frac {\\partial f}{\\partial x} - \\frac {d}{d t} \\frac {\\partial f}{\\partial \\dot {x}} = 0\n$$", "source": "mit-ocw", "source_doc_id": "0004-kinematics-handout-42d52bee-882cb5f1", "source_title": "MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-kinematics-handout-42d52bee-882cb5f1:page-0004", "text": "# Implications: time dilation\n\n- In the frame  $S$, a clock is at rest at the origin ticking at time intervals that are  $\\Delta t = 1$ seconds long, so the two consecutive ticks at  $t = 0$ and  $t = \\Delta t$ have coordinates\n\n$$\n\\mathbf {x} _ {1} = \\left( \\begin{array}{c} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{array} \\right), \\quad \\mathbf {x} _ {2} = \\left( \\begin{array}{c} 0 \\\\ 0 \\\\ 0 \\\\ c \\Delta t \\end{array} \\right).\n$$\n\n- In the frame  $S'$, the coordinates are\n\n$$\n\\mathbf {x} _ {1} ^ {\\prime} = \\left( \\begin{array}{c c c c} \\gamma & 0 & 0 & - \\gamma \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ - \\gamma \\beta & 0 & 0 & \\gamma \\end{array} \\right) \\left( \\begin{array}{c} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{array} \\right) = \\left( \\begin{array}{c} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{array} \\right),\n$$\n\n$$\n\\mathbf {x} _ {2} ^ {\\prime} = \\left( \\begin{array}{c c c c} \\gamma & 0 & 0 & - \\gamma \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ - \\gamma \\beta & 0 & 0 & \\gamma \\end{array} \\right) \\left( \\begin{array}{c} 0 \\\\ 0 \\\\ 0 \\\\ c \\Delta t \\end{array} \\right) = \\left( \\begin{array}{c} - \\gamma v \\Delta t \\\\ 0 \\\\ 0 \\\\ \\gamma c \\Delta t \\end{array} \\right)\n$$\n\n- So in  $S'$, the clock appears to tick at intervals  $\\Delta t' = \\gamma \\Delta t > \\Delta t$, i.e., slower! (Draw Minkowski diagram.)", "source": "mit-ocw", "source_doc_id": "0004-kinematics-handout-42d52bee-882cb5f1", "source_title": "MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-kinematics-handout-42d52bee-882cb5f1:page-0005", "text": "# Time dilation, cont'd\n\n- The light clock movie says it all: http://www.anu.edu.au/Physics/qt/\n\n- Cosmic ray muon puzzle\n\n- – Created about 10km above ground\n- Half life  $1.56 \\times 10^{-6}$ second\n- In this time, light travels  $0.47 \\mathrm{~km}$\n- So how can they reach the ground?\n- $v\\approx 0.99c$ gives  $\\gamma \\approx 7$\n- $v \\approx 0.9999c$ gives  $\\gamma \\approx 71$\n\n- Leads to twin paradox", "source": "mit-ocw", "source_doc_id": "0004-kinematics-handout-42d52bee-882cb5f1", "source_title": "MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-kinematics-handout-42d52bee-882cb5f1:page-0006", "text": "Consider two frames in relative motion. For t = 0, the Lorentz transformation gives   $x' = \\gamma x$ , where   $\\gamma > 1$ .\n\nQuestion: How long does a yard stick at rest in the unprimed frame look in the primed frame?\n\n- 1. Longer than one yard\n- 2. Shorter than one yard\n- 3. One yard\n\n# Implications: relativity of simultaneity\n\n- Consider two events simultaneous in frame  $S$:\n\n$$\n\\mathbf {x} _ {1} = \\left( \\begin{array}{c} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{array} \\right), \\quad \\mathbf {x} _ {2} = \\left( \\begin{array}{c} L \\\\ 0 \\\\ 0 \\\\ 0 \\end{array} \\right).\n$$\n\n- In the frame  $S'$, they are\n\n$$\n\\begin{array}{l} \\mathbf {x} _ {1} ^ {\\prime} = \\left( \\begin{array}{c c c c} \\gamma & 0 & 0 & - \\gamma \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ - \\gamma \\beta & 0 & 0 & \\gamma \\end{array} \\right) \\left( \\begin{array}{c} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{array} \\right) = \\left( \\begin{array}{c} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{array} \\right) \\\\ \\mathbf {x} _ {2} ^ {\\prime} = \\left( \\begin{array}{c c c c} \\gamma & 0 & 0 & - \\gamma \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ - \\gamma \\beta & 0 & 0 & \\gamma \\end{array} \\right) \\left( \\begin{array}{c} L \\\\ 0 \\\\ 0 \\\\ 0 \\end{array} \\right) = \\left( \\begin{array}{c} \\gamma L \\\\ 0 \\\\ 0 \\\\ - \\gamma \\beta \\mathrm{L} \\end{array} \\right) \\\\ \\end{array}\n$$\n\n- So in  $S'$, the second event happened first!\n\n- So  $S$-clocks appear unsynchronized in  $S'$ - those with larger  $x$ run further ahead", "source": "mit-ocw", "source_doc_id": "0004-kinematics-handout-42d52bee-882cb5f1", "source_title": "MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 5, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "Page 6"], "page_start": 6, "page_end": 6, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-kinematics-handout-42d52bee-882cb5f1:page-0007", "text": "# Implications: length contraction\n\n- Trickier than time dilation, opposite result (interval appears shorter, not longer)\n- In the frame  $S$, a yardstick of length  $L$ is at rest along the  $x$-axis with its endpoints tracing out world lines with coordinates\n\n$$\n\\mathbf {x} _ {1} = \\left( \\begin{array}{c} 0 \\\\ 0 \\\\ 0 \\\\ c t \\end{array} \\right), \\quad \\mathbf {x} _ {2} = \\left( \\begin{array}{c} L \\\\ 0 \\\\ 0 \\\\ c t \\end{array} \\right).\n$$\n\n- In the frame  $S'$, these world lines are\n\n$$\n\\mathbf {x} _ {1} ^ {\\prime} = \\left( \\begin{array}{c} x _ {1} ^ {\\prime} \\\\ y _ {1} ^ {\\prime} \\\\ z _ {1} ^ {\\prime} \\\\ c t _ {1} ^ {\\prime} \\end{array} \\right) = \\left( \\begin{array}{c c c c} \\gamma & 0 & 0 & - \\gamma \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ - \\gamma \\beta & 0 & 0 & \\gamma \\end{array} \\right) \\left( \\begin{array}{c} 0 \\\\ 0 \\\\ 0 \\\\ c t \\end{array} \\right) = \\left( \\begin{array}{c} - \\gamma \\beta c t \\\\ 0 \\\\ 0 \\\\ \\gamma c t \\end{array} \\right)\n$$\n\n$$\n\\begin{array}{r l r} {\\mathbf {x} _ {2} ^ {\\prime}} & = & {\\left( \\begin{array}{c} x _ {2} ^ {\\prime} \\\\ y _ {2} ^ {\\prime} \\\\ z _ {2} ^ {\\prime} \\\\ c t _ {2} ^ {\\prime} \\end{array} \\right) = \\left( \\begin{array}{c c c c} \\gamma & 0 & 0 & - \\gamma \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ - \\gamma \\beta & 0 & 0 & \\gamma \\end{array} \\right) \\left( \\begin{array}{c} L \\\\ 0 \\\\ 0 \\\\ c t \\end{array} \\right) = \\left( \\begin{array}{c} \\gamma L - \\gamma \\beta c t \\\\ 0 \\\\ 0 \\\\ \\gamma c t - \\gamma \\beta L \\end{array} \\right)} \\end{array}\n$$\n\n- An observer in  $S'$ measures length as  $x_2' - x_1'$ at the same time  $t'$, - not at the same time  $t$.\n- Let's measure at  $t' = 0$.\n- $t_1' = 0$ when  $t = 0$ — at this time,  $x_1' = 0$\n- $t_2' = 0$ when  $ct = \\beta L$ - at this time,  $\\mathbf{x}_2' = \\gamma L - \\gamma \\beta^2 L = L / \\gamma$\n- So in  $S'$-frame, measured length is  $L' = L / \\gamma$, i.e., shorter\n- Let's work out the new world lines of the yard stick endpoints\n- $\\mathbf{x}_1' + \\beta ct_1' = 0$, so left endpoint world line is\n\n$$\nx _ {1} ^ {\\prime} = - v t _ {1} ^ {\\prime}\n$$\n\n-  $\\mathbf{x}_2' - \\gamma L + \\beta(ct_2' + \\gamma \\beta L) = 0$, so right endpoint world line is\n\n$$\nx _ {2} ^ {\\prime} = \\gamma L - \\beta (c t _ {2} ^ {\\prime} + \\gamma \\beta L) = \\frac {L}{\\gamma} - v t _ {2} ^ {\\prime}\n$$\n\n- Length in  $S'$ is\n\n$$\nx _ {2} ^ {\\prime} - x _ {1} ^ {\\prime} = \\frac {L}{\\gamma} + v (t _ {1} ^ {\\prime} - t _ {2} ^ {\\prime}) = \\frac {L}{\\gamma}\n$$\n\nsince both endpoints measured at same time ( $t_1' = t_2'$)\n\n- Draw Minkowski diagram of this", "source": "mit-ocw", "source_doc_id": "0004-kinematics-handout-42d52bee-882cb5f1", "source_title": "MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 6, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "Page 7"], "page_start": 7, "page_end": 7, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-kinematics-handout-42d52bee-882cb5f1:page-0008", "text": "# Superluminal communication?\n\n- Velocity addition formula shows that it's impossible to accelerate something past the speed of light\n- But could there be another way, say a type of radiation that moves faster than light?\n- Can an event A influence another event B at spacelike separation (hence transmitting information faster than the speed of light)?\n- There is another frame where B happened before A! (PS3)\n- Draw Minkowski diagram of this\n- By inertial frame invariance, B can then send a signal that arrives back to A before she sent her initial signal, telling her not to send it.\n- Implication:  $c$ isn't merely the speed of light, but the limiting speed for anything\n\n# \"Everything is relative\" — or is it?\n\n- All observers agree on rest length\n- All observers agree on proper time\n- All observers (as we'll see later) agree on rest mass", "source": "mit-ocw", "source_doc_id": "0004-kinematics-handout-42d52bee-882cb5f1", "source_title": "MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 7, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "Page 8"], "page_start": 8, "page_end": 8, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-kinematics-handout-42d52bee-882cb5f1:page-0009", "text": "# Transformation toolbox: the inverse Lorentz transform\n\n- Since  $\\mathbf{x}' = \\boldsymbol{\\Lambda}(v)\\mathbf{x}$ and  $\\mathbf{x} = \\boldsymbol{\\Lambda}(-v)\\mathbf{x}'$, we get the consistency requirement\n\n$$\n\\mathbf {x} = \\boldsymbol {\\Lambda} (- v) \\mathbf {x} ^ {\\prime} = \\boldsymbol {\\Lambda} (- v) \\boldsymbol {\\Lambda} (v) \\mathbf {x}\n$$\n\nfor any event x, so we must have   $\\mathbf{\\Lambda}(-v) = \\mathbf{\\Lambda}(v)^{-1}$ , the matrix inverse of   $\\mathbf{\\Lambda}(v)$ .\n\n- Is it?\n\n$$\n\\boldsymbol {\\Lambda} (- \\mathbf {v}) \\boldsymbol {\\Lambda} (\\mathbf {v}) = \\left( \\begin{array}{c c c c} \\gamma & 0 & 0 & \\gamma \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ \\gamma \\beta & 0 & 0 & \\gamma \\end{array} \\right) \\left( \\begin{array}{c c c c} \\gamma & 0 & 0 & - \\gamma \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ - \\gamma \\beta & 0 & 0 & \\gamma \\end{array} \\right) = \\left( \\begin{array}{c c c c} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{array} \\right),\n$$\n\ni.e., yes!\n\n# Transformation toolbox: velocity addition\n\n- If the frame  $S'$ has velocity  $v_1$ relative to  $S$ and the frame  $S''$ has velocity  $v_2$ relative to  $S'$ (both in the x-direction), then what is the speed  $v_3$ of  $S''$ relative to  $S$?\n- $\\mathbf{x}' = \\boldsymbol{\\Lambda}(v_1)\\mathbf{x}$ and  $\\mathbf{x}'' = \\boldsymbol{\\Lambda}(v_2)\\mathbf{x}' = \\boldsymbol{\\Lambda}(v_2)\\boldsymbol{\\Lambda}(v_1)\\mathbf{x}$, so\n- $\\mathbf{\\Lambda}(\\mathbf{v}_3) = \\mathbf{\\Lambda}(v_2)\\mathbf{\\Lambda}(v_1)$, i.e.\n\n$$\n\\left( \\begin{array}{c c c c} \\gamma_ {3} & 0 & 0 & - \\gamma_ {3} \\beta_ {3} \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ - \\gamma_ {3} \\beta_ {3} & 0 & 0 & \\gamma_ {3} \\end{array} \\right) = \\left( \\begin{array}{c c c c} \\gamma_ {2} & 0 & 0 & - \\gamma_ {2} \\beta_ {2} \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ - \\gamma_ {2} \\beta_ {2} & 0 & 0 & \\gamma_ {2} \\end{array} \\right) \\left( \\begin{array}{c c c c} \\gamma_ {1} & 0 & 0 & - \\gamma_ {1} \\beta_ {1} \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ - \\gamma_ {1} \\beta_ {1} & 0 & 0 & \\gamma_ {1} \\end{array} \\right)\n$$\n\n$$\n= \\gamma_ {1} \\gamma_ {2} \\left( \\begin{array}{c c c c} 1 + \\beta_ {1} \\beta_ {2} & 0 & 0 & - [ \\beta_ {1} + \\beta_ {2} ] \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ - [ \\beta_ {1} + \\beta_ {2} ] & 0 & 0 & 1 + \\beta_ {1} \\beta_ {2} \\end{array} \\right)\n$$\n\n• Take ratio between   $(1,4)$  and   $(1,1)$  elements:\n\n$$\n\\beta_ {3} = - \\frac {\\mathbf {\\Lambda} (v _ {3}) _ {4 1}}{\\mathbf {\\Lambda} (v _ {3}) _ {1 1}} = \\frac {\\beta_ {1} + \\beta_ {2}}{1 + \\beta_ {1} \\beta_ {2}}.\n$$\n\n- In other words,\n\n$$\nv _ {3} = \\frac {v _ {1} + v _ {2}}{1 + \\frac {v _ {1} v _ {2}}{c ^ {2}}}.\n$$", "source": "mit-ocw", "source_doc_id": "0004-kinematics-handout-42d52bee-882cb5f1", "source_title": "MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 8, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "Page 9"], "page_start": 9, "page_end": 9, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-kinematics-handout-42d52bee-882cb5f1:page-0010", "text": "# Transformation toolbox: perpendicular velocity addition\n\n- Here's an alternative derivation of velocity addition that easily gives the non-parallel components too (but 4-vector method on next page is simpler)\n- If the frame  $S'$ has velocity  $v$ in the  $x$-direction relative to  $S$ and a particle has velocity  $\\mathbf{u}' = (u_x', u_y', u_z')$ in  $S'$, then what is its velocity  $\\mathbf{u}$ in  $S$?\n- Applying the inverse Lorentz transformation\n\n$$\nx = \\gamma (x ^ {\\prime} + v t ^ {\\prime})\n$$\n\n$$\ny = y ^ {\\prime}\n$$\n\n$$\nz = z ^ {\\prime}\n$$\n\n$$\n{ t } { = } { \\gamma ( t ^ { \\prime } + v x ^ { \\prime } / c ^ { 2 } ) }\n$$\n\nto two nearby points on the particle's world line and subtracting gives\n\n$$\nd x = \\gamma (d x ^ {\\prime} + v d t ^ {\\prime})\n$$\n\n$$\nd y = d y ^ {\\prime}\n$$\n\n$$\nd z = d z ^ {\\prime}\n$$\n\n$$\n{ d t } { = } { \\gamma ( d t ^ { \\prime } + v d x ^ { \\prime } / c ^ { 2 } ) . }\n$$\n\n$$\nd x = \\gamma (d x ^ {\\prime} + v d t ^ {\\prime})\n$$\n\n$$\nd y = d y ^ {\\prime}\n$$\n\n$$\nd z = d z ^ {\\prime}\n$$\n\n$$\n{ d t } { = } { \\gamma ( d t ^ { \\prime } + v d x ^ { \\prime } / c ^ { 2 } ) . }\n$$\n\n- Answer:\n\n$$\nu _ {x} = \\frac {d x}{d t} = \\frac {\\gamma \\left(d x ^ {\\prime} + v d t ^ {\\prime}\\right)}{\\gamma \\left(d t ^ {\\prime} + \\frac {v d x ^ {\\prime}}{c ^ {2}}\\right)} = \\frac {\\frac {d x ^ {\\prime}}{d t ^ {\\prime}} + v}{1 + \\frac {v}{c ^ {2}} \\frac {d x ^ {\\prime}}{d t ^ {\\prime}}} = \\frac {u _ {x} ^ {\\prime} + v}{1 + \\frac {u _ {x} ^ {\\prime} v}{c ^ {2}}}\n$$\n\n$$\n{u _ {y}} = {\\frac {d y}{d t} = \\frac {d y ^ {\\prime}}{\\gamma (d t ^ {\\prime} + \\frac {v d x ^ {\\prime}}{c ^ {2}})} = \\frac {\\gamma^ {- 1} \\frac {d y ^ {\\prime}}{d t ^ {\\prime}}}{1 + \\frac {v}{c ^ {2}} \\frac {d x ^ {\\prime}}{d t ^ {\\prime}}} = \\frac {u _ {y} ^ {\\prime} \\sqrt {1 - \\frac {v ^ {2}}{c ^ {2}}}}{1 + \\frac {u _ {x} ^ {\\prime} v}{c ^ {2}}}}\n$$\n\n$$\nu _ {z} = \\frac {d z}{d t} = \\frac {d z ^ {\\prime}}{\\gamma (d t ^ {\\prime} + \\frac {v d x ^ {\\prime}}{c ^ {2}})} = \\frac {\\gamma^ {- 1} \\frac {d z ^ {\\prime}}{d t ^ {\\prime}}}{1 + \\frac {v}{c ^ {2}} \\frac {d x ^ {\\prime}}{d t ^ {\\prime}}} = \\frac {u _ {z} ^ {\\prime} \\sqrt {1 - \\frac {v ^ {2}}{c ^ {2}}}}{1 + \\frac {u _ {x} ^ {\\prime} v}{c ^ {2}}}\n$$", "source": "mit-ocw", "source_doc_id": "0004-kinematics-handout-42d52bee-882cb5f1", "source_title": "MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 9, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "Page 10"], "page_start": 10, "page_end": 10, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-kinematics-handout-42d52bee-882cb5f1:page-0011", "text": "# Transformation toolbox: velocity as a 4-vector\n\n- For a particle moving along its world-line, define its velocity 4-vector\n\n$$\n\\mathbf {U} \\equiv \\frac {d \\mathbf {X}}{d \\tau} = \\gamma_ {u} \\left( \\begin{array}{c} u _ {x} \\\\ u _ {y} \\\\ u _ {z} \\\\ c \\end{array} \\right),\n$$\n\nwhere\n\n$$\n\\gamma_ {u} \\equiv \\frac {1}{\\sqrt {1 - \\frac {u ^ {2}}{c ^ {2}}}}\n$$\n\n- This is the derivative of its 4-vector  $\\mathbf{x}$ w.r.t. its proper time  $\\tau$, since  $d\\tau = dt / \\gamma_{u}$\n\n•   $U' = \\Lambda U:$\n\n$$\n\\mathbf {U} ^ {\\prime} = \\frac {d \\mathbf {X} ^ {\\prime}}{d \\tau^ {\\prime}} = \\frac {d \\boldsymbol {\\Lambda} \\mathbf {X}}{d \\tau} = \\boldsymbol {\\Lambda} \\frac {d \\mathbf {X}}{d \\tau} = \\boldsymbol {\\Lambda} \\mathbf {U},\n$$\n\nsince the proper time interval   $d\\tau$  is Lorentz-invariant\n\n- This means that all velocity 4-vectors are normalized so that\n\n$$\n\\mathbf {U} ^ {t} \\boldsymbol {\\eta} \\mathbf {U} = - c ^ {2}.\n$$\n\n- This immediately gives the velocity addition formulas:\n\n$$\n\\begin{array}{l} \\mathbf {U} ^ {\\prime} = \\gamma_ {u ^ {\\prime}} \\left( \\begin{array}{c} u _ {x} ^ {\\prime} \\\\ u _ {y} ^ {\\prime} \\\\ u _ {z} ^ {\\prime} \\\\ c \\end{array} \\right) = \\boldsymbol {\\Lambda} (- \\mathbf {v}) \\mathbf {U} = \\gamma_ {u} \\left( \\begin{array}{c c c c} \\gamma & 0 & 0 & \\gamma \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ \\gamma \\beta & 0 & 0 & \\gamma \\end{array} \\right) \\left( \\begin{array}{c} u _ {x} \\\\ u _ {y} \\\\ u _ {z} \\\\ c \\end{array} \\right) \\\\ = \\left( \\begin{array}{c} \\gamma_ {u} \\gamma [ u _ {x} + v ] \\\\ \\gamma_ {u} u _ {y} \\\\ \\gamma_ {u} y _ {z} \\\\ \\gamma_ {u} \\gamma [ 1 + \\frac {u _ {x} v}{c ^ {2}} ] c \\end{array} \\right) = \\gamma_ {u ^ {\\prime}} \\left( \\begin{array}{c} \\frac {u _ {x} + v}{1 + u _ {x} v / c ^ {2}} \\\\ \\frac {u _ {y} / \\gamma}{1 + u _ {x} v / c ^ {2}} \\\\ \\frac {u _ {z} / \\gamma}{1 + u _ {x} v / c ^ {2}} \\\\ c \\end{array} \\right), \\\\ \\end{array}\n$$\n\nwhere   $\\gamma_{u'} = \\gamma_u \\gamma \\left[1 + \\frac{u_x v}{c^2}\\right]$  — this last equation follows from the fact that the 4-vector normalization in Lorentz invariant, i.e.,   $u'^t \\eta u' = u^t \\eta u = -1$ .\n\n- The 1st 3 components give the velocity addition equations we derived previously.", "source": "mit-ocw", "source_doc_id": "0004-kinematics-handout-42d52bee-882cb5f1", "source_title": "MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 10, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "Page 11"], "page_start": 11, "page_end": 11, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-kinematics-handout-42d52bee-882cb5f1:page-0012", "text": "# Transformation toolbox: boosts as generalized rotations\n\n- A “boost” is a Lorentz transformation with no rotation\n- A rotation around the  $z$-axis by angle  $\\theta$ is given by the transformation\n\n$$\n\\left( \\begin{array}{c c c c} \\cos \\theta & \\sin \\theta & 0 & 0 \\\\ - \\sin \\theta & \\cos \\theta & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{array} \\right)\n$$\n\n- We can think of a boost in the  $x$-direction as a rotation by an imaginary angle in the  $(x, ct)$-plane:\n\n$$\n\\mathbf {\\Lambda} (- v) = \\left( \\begin{array}{c c c c} \\gamma & 0 & 0 & \\gamma \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ \\gamma \\beta & 0 & 0 & \\gamma \\end{array} \\right) = \\left( \\begin{array}{c c c c} \\cosh \\eta & 0 & 0 & \\sinh \\eta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ \\sinh \\eta & 0 & 0 & \\cosh \\eta \\end{array} \\right),\n$$\n\nwhere  $\\eta \\equiv \\tanh^{-1}\\beta$ is called the rapidity.\n\n- Proof: use hyperbolic trig identities on next page\n- Implication: for multiple boosts in same direction, rapidities add and hence the order doesn't matter\n\n# Hyperbolic trig reminders\n\n$$\n\\cosh x = \\frac {e ^ {x} + e ^ {- x}}{2}\n$$\n\n$$\n\\sinh x = \\frac {e ^ {x} - e ^ {- x}}{2}\n$$\n\n$$\n\\tanh x = \\frac {e ^ {x} - e ^ {- x}}{e ^ {x} + e ^ {- x}}\n$$\n\n$$\n\\cosh^ {- 1} x = \\ln (x + \\sqrt {x ^ {2} - 1})\n$$\n\n$$\n\\sinh^ {- 1} x = \\ln (x + \\sqrt {x ^ {2} + 1})\n$$\n\n$$\n\\tanh ^ {- 1} x = \\frac {1}{2} \\ln \\left(\\frac {1 + x}{1 - x}\\right)\n$$\n\n$$\n\\cosh \\tanh ^ {- 1} x = \\frac {1}{\\sqrt {1 - x ^ {2}}}\n$$\n\n$$\n\\sinh \\tanh ^ {- 1} x = \\frac {x}{\\sqrt {1 - x ^ {2}}}\n$$\n\n$$\n\\cosh^ {2} x - \\sinh^ {2} x = 1\n$$", "source": "mit-ocw", "source_doc_id": "0004-kinematics-handout-42d52bee-882cb5f1", "source_title": "MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 11, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "Page 12"], "page_start": 12, "page_end": 12, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-kinematics-handout-42d52bee-882cb5f1:page-0013", "text": "# The Lorentz invariant\n\n• The Minkowski metric\n\n$$\n\\boldsymbol {\\eta} = \\left( \\begin{array}{c c c c} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & - 1 \\end{array} \\right)\n$$\n\nis left invariant by all Lorentz matrices  $\\Lambda$:\n\n$$\n\\boldsymbol {\\Lambda} ^ {t} \\eta \\boldsymbol {\\Lambda} = \\eta\n$$\n\n(indeed, this equation is often used to define the set of Lorentz matrices — for comparison,  $\\mathbf{\\Lambda}^t\\mathbf{I}\\mathbf{\\Lambda} = \\mathbf{I}$ would define rotation matrices)\n\n- Proof: Show that works for boost along  $x$-axis. Show that works for rotation along  $y$-axis or  $z$-axis. General case is equivalent to applying such transformations in succession.\n- All Lorentz transforms leave the quantity\n\n$$\n\\mathbf {x} ^ {t} \\pmb {\\eta} \\mathbf {x} = x ^ {2} + y ^ {2} + z ^ {2} - (c t) ^ {2}\n$$\n\ninvariant\n\n- Proof:\n\n$$\n\\mathbf {x} ^ {\\prime t} \\boldsymbol {\\eta} \\mathbf {x} ^ {\\prime} = (\\boldsymbol {\\Lambda} \\mathbf {x}) ^ {t} \\boldsymbol {\\eta} (\\boldsymbol {\\Lambda} \\mathbf {x}) = \\mathbf {x} ^ {t} (\\boldsymbol {\\Lambda} ^ {t} \\boldsymbol {\\eta} \\boldsymbol {\\Lambda}) \\mathbf {x} = \\mathbf {x} ^ {t} \\boldsymbol {\\eta} \\mathbf {x}\n$$\n\n- (More generally, the same calculation shows that  $\\mathbf{x}^t\\boldsymbol{\\eta}\\mathbf{y}$ is invariant)\n- So just as the usual Euclidean squared length   $|r|^{2} = r \\cdot r = r^{t} r = r^{t} Ir$  of a 3-vector is rotationally invariant, the generalized “length”   $x^{t} \\eta x$  of a 4-vector is Lorentz-invariant.\n- It can be positive or negative\n- For events  $\\mathbf{x}_1$ and  $\\mathbf{x}_2$, their Lorentz-invariant separation is defined as\n\n$$\n\\Delta \\sigma^ {2} \\equiv \\Delta \\mathbf {x} ^ {t} \\boldsymbol {\\eta} \\Delta \\mathbf {x} = \\Delta x ^ {2} + \\Delta y ^ {2} + \\Delta z ^ {2} - (c \\Delta t) ^ {2}\n$$\n\n- A separation  $\\Delta \\sigma^2 = 0$ is called null\n\n- A separation  $\\Delta \\sigma^2 > 0$ is called spacelike, and\n\n$$\n\\Delta \\sigma \\equiv \\sqrt {\\Delta \\sigma^ {2}}\n$$\n\nis called the proper distance (the distance measured in a frame where the events are simultaneous)\n\n- A separation  $\\Delta \\sigma^2 < 0$ is called timelike, and\n\n$$\n\\Delta \\tau \\equiv \\sqrt {- \\Delta \\sigma^ {2}}\n$$\n\nis called the proper time interval (the time interval measured in a frame where the events are at the same place)\n\n- More generally, any 4-vector is either null, spacelike of timelike.\n- The velocity 4-vector  $\\mathbf{U}$ is always timelike.", "source": "mit-ocw", "source_doc_id": "0004-kinematics-handout-42d52bee-882cb5f1", "source_title": "MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 12, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "Page 13"], "page_start": 13, "page_end": 13, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-kinematics-handout-42d52bee-882cb5f1:page-0014", "text": "# Transforming a wave vector\n\n- A plane wave\n\n$$\nE (\\mathbf {x}) = \\sin (k _ {x} x + k _ {y} y + k _ {z} z - \\omega t) \\tag {1}\n$$\n\nis defined by the four numbers\n\n$$\n\\mathbf {K} \\equiv \\left( \\begin{array}{c} k _ {x} \\\\ k _ {y} \\\\ k _ {z} \\\\ \\omega / c \\end{array} \\right).\n$$\n\n- If the wave propagates with the speed of light  $c$ (like for an electromagnetic or gravitational wave), then the frequency is determined by the 3D wave vector  $(k_x, k_y, k_z)$ through the relation  $\\omega / c = k$, where  $k \\equiv \\sqrt{k_x^2 + k_y^2 + k_z^2}$\n\n- How does the 4-vector  $\\mathbf{K}$ transform under Lorentz transformations? Let's see.\n\n- Using the Minkowski matrix, we can rewrite equation (1) as\n\n$$\nE (\\mathbf {X}) = \\sin (\\mathbf {K} ^ {t} \\pmb {\\eta} \\mathbf {X}).\n$$\n\n- Let's Lorentz transform this:  $\\mathbf{X} \\to \\mathbf{X}'$,  $\\mathbf{K} \\to \\mathbf{K}'$. Using that  $\\mathbf{X}' = \\boldsymbol{\\Lambda}\\mathbf{X}$, let's determine  $\\mathbf{K}'$.\n\n$$\nE ^ {\\prime} = \\sin (\\mathbf {K} ^ {\\prime t} \\boldsymbol {\\eta} \\mathbf {X} ^ {\\prime}) = \\sin (\\mathbf {K} ^ {\\prime t} \\boldsymbol {\\eta} \\boldsymbol {\\Lambda} \\mathbf {X}) = \\sin [ (\\boldsymbol {\\Lambda} ^ {- 1} \\mathbf {K} ^ {\\prime}) ^ {t} (\\boldsymbol {\\Lambda} ^ {t} \\boldsymbol {\\eta} \\boldsymbol {\\Lambda}) \\mathbf {X} ] = \\sin [ (\\boldsymbol {\\Lambda} ^ {- 1} \\mathbf {K} ^ {\\prime}) ^ {t} \\boldsymbol {\\eta} \\mathbf {X} ].\n$$\n\n- This equals  $E$ if  $\\mathbf{\\Lambda}^{-1}\\mathbf{K}' = \\mathbf{K}$, i.e., if the wave 4-vector transforms just as a normal 4-vector:\n\n$$\n\\mathbf {K} ^ {\\prime} = \\boldsymbol {\\Lambda} \\mathbf {K}\n$$\n\n- This argument assumed that  $E' = E$. Later we'll see that the electric and magnetic fields do in fact change under Lorentz transforms, but not in a way that spoils the above derivation (in short, the phase of the wave,  $\\mathbf{K}^t\\eta \\mathbf{X}$, must be Lorentz invariant)\n\n- So a plane wave  $\\mathbf{K}$ in  $S$ is also a plane wave in  $S'$, and the wave 4-vector transforms in exactly the same way as  $\\mathbf{X}$ does.", "source": "mit-ocw", "source_doc_id": "0004-kinematics-handout-42d52bee-882cb5f1", "source_title": "MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 13, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "Page 14"], "page_start": 14, "page_end": 14, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-kinematics-handout-42d52bee-882cb5f1:page-0015", "text": "# Aberration and Doppler effects\n\n- Consider a plane wave propagating with speed  $c$ in the frame  $S$:\n\n$$\n\\mathbf {K} = k \\left( \\begin{array}{c} \\sin \\theta \\cos \\phi \\\\ \\sin \\theta \\sin \\phi \\\\ \\cos \\theta \\\\ 1 \\end{array} \\right),\n$$\n\nwhere ck is the wave frequency and the angles   $\\theta$  and   $\\phi$  give the propagation direction in polar coordinates.\n\n- Let's Lorentz transform this into a frame  $S'$ moving with speed  $v$ relative to  $S$ in the  $z$-direction:  $\\mathbf{k}' = \\boldsymbol{\\Lambda}\\mathbf{k}$, i.e.,\n\n$$\n\\begin{array}{l} \\mathbf {K} ^ {\\prime} = k ^ {\\prime} \\left( \\begin{array}{c} \\sin \\theta^ {\\prime} \\cos \\phi^ {\\prime} \\\\ \\sin \\theta^ {\\prime} \\sin \\phi^ {\\prime} \\\\ \\cos \\theta^ {\\prime} \\\\ 1 \\end{array} \\right) = k \\left( \\begin{array}{c c c c} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & \\gamma & - \\gamma \\beta \\\\ 0 & 0 & - \\gamma \\beta & \\gamma \\end{array} \\right) \\left( \\begin{array}{c} \\sin \\theta \\cos \\phi \\\\ \\sin \\theta \\sin \\phi \\\\ \\cos \\theta \\\\ 1 \\end{array} \\right) \\\\ = k \\left( \\begin{array}{c} \\sin \\theta \\cos \\phi \\\\ \\sin \\theta \\sin \\phi \\\\ \\gamma (\\cos \\theta - \\beta) \\\\ \\gamma (1 - \\beta \\cos \\theta) \\end{array} \\right), \\\\ \\end{array}\n$$\n\nSO\n\n$$\n\\phi^ {\\prime} = \\phi\n$$\n\n$$\n\\cos \\theta^ {\\prime} = \\frac {\\cos \\theta - \\beta}{1 - \\beta \\cos \\theta}\n$$\n\n$$\nk ^ {\\prime} = k \\gamma (1 - \\beta \\cos \\theta)\n$$\n\n- This matches equations (1)-(4) in the Weiskopf et al ray tracing handout\n- The change in the angle  $\\theta$ is known as aberration\n- The change in frequency  $ck$ is known as the Doppler shift — note that since  $k = 2\\pi/\\lambda$, we have  $\\lambda'/\\lambda = k/k'$.\n- If we instead take the ratio  $\\sqrt{k'_{x}^{2} + k'_{y}^{2}} / k_{z}'$ above, we obtain the mathematically equivalent form of the aberration formula given by Resnick (2-27b):\n\n$$\n\\tan \\theta^ {\\prime} = \\frac {\\sin \\theta}{\\gamma (\\cos \\theta - \\beta)}\n$$", "source": "mit-ocw", "source_doc_id": "0004-kinematics-handout-42d52bee-882cb5f1", "source_title": "MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 14, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "Page 15"], "page_start": 15, "page_end": 15, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-kinematics-handout-42d52bee-882cb5f1:page-0016", "text": "- Examine classical limits\n- Transverse Doppler effect:  $\\cos \\theta = 0$ gives  $\\omega' = \\omega \\gamma$, i.e., simple time dilation (classically,  $\\omega' = \\omega$, i.e., no transverse effect)\n- Longitudinal doppler effect:  $\\cos \\theta = 1$ gives\n\n$$\n\\frac {\\omega^ {\\prime}}{\\omega} = \\gamma (1 - \\beta) = \\sqrt {\\frac {1 - \\beta}{1 + \\beta}}.\n$$\n\n- For comparison, classical physics, moving observer:\n\n$$\n\\frac {\\omega^ {\\prime}}{\\omega} = 1 - \\beta .\n$$\n\n- For comparison, classical physics, moving source:\n\n$$\n\\frac {\\omega^ {\\prime}}{\\omega} = \\frac {1}{1 + \\beta}\n$$", "source": "mit-ocw", "source_doc_id": "0004-kinematics-handout-42d52bee-882cb5f1", "source_title": "MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 15, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "Page 16"], "page_start": 16, "page_end": 16, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-kinematics-handout-42d52bee-882cb5f1:page-0017", "text": "# Accelerated motion & proper time\n\n- Consider a clock moving along a curve  $\\mathbf{r}(t)$ though spacetime, as measured in a frame  $S$. During an infinitesimal time interval between  $t$ and  $t + dt$, it moves with velocity  $\\mathbf{u}(t) = \\dot{\\mathbf{r}}(t)$ and measures a proper time interval\n\n$$\nd \\tau = \\frac {d t}{\\gamma_ {u}} = \\sqrt {1 - \\frac {| \\dot {\\bf r} (t) | ^ {2}}{c ^ {2}}} d t.\n$$\n\n- The proper time interval (a.k.a. wristwatch time) measured by the clock as it moves from event A to event B along this path is\n\n$$\n\\Delta \\tau = \\int_ {t _ {A}} ^ {t _ {B}} d \\tau = \\int_ {t _ {A}} ^ {t _ {B}} \\sqrt {1 - \\frac {| \\dot {\\bf r} (t) | ^ {2}}{c ^ {2}}} d t\n$$\n\n- If the two events are at the same position in  $S$, i.e., if  $\\mathbf{r}(t_A) = \\mathbf{r}(t_B)$, then the path  $\\mathbf{r}(t)$ between the two events that maximizes  $\\Delta \\tau$ is clearly the straight line  $\\mathbf{r}(t) = \\mathbf{r}(t_A)$ where the clock never moves, giving  $\\mathbf{u} = \\mathbf{0}$ and  $\\Delta \\tau = \\Delta t = t_B - t_A$.\n\n- For any two events with timelike separation, the proper time is again maximized when the path between the two points is a straight line though spacetime.\n\nProof: Lorentz transform to a frame   $S'$  where A and B are at the same position, conclude the path is a straight line in   $S'$  and use the fact that the Lorentz transform of a straight line through spacetime is always a straight line through spacetime.\n\n- One can also deduce this with calculus of variations, which is overkill for this simple case.\n\n# Calculus of variations\n\n- The much more general optimization problem of finding the path  $x(t)$ that minimizes or maximizes a quantity\n\n$$\nS [ x ] \\equiv \\int_ {t _ {0}} ^ {t _ {1}} f [ t, x (t), \\dot {x} (t) ] d t\n$$\n\nsubject to the constraints that  $x(t_0) = x_0$ and  $x(t_1) = x_1$ reduces to solving the differential equation known as the Euler-Lagrange equation:\n\n$$\n{\\frac {\\partial f}{\\partial x}} - {\\frac {d}{d t}} {\\frac {\\partial f}{\\partial {\\dot {x}}}} = 0.\n$$\n\n- Here the meaning of  $\\frac{\\partial f}{\\partial \\dot{x}}$ is simply the partial derivative of  $f$ with respect to its third argument, i.e., just treat  $\\dot{x}$ as a variable totally independent of  $x$ when evaluating this derivative.", "source": "mit-ocw", "source_doc_id": "0004-kinematics-handout-42d52bee-882cb5f1", "source_title": "MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 16, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "Page 17"], "page_start": 17, "page_end": 17, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-kinematics-handout-42d52bee-882cb5f1:page-0018", "text": "# Metrics and geodesics\n\n- In an  $n$-dimensional space, the metric is a (usually position-dependent)  $n \\times n$ symmetric matrix  $\\mathbf{g}$ that defines the way distances are measured. The length of a curve is  $\\int d\\sigma$, where\n\n$$\nd \\sigma^ {2} = d \\mathbf {r} ^ {t} \\mathbf {g} d \\mathbf {r},\n$$\n\nand  $\\mathbf{r}$ are whatever coordinates you're using in the space. If you change coordinates, the metric is transformed so that  $d\\sigma$ stays the same ( $d\\sigma$ is invariant under all coordinate transformations).\n\n• Example: 2D Euclidean space in Cartesian coordinates.\n\n$$\n\\mathbf {g} = \\left( \\begin{array}{c c} 1 & 0 \\\\ 0 & 1 \\end{array} \\right),\n$$\n\n$$\nd \\sigma^ {2} = d \\mathbf {r} ^ {t}   \\mathbf {g}   d \\mathbf {r} = \\left( \\begin{array}{c c} d x & d y \\end{array} \\right) \\left( \\begin{array}{c c} 1 & 0 \\\\ 0 & 1 \\end{array} \\right) \\binom{d x}{d y} = d x ^ {2} + d y ^ {2},\n$$\n\n$$\n\\int d \\sigma = \\int \\sqrt {d {\\bf r} ^ {t} {\\bf g} d {\\bf r}} = \\sqrt {d x ^ {2} + d y ^ {2}} = \\sqrt {1 + y ^ {\\prime} (x) ^ {2}} d x.\n$$\n\nApplying the Euler-Lagrange equation to this shows that the shortest path between any two points is a straight line.\n\n- Example: 4D Minkowski space in Cartesian coordinates ( $c = 1$ for simplicity)\n\n$$\n\\mathbf {g} = \\boldsymbol {\\eta} = \\left( \\begin{array}{c c c c} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & - 1 \\end{array} \\right),\n$$\n\n$$\n{d \\tau^ {2}} = {- d \\sigma^ {2} = d {\\bf x} ^ {t} {\\bf g} d {\\bf x} =}\n$$\n\n$$\n= \\left( \\begin{array}{c c c c c} d x & d y & d z & d t \\end{array} \\right) \\left( \\begin{array}{c c c c} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & - 1 \\end{array} \\right) \\left( \\begin{array}{c} d x \\\\ d y \\\\ d z \\\\ d t \\end{array} \\right)\n$$\n\n$$\n= d t ^ {2} - d x ^ {2} - x y ^ {2} - d z ^ {2},\n$$\n\n$$\n\\begin{array}{l} \\Delta \\tau = \\int d \\tau = \\int \\sqrt {d t ^ {2} - d x ^ {2} - d y ^ {2} - d z ^ {2}} = \\int \\sqrt {1 - \\dot {x} ^ {2} - \\dot {y} ^ {2} - \\dot {z} ^ {2}} d t \\\\ = \\int \\sqrt {1 - u ^ {2}} d t = \\int \\frac {d t}{\\gamma}. \\\\ \\end{array}\n$$\n\nApplying the Euler-Lagrange equation to this shows that the extremal interval between any two events is a straight line though spacetime.", "source": "mit-ocw", "source_doc_id": "0004-kinematics-handout-42d52bee-882cb5f1", "source_title": "MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 17, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2006, Relativistic Kinematics: Max Tegmark", "Page 18"], "page_start": 18, "page_end": 18, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0005-relativistic-dynamics-handout-bc253f0e-b3138e4c:page-0001", "text": "MIT Course 8.033, Fall 2005, Relativistic dynamics\n\nMax Tegmark\n\nLast revised October 25 2005\n\n# Topics\n\n- Formula summary\n- Momentum & energy\n- Acceleration & force (optional)\n- Transformation of force (optional)\n- Transformation of acceleration (optional)", "source": "mit-ocw", "source_doc_id": "0005-relativistic-dynamics-handout-bc253f0e-b3138e4c", "source_title": "Topics: Dynamics toolbox: formula summary", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Topics: Dynamics toolbox: formula summary", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0005-relativistic-dynamics-handout-bc253f0e-b3138e4c:page-0002", "text": "# Dynamics toolbox: formula summary\n\n• Mass-energy unification:\n\n$$\nE = m c ^ {2} = m _ {0} \\gamma c ^ {2}\n$$\n\n- Momentum 4-vector:\n\n$$\n\\mathbf {P} \\equiv m _ {0} \\mathbf {U} = \\left( \\begin{array}{c} p _ {x} \\\\ p _ {y} \\\\ p _ {z} \\\\ E / c \\end{array} \\right)\n$$\n\n- Energy formula:\n\n$$\nE = \\sqrt {(m _ {0} c ^ {2}) ^ {2} + (c p) ^ {2}}\n$$\n\n- Velocity formula:\n\n$$\n\\beta = \\frac {c p}{E}\n$$\n\n- Optional material:\n\n- Acceleration 4-vector:\n\n$$\n\\mathbf {A} \\equiv \\frac {d \\mathbf {U}}{d \\tau} = \\gamma_ {u} ^ {2} \\binom{\\mathbf {a}}{0} + \\gamma_ {u} ^ {4} \\frac {\\mathbf {u} \\cdot \\mathbf {a}}{c ^ {2}} \\binom{\\mathbf {u}}{c}\n$$\n\n- Force 4-vector:\n\n$$\n\\mathbb {F} \\equiv \\frac {d}{d \\tau} \\mathbf {P} = \\gamma_ {u} \\binom{\\mathbf {F}}{P / c} = m _ {0} \\mathbf {A}\n$$\n\n- Power:\n\n$$\nP = \\dot {E} = \\mathbf {u} \\cdot \\mathbf {F} = m _ {0} \\gamma_ {u} ^ {3} \\mathbf {u} \\cdot \\mathbf {a}\n$$\n\n- Force 3-vector:\n\n$$\n\\frac {\\mathbf {F}}{m _ {0} \\gamma_ {u}} = \\mathbf {a} + \\gamma_ {u} ^ {2} \\frac {\\mathbf {u} \\cdot \\mathbf {a}}{c ^ {2}} \\mathbf {u} = \\left\\{ \\begin{array}{l l} \\gamma_ {u} ^ {2} \\mathbf {a} & (\\mathbf {u} \\& \\mathbf {a p a r a l l e l}) \\\\ \\mathbf {a} & (\\mathbf {u} \\& \\mathbf {a p e r p e n d i c u l a r}) \\end{array} \\right.\n$$\n\n- Acceleration 3-vector:\n\n$$\nm \\mathbf {a} = \\mathbf {F} - \\frac {P \\mathbf {u}}{c ^ {2}}\n$$\n\n- Force transformation:\n\n$$\nF _ {x} ^ {\\prime} = \\frac {F _ {x} - \\frac {v}{c ^ {2}} P}{1 - \\frac {u _ {x} v}{c ^ {2}}},\n$$\n\n$$\n{F _ {y} ^ {\\prime}} = {\\frac {F _ {y}}{\\gamma \\left(1 - \\frac {u _ {x} v}{c ^ {2}}\\right)},}\n$$\n\n$$\nF _ {z} ^ {\\prime} = \\frac {F _ {z}}{\\gamma \\left(1 - \\frac {u _ {x} v}{c ^ {2}}\\right)},\n$$\n\n$$\nP ^ {\\prime} = \\frac {P - v F _ {x}}{1 - \\frac {u _ {x} v}{c ^ {2}}}\n$$", "source": "mit-ocw", "source_doc_id": "0005-relativistic-dynamics-handout-bc253f0e-b3138e4c", "source_title": "Topics: Dynamics toolbox: formula summary", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Topics: Dynamics toolbox: formula summary", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0005-relativistic-dynamics-handout-bc253f0e-b3138e4c:page-0003", "text": "# Momentum & energy toolbox:\n\n- Relativistic mass:\n\n$$\nm = \\gamma m _ {0}\n$$\n\n• Mass-energy unification:\n\n$$\nE = m c ^ {2}\n$$\n\n- Momentum 4-vector (momentum-energy unification):\n\n$$\n\\mathbf {P} \\equiv m _ {0} \\mathbf {U} = m _ {0} \\frac {d \\mathbf {X}}{d \\tau} = m _ {0} \\gamma_ {u} \\left( \\begin{array}{c} u _ {x} \\\\ u _ {y} \\\\ u _ {z} \\\\ c \\end{array} \\right) = m \\left( \\begin{array}{c} u _ {x} \\\\ u _ {y} \\\\ u _ {z} \\\\ c \\end{array} \\right) = \\left( \\begin{array}{c} p _ {x} \\\\ p _ {y} \\\\ p _ {z} \\\\ E / c \\end{array} \\right),\n$$\n\n(Use upper case X, U and P for the 4-vectors to avoid confusion with the x, u and p 3-vectors.)\n\n- Handy velocity formula follows straight from this:\n\n$$\n\\beta = \\frac {c p}{E}\n$$\n\n- Rest energy:\n\n$$\nE _ {0} = m _ {0} c ^ {2}\n$$\n\nis total energy of particle in the frame where it is at rest\n\n- Kinetic enery:\n\n$$\nK = E - E _ {0} = m c ^ {2} - m _ {0} c ^ {2} = m _ {0} c ^ {2} (\\gamma - 1) = \\frac {1}{2} m _ {0} u ^ {2} + O \\left(\\frac {u ^ {4}}{c}\\right)\n$$\n\n- Rest mass invariant:\n\n$$\nm _ {0} = \\frac {1}{c} \\sqrt {- \\mathbf {P} ^ {t} \\pmb {\\eta} \\mathbf {P}} = \\frac {1}{c ^ {2}} \\sqrt {E ^ {2} - c ^ {2} p ^ {2}},\n$$\n\ngiving the handy relations\n\n$$\nE = \\sqrt {(m _ {0} c ^ {2}) ^ {2} + (c p) ^ {2}},\n$$\n\n$$\np \\equiv | \\mathbf {p} | = \\sqrt {\\frac {E ^ {2}}{c ^ {2}} - (m _ {0} c) ^ {2}}.\n$$\n\n- Low-speed limit  $|\\beta| \\ll 1$:\n\n$$\nE \\approx m _ {0} c ^ {2} + \\frac {1}{2} m _ {0} u ^ {2},\n$$\n\n$$\np = m _ {0} \\gamma u \\approx m _ {0} u.\n$$\n\n- High-speed limit  $|\\beta| \\approx 1 (\\gamma \\gg 1, E \\gg E_0)$:\n\n$$\nE \\approx c p\n$$\n\nThis becomes exact   $(E = cp)$  for particles moving with speed of light, like photons and gravitons.", "source": "mit-ocw", "source_doc_id": "0005-relativistic-dynamics-handout-bc253f0e-b3138e4c", "source_title": "Topics: Dynamics toolbox: formula summary", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Topics: Dynamics toolbox: formula summary", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0005-relativistic-dynamics-handout-bc253f0e-b3138e4c:page-0004", "text": "-  $-\\mathbf{P}^t\\pmb{\\eta}\\mathbf{P} = (E / c)^2 - p^2$ is invariant also for system of particles, since\n\n$$\n\\mathbf {P} _ {\\mathrm{tot}} ^ {\\prime} \\equiv \\sum_ {i} \\mathbf {P} _ {i} ^ {\\prime} = \\sum_ {i} \\boldsymbol {\\Lambda} \\mathbf {P} _ {i} = \\boldsymbol {\\Lambda} \\left(\\sum_ {i} \\mathbf {P} _ {i}\\right) = \\boldsymbol {\\Lambda} \\mathbf {P} _ {\\mathrm{tot}}.\n$$\n\n- We derived   $p = m_{0}\\gamma u$  only for 1-dimensional collision. But any collision is 1-dimensional in the frame where the total momentum is zero!", "source": "mit-ocw", "source_doc_id": "0005-relativistic-dynamics-handout-bc253f0e-b3138e4c", "source_title": "Topics: Dynamics toolbox: formula summary", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Topics: Dynamics toolbox: formula summary", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0005-relativistic-dynamics-handout-bc253f0e-b3138e4c:page-0005", "text": "# Acceleration & force (optional!)\n\n- The acceleration 4-vector A and the Force 4-vector F are less useful than their 4-vector cousins X, U, P and K. We'll use F mainly for deriving the force transformation law, which will in turn give us the transformation law for electromagnetic fields. We'll use upper case A for the acceleration 4-vector to avoid confusion with the the acceleration 3-vector a, and the annoying symbol F for the force 4-vector to avoid confusion with the the force 3-vector F.\n\n- Acceleration 4-vector:\n\n$$\n\\begin{array}{l} \\mathbf {A} \\equiv \\frac {d \\mathbf {U}}{d \\tau} = \\gamma_ {u} \\frac {d \\mathbf {U}}{d t} = \\gamma_ {u} \\frac {d}{d t} \\gamma_ {u} \\left( \\begin{array}{c} u _ {x} \\\\ u _ {y} \\\\ u _ {z} \\\\ c \\end{array} \\right) = \\gamma_ {u} \\frac {d}{d t} \\gamma_ {u} \\binom{\\mathbf {u}}{c} \\\\ = \\gamma_ {u} ^ {2} \\binom{\\dot {\\mathbf {u}}}{0} + \\gamma_ {u} \\dot {\\gamma} _ {u} \\binom{\\mathbf {u}}{c} = \\gamma_ {u} \\binom{\\mathbf {a} + \\dot {\\gamma} _ {u} \\mathbf {u}}{\\dot {\\gamma} _ {u} c} \\\\ = \\gamma_ {u} ^ {2} \\binom{\\mathbf {a}}{0} + \\gamma_ {u} ^ {4} \\frac {\\mathbf {u} \\cdot \\mathbf {a}}{c ^ {2}} \\binom{\\mathbf {u}}{c}, \\\\ \\end{array}\n$$\n\nwhere in the last step, we have used the fact that\n\n$$\n\\dot {\\gamma} _ {u} = \\gamma_ {u} ^ {3} \\frac {\\mathbf {u} \\cdot \\mathbf {a}}{c ^ {2}}.\n$$\n\n- Force 4-vector:\n\n$$\n\\mathbb {F} \\equiv \\frac {d}{d \\tau} \\mathbf {P} = \\gamma_ {u} \\frac {d}{d t} \\mathbf {P} = \\gamma_ {u} \\frac {d}{d t} m _ {0} \\mathbf {U} = m _ {0} \\frac {d}{d \\tau} \\mathbf {U},\n$$\n\nso by definition, we have\n\n$$\n\\mathbb {F} = m _ {0} \\mathbf {A}.\n$$\n\n(Note that this does not apply the Newtonian result  $\\mathbf{F} = ma!$)\n\n- Interpretation of Force 4-vector:\n\n$$\n\\mathbb {F} = \\gamma_ {u} \\frac {d}{d t} \\mathbf {P} = \\gamma_ {u} \\binom{\\dot {\\mathbf {p}}}{\\dot {E} / c} = \\gamma_ {u} \\binom{\\mathbf {F}}{P / c},\n$$\n\nwhere   $F = \\dot{p}$  is the familiar force 3-vector and   $P = \\dot{E}$  is the power, the energy change per unit time (in Watts).\n\n• Work-energy theorem:\n\n$$\nd E = \\mathbf {F} \\cdot d \\mathbf {r} = \\mathbf {F} \\cdot \\frac {d \\mathbf {r}}{d t} d t = \\mathbf {F} \\cdot \\mathbf {u} d t,\n$$\n\nso the power satisfies\n\n$$\nP = \\dot {E} = \\mathbf {u} \\cdot \\mathbf {F}.\n$$\n\n- Force 3-vector explicitly: Dividing the above equation  $\\mathbb{F} = m_0\\mathbf{A}$ by  $\\gamma_{u}$ gives\n\n$$\n\\frac {\\mathbf {F}}{m _ {0} \\gamma_ {u}} = \\mathbf {a} + \\gamma_ {u} ^ {2} \\frac {\\mathbf {u} \\cdot \\mathbf {a}}{c ^ {2}} \\mathbf {u}.\n$$", "source": "mit-ocw", "source_doc_id": "0005-relativistic-dynamics-handout-bc253f0e-b3138e4c", "source_title": "Topics: Dynamics toolbox: formula summary", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Topics: Dynamics toolbox: formula summary", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0005-relativistic-dynamics-handout-bc253f0e-b3138e4c:page-0006", "text": "- Special case where  $\\mathbf{u}$ and  $\\mathbf{a}$ are parallel, e.g., for linear motion:\n\n$$\n\\frac {\\mathbf {F}}{m _ {0} \\gamma_ {u}} = \\mathbf {a} + \\gamma_ {u} ^ {2} \\frac {u ^ {2} \\mathbf {a}}{c ^ {2}} = \\left(1 + \\gamma_ {u} ^ {2} \\beta^ {2}\\right) \\mathbf {a} = \\gamma_ {u} ^ {2} \\mathbf {a}.\n$$\n\n- Special case where  $\\mathbf{u}$ and  $\\mathbf{a}$ are perpendicular, eg, for circular motion:\n\n$$\n\\frac {\\mathbf {F}}{m _ {0} \\gamma_ {u}} = \\mathbf {a}\n$$\n\n- Note that in relativity,  $\\mathbf{F}$ and  $\\mathbf{a}$ are generally not parallel, but that they are parallel for these two special cases.\n\n- Acceleration 3-vector explicitly:\n\n$$\n\\mathbf {a} = \\frac {\\mathbf {F}}{m _ {0} \\gamma_ {u}} - \\frac {\\mathbf {u} \\cdot \\mathbf {F}}{m _ {0} \\gamma_ {u} c ^ {2}} \\mathbf {u} = \\frac {\\mathbf {F}}{m} - \\frac {P}{m c ^ {2}} \\mathbf {u}.\n$$\n\nThe last term (the departure from F = ma) is seen to have the form of a friction term proportional to the power put into the particle. Derivation: the three steps below.\n\n$$\n\\dot {\\gamma} _ {u} = \\frac {d}{d t} \\frac {m _ {0} \\gamma_ {u} c ^ {2}}{m _ {0} c ^ {2}} = \\frac {d}{d t} \\frac {E}{m _ {0} c ^ {2}} = \\frac {\\dot {E}}{m _ {0} c ^ {2}} = \\frac {\\mathbf {u} \\cdot \\mathbf {F}}{m _ {0} c ^ {2}} = \\frac {P}{m _ {0} c ^ {2}}.\n$$\n\nCombining this with the other expression for  $\\dot{\\gamma}_u$ above gives\n\n$$\n\\mathbf {u} \\cdot \\mathbf {a} = \\frac {\\mathbf {u} \\cdot \\mathbf {F}}{\\gamma_ {u} ^ {3} m _ {0}}.\n$$\n\nThe above equation for  $\\mathbf{F}$ now becomes\n\n$$\n\\frac {\\mathbf {F}}{m _ {0} \\gamma_ {u}} = \\mathbf {a} + \\frac {P}{m _ {0} \\gamma_ {u} c ^ {2}} \\mathbf {u} = \\mathbf {a} + \\gamma_ {u} ^ {2} \\frac {\\mathbf {u} \\cdot \\mathbf {a}}{c ^ {2}} \\mathbf {u}.\n$$", "source": "mit-ocw", "source_doc_id": "0005-relativistic-dynamics-handout-bc253f0e-b3138e4c", "source_title": "Topics: Dynamics toolbox: formula summary", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 5, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Topics: Dynamics toolbox: formula summary", "Page 6"], "page_start": 6, "page_end": 6, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0005-relativistic-dynamics-handout-bc253f0e-b3138e4c:page-0007", "text": "# Transformation of force\n\n- Let's compute the transformation law for force by transforming to a frame  $S'$ moving with velocity  $v$ in the  $x$-direction relative to  $S$:\n\n$$\n\\begin{array}{l} \\mathbb {F} ^ {\\prime} = \\gamma_ {u ^ {\\prime}} \\left( \\begin{array}{c} F _ {x} ^ {\\prime} \\\\ F _ {y} ^ {\\prime} \\\\ F _ {z} ^ {\\prime} \\\\ P ^ {\\prime} / c \\end{array} \\right) = \\boldsymbol {\\Lambda} \\mathbb {F} = \\gamma_ {u} \\left( \\begin{array}{c c c c} \\gamma & 0 & 0 & - \\gamma \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ - \\gamma \\beta & 0 & 0 & \\gamma \\end{array} \\right) \\left( \\begin{array}{c} F _ {x} \\\\ F _ {y} \\\\ F _ {z} \\\\ P / c \\end{array} \\right) \\\\ = \\gamma_ {u} \\left( \\begin{array}{c} \\gamma [ F _ {x} - \\beta P / c ] \\\\ F _ {y} \\\\ F _ {z} \\\\ \\gamma [ P / c - \\beta F _ {x} ] \\end{array} \\right) = \\frac {\\gamma_ {u ^ {\\prime}}}{\\gamma (1 - \\frac {u _ {x} v}{c ^ {2}})} \\left( \\begin{array}{c} \\gamma [ F _ {x} - \\beta P / c ] \\\\ F _ {y} \\\\ F _ {z} \\\\ \\gamma [ P / c - \\beta F _ {x} ] \\end{array} \\right). \\\\ \\end{array}\n$$\n\nIn the last step, we used the relation   $\\gamma_{u'} = \\gamma_u \\gamma [1 - u_x v/c^2]$  which we proved earlier when transforming the velocity 4-vector U — it followed from the fact that its normalization is Lorentz invariant, i.e.,   $U'^t \\eta U' = U^t \\eta U$ .\n\n- The 4 components now give our desired force transformation equations:\n\n$$\nF _ {x} ^ {\\prime} = \\frac {F _ {x} - \\frac {v}{c ^ {2}} P}{1 - \\frac {u _ {x} v}{c ^ {2}}},\n$$\n\n$$\n{F _ {y} ^ {\\prime}} = {\\frac {F _ {y}}{\\gamma \\left(1 - \\frac {u _ {x} v}{c ^ {2}}\\right)},}\n$$\n\n$$\nF _ {z} ^ {\\prime} = \\frac {F _ {z}}{\\gamma \\left(1 - \\frac {u _ {x} v}{c ^ {2}}\\right)},\n$$\n\n$$\nP ^ {\\prime} = \\frac {P - v F _ {x}}{1 - \\frac {u _ {x} v}{c ^ {2}}},\n$$\n\nwhere  $P = \\mathbf{u}\\cdot \\mathbf{F}$ as usual.\n\n- If we take  $S$ to be the rest frame of the particle, then  $\\mathbf{u} = 0$,  $P = \\mathbf{u} \\cdot \\mathbf{F} = 0$ and this simplifies to  $F_x' = F_x$,  $F_y' = F_y / \\gamma$,  $F_z' = F_z / \\gamma$, so in the frame  $S'$ where the particle is moving, the force is unaffected in the parallel direction and suppressed by  $\\gamma$ in the transverse directions.", "source": "mit-ocw", "source_doc_id": "0005-relativistic-dynamics-handout-bc253f0e-b3138e4c", "source_title": "Topics: Dynamics toolbox: formula summary", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 6, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Topics: Dynamics toolbox: formula summary", "Page 7"], "page_start": 7, "page_end": 7, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0005-relativistic-dynamics-handout-bc253f0e-b3138e4c:page-0008", "text": "# Transformation of acceleration\n\n- We could derive expressions using an approach like for force, but the results are so messy that it's not particularly useful — it's better to deal with explicit problems as needed.\n- Here's a useful special case that you get to derive on a problem set (probably PS7): For an arbitrary acceleration  $\\mathbf{a}$ in  $S$, the acceleration  $\\mathbf{a}'$ in  $S'$ is related to  $\\mathbf{a}$ via\n\n$$\n{a _ {x}} = {\\frac {a _ {x} ^ {\\prime}}{\\gamma^ {3} (1 + v u _ {x} ^ {\\prime} / c ^ {2}) ^ {3}}}\n$$\n\n$$\n{a _ {y}} = {\\frac {a _ {y} ^ {\\prime}}{\\gamma^ {2} (1 + v u _ {x} ^ {\\prime} / c ^ {2}) ^ {2}},}\n$$\n\nwith the important caveat that the expression for   $a_{y}$  is only valid for the case where either   $u_{y}^{\\prime}=0$  or   $a_{x}^{\\prime}=0$ .", "source": "mit-ocw", "source_doc_id": "0005-relativistic-dynamics-handout-bc253f0e-b3138e4c", "source_title": "Topics: Dynamics toolbox: formula summary", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 7, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Topics: Dynamics toolbox: formula summary", "Page 8"], "page_start": 8, "page_end": 8, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0006-particle-physics-handout-7d6df712-1d7aa402:page-0001", "text": "# MIT Course 8.033, Fall 2005, Particle physics\n\n# Max Tegmark\n\n# Last revised October 22 2006\n\n# Topics\n\n- Important particles\n- Nuclear physics terminology\n- Rest mass & binding energy\n- Photons\n- Particle physics processes\n- Examples: photon emission & absorption\n- Example: Compton scattering", "source": "mit-ocw", "source_doc_id": "0006-particle-physics-handout-7d6df712-1d7aa402", "source_title": "MIT Course 8.033, Fall 2005, Particle physics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Particle physics: Max Tegmark", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0006-particle-physics-handout-7d6df712-1d7aa402:page-0002", "text": "# Rest energies of important particles\n\n| Particle | Symbol | Rest energy |\n| --- | --- | --- |\n| electron | $e^{-}$ | 0.511 MeV |\n| muon | $\\mu^{-}$ | 105.6 MeV |\n| tau | $\\tau^{-}$ | 1777 MeV |\n| proton | $p^{+}$ | 938.26 MeV |\n| neutron | $n$ | 939.55 MeV |\n| charged pion | $\\pi^{+}, \\pi^{-}$ | 139.6 MeV |\n| neutral pion | $\\pi^{0}$ | 135.0 MeV |\n| neutrinos | $\\nu_{e}, \\nu_{\\mu}, \\mu_{\\tau}$ | < 0.14 eV |\n| photon | $\\gamma$ | 0 MeV |\n| graviton | $g$ | 0 MeV |\n| C $^{12}$ /12 | amu | 931.5 MeV |\n\n- For more, see particle physics handouts.\n- Open question: why?\n\nWhy is proton/electron mass ratio 1836, say?\n\n# Nuclear physics terminology\n\n- The atomic number  $Z$ of a nucleus is its number of protons.\n- The atomic weight  $A$ of a nucleus is its number of nucleons (protons + neutrons).\n- $Z$ determines the name of the element (its order in the periodic table).\n- Nuclei with same  $Z$ and different  $A$ are said to be different isotopes of the same element.\n- Notation example:  $\\mathrm{Fe}^{56}$ means  $Z = 26$ (iron) and  $A = 56$.\n- The mass excess for a nucleus is  $m_0 - A$ amu, i.e., its rest mass minus the number of nucleons times amu.\n- By this definition, the mass excess of  $\\mathrm{C}^{12}$ is zero.\n- Historically (before people knew exactly what they were), Helium nuclei, electrons and energetic photons were called   $\\alpha$ -particles,   $\\beta$ -particles and   $\\gamma$ -particles, respectively, and linguistic vestiges of this live on:\n\n- The process  $n \\to p^{+} + e^{-} + \\bar{\\nu}$ is called  $\\beta$-decay.\n\n- High energy photons are denoted  $\\gamma$-rays, and photons are denoted  $\\gamma$ (which is of course confusing in 8.033)!", "source": "mit-ocw", "source_doc_id": "0006-particle-physics-handout-7d6df712-1d7aa402", "source_title": "MIT Course 8.033, Fall 2005, Particle physics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Particle physics: Max Tegmark", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0006-particle-physics-handout-7d6df712-1d7aa402:page-0003", "text": "# Rest mass and binding energy\n\n- The rest energy of an object is its energy in the frame where it has zero momentum.\n\n- This rest energy is the sum of all energy contributions, both positive (like rest masses and kinetic energies of its constituent particles) and negative (like potential energy from force holding constituents together).\n\n- The binding energy of a nucleus is rest energy of its neutrons and protons free minus rest energy of the nucleus.\n\n- Electric repulsion between protons increases mass of nucleus.\n\n- Attraction between nucleons (strong force) decreases mass of nucleus.\n\n- Only nuclei whose  $(Z, A)$ give positive binding energy can exist\n\n- Semi-empirical relationship (von Weizsäcker 1935):\n\n$$\n\\frac {E _ {\\mathrm{binding}}}{c ^ {2}} \\approx \\left[ 1 5. 8 A - 1 8. 3 A ^ {2 / 3} - 0. 7 1 4 \\frac {Z ^ {2}}{A ^ {1 / 3}} - 2 3. 2 \\frac {(A - 2 Z) ^ {2}}{A} + (- 1) ^ {Z} \\frac {1 2}{A ^ {1 / 2}} \\right] \\mathrm{MeV}\n$$\n\nThe last term is omitted if  $A$ is an odd number.\n\n- Much work remains to be done in this field!\n\n# Photoelectric effect\n\n- Einstein's model was that\n\n- the photon carries energy  $h\\nu$\n- a certain work  $W_{e}$ is required to liberate an electron from the metal\n\n- This explained both of Lenard's 1902 observations:\n\n- light with frequency  $h\\nu < W_e$ liberates no electrons at all\n- light with frequency  $h\\nu > W_e$ liberates electrons with kinetic energy  $h\\nu - W_e$.\n- increasing the intensity of the light (the photon flux) didn't affect the existence of liberated electrons or their kinetic energy\n\n- Bottom line: we can treat the photon as just another particle.", "source": "mit-ocw", "source_doc_id": "0006-particle-physics-handout-7d6df712-1d7aa402", "source_title": "MIT Course 8.033, Fall 2005, Particle physics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Particle physics: Max Tegmark", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0006-particle-physics-handout-7d6df712-1d7aa402:page-0004", "text": "# Working with photons:\n\n- Photon 4-vector:\n\n$$\n\\mathbf {P} = \\hbar \\binom{\\mathbf {k}}{k},\n$$\n\nwhere   $k = \\omega/c$ .\n\n- So  $p = E/c$ for photons.\n\n• Comparing P with the wave 4-vector K shows that\n\n$$\n\\mathbf {P} = \\hbar \\mathbf {K}.\n$$\n\nThis relation in fact holds for all particles, even massive ones — as you'll see when you get to wave-particle duality in quantum mechanics. If you take a field theory course, you'll see this pop right out of the so-called Klein-Gordon equation.\n\n- Doppler effect is just special case of P-transformation for zero rest mass — show on PS6.", "source": "mit-ocw", "source_doc_id": "0006-particle-physics-handout-7d6df712-1d7aa402", "source_title": "MIT Course 8.033, Fall 2005, Particle physics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Particle physics: Max Tegmark", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0006-particle-physics-handout-7d6df712-1d7aa402:page-0005", "text": "# Particle physics processes\n\n- We know of four fundamental interactions: gravitational, electromagnetic, weak and strong. In particle physics, the first is negligible.\n- See the handouts for summaries of particles and interactions.\n- Summary of particle physics processes we consider:\n\n- Absorption (two particles in, one out)\n- Emission/decay (one particle in, two out)\n- Collision/scattering/annihilation/creation (two particles in, two out)\n\n- Footnote: if you take a course in quantum field theory, you'll find that two in, two out (\"four-vertex\") interactions can generally be reduced to two separate three-vertex interactions, where the momentum and energy transfer between the two colliding particles is mediated by an intermediate particle. For instance, an elastic collision between two electrons can be reduced to a photon exchange: one electron emits a photon that's later absorbed by the other.\n\n- Which processes are allowed in nature? All that aren't forbidden by a conservation law, e.g.,\n\n- Energy-momentum conservation (P conserved)\n- Charge conservation\n- Baryon number conservation\n- Lepton number conservation\n- Parity conservation (except in weak interactions)\n\n- Everything is provisional:\n\n- Momentum conservation appeared to be violated in  $\\beta$-decay, but was rescued with neutrino discovery (proposed by Wolfgang Pauli 1931, detected by Fred Reines & Clyde Cowan 1956).\n- Parity conservation was believed to be universally valid until the shock of 1956 (Yang, Lee, Wu).\n- Many physicists believe (but haven't shown) that lepton and/or baryon number is violated ever so slightly, e.g., that protons decay if you wait  $\\gg 10^{32}$ years.\n\n- There's more to it: computing lifetimes and scattering probabilities requires quantum field theory - in this course, we'll limit ourselves to drawing conclusions from energy-momentum conservation.", "source": "mit-ocw", "source_doc_id": "0006-particle-physics-handout-7d6df712-1d7aa402", "source_title": "MIT Course 8.033, Fall 2005, Particle physics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Particle physics: Max Tegmark", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0006-particle-physics-handout-7d6df712-1d7aa402:page-0006", "text": "# Common interaction processes\n\n- Chemical reactions: atoms get rearranged in new ways, perhaps emitting or absorbing photons and electrons. Non-relativistic.\n- Nuclear reactions: nucleons get rearranged in new ways, perhaps emitting or absorbing photons, electrons, positrons and neutrinos (electron/positrons and neutrinos must be involved whenever there are conversions between protons and neutrons, to conserve charge and lepton number).\n- Elementary particle interactions: energy, momentum, charge, lepton number etc. gets rearranged in new ways, corresponding to scattering, destruction and creation of particles.\n\n# Examples:\n\n- Molecule + molecule → new molecules + γ (chemical reaction)\n- $\\gamma + \\text{atom} \\rightarrow \\text{exited atom (excitation)}$\n- $\\gamma + \\text{atom} \\rightarrow e^{-} + \\text{atom (ionization; photoelectric effect)}$\n- Nucleus + nucleus → new nuclei +  $\\gamma / e^{-} / \\nu$ (nuclear reaction)\n- $n \\to p^{+} + e^{-} + \\bar{\\nu}$ (beta decay)\n- $\\gamma + \\gamma \\rightarrow e^{-} + e^{+}$ (pair creation)\n- $\\gamma + \\text{particle} \\rightarrow \\text{particle} + e^{-} + e^{+}$\n- $\\gamma + e^{-} \\rightarrow \\gamma + e^{-}$ (Compton scattering)", "source": "mit-ocw", "source_doc_id": "0006-particle-physics-handout-7d6df712-1d7aa402", "source_title": "MIT Course 8.033, Fall 2005, Particle physics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 5, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Particle physics: Max Tegmark", "Page 6"], "page_start": 6, "page_end": 6, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0006-particle-physics-handout-7d6df712-1d7aa402:page-0007", "text": "# Photon emission & absorption:\n\n- Photon absorption  $(X + \\gamma \\rightarrow X^{*})$: If a particle at rest with mass  $m_0$ absorbs a photon of frequency  $\\omega$, it acquires a speed\n\n$$\n\\beta^ {\\prime} = \\frac {\\hbar \\omega}{m _ {0} c ^ {2} + \\hbar \\omega}.\n$$\n\n- Photon emission with recoil   $(X^{*} \\rightarrow X + \\gamma)$ : if a particle emits energy   $\\hbar\\omega$  as a photon, thereby reducing its rest mass from   $m_{0}$  to   $m_{0}^{\\prime} \\equiv m_{0} - Q_{0}/c^{2}$ , then\n\n$$\n\\hbar \\omega = \\left(1 - \\frac {Q _ {0}}{2 m _ {0} c ^ {2}}\\right) Q _ {0}.\n$$\n\nThus the photon energy   $\\hbar\\omega < Q_{0}$  because of recoil, whereby some of the released energy   $Q_{0}$  turns into kinetic energy of the recoiling particle.\n\n- This works in reverse too: to increase its rest energy by   $Q_{0}$ , the particle needs to absorb a photon with energy   $\\hbar\\omega > Q_{0}$  to compensate for the recoil.\n- This recoil effect (the term   $Q_{0}/2m_{0}c^{2}$  in the parenthesis above) is normally negligibly small   $\\sim10^{-8}$  for typical atomic transition energies (  $\\sim10eV$ ) — for comparison, Doppler line broadening is of order   $\\beta\\sim10^{-6}$  for room temperature atoms moving with thermal velocities of hundreds of meters per second.\n- However, it is important for nuclear transition energies, which are of order a thousand times larger (Moon's experiment 1951).\n- Mössbauer effect (1961 Nobel Prize for Ph.D. thesis work) all but eliminates recoil, making  $m_0$ the rest mass of the whole crystal rather than one particle. Allows measuring 2cm/s Doppler shifts!\n- Pound & Rebka experiment from Harvard Tower 1960 used this to detect tiny  $\\sim 10^{-14}$ gravitational redshift.", "source": "mit-ocw", "source_doc_id": "0006-particle-physics-handout-7d6df712-1d7aa402", "source_title": "MIT Course 8.033, Fall 2005, Particle physics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 6, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Particle physics: Max Tegmark", "Page 7"], "page_start": 7, "page_end": 7, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0006-particle-physics-handout-7d6df712-1d7aa402:page-0008", "text": "# Compton scattering\n\n- Compton scattering  $(\\gamma + e^{-} \\rightarrow \\gamma + e^{-})$ with electron initially at rest:\n\n$$\nh \\nu^ {\\prime} = \\frac {h \\nu}{1 + \\frac {h \\nu}{m _ {e} c ^ {2}} (1 - \\cos \\theta)}\n$$\n\n- Such an elastic photon-electron collision is called Compton scattering when the photon transfers energy to the electron and inverse Compton scattering when the electron transfers energy to the photon.\n- The former occurs when shining x-rays at matter.\n- The latter occurs frequently in astrophysics.\n- Them two are of course equivalent in special relativity, since you can always Lorentz transform into a frame where, before the collision, either the electron has much more energy than the photon or vice versa.\n\n| Property | Independent of velocity? |  |\n| --- | --- | --- |\n|  | Classically? | Relativistically? |\n| Charge q | Y | Y |\n| Spin | Y | Y |\n| Lepton number | Y | Y |\n| Duration Δt | Y | N |\n| Length L | Y | N |\n| Mass m | Y | N |\n| Proper duration Δτ | Y | Y |\n| Proper length L0 | Y | Y |\n| Rest mass m0 | Y | Y |\n| Momentum p | N | N |\n| Energy E | N | N |", "source": "mit-ocw", "source_doc_id": "0006-particle-physics-handout-7d6df712-1d7aa402", "source_title": "MIT Course 8.033, Fall 2005, Particle physics: Max Tegmark", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 7, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2005, Particle physics: Max Tegmark", "Page 8"], "page_start": 8, "page_end": 8, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-electromagnetism-handout-30b7ee57-99b26a7c:page-0001", "text": "# Key formula summary\n\n- Lorentz force law:\n\n$$\n\\mathbf {F} = q (\\mathbf {E} + \\frac {1}{c} \\mathbf {u} \\times \\mathbf {B})\n$$\n\n- Lorentz transforming the electromagnetic field:\n\n$$\nE _ {x} ^ {\\prime} = E _ {x}\n$$\n\n$$\nE _ {y} ^ {\\prime} = \\gamma (E _ {y} - \\beta B _ {z})\n$$\n\n$$\nE _ {z} ^ {\\prime} = \\gamma (E _ {z} + \\beta B _ {y})\n$$\n\n$$\nB _ {x} ^ {\\prime} = B _ {x}\n$$\n\n$$\nB _ {y} ^ {\\prime} = \\gamma (B _ {y} + \\beta E _ {z})\n$$\n\n$$\nB _ {z} ^ {\\prime} = \\gamma (B _ {z} - \\beta E _ {y}).\n$$\n\n- Current 4-vector:\n\n$$\n\\mathbb {J} \\equiv \\left( \\begin{array}{c} J _ {x} \\\\ J _ {y} \\\\ J _ {z} \\\\ \\rho c \\end{array} \\right) = \\rho_ {0} \\mathbf {U},\n$$\n\nwhere the proper charge density   $\\rho_{0}$  is the local charge density in a frame where J = 0.\n\n- Electric field from stationary charge  $q$ (Coulomb's law):\n\n$$\n\\mathbf {E} = \\frac {q}{r ^ {2}} \\hat {\\mathbf {r}} = \\frac {q}{\\mathbf {x} ^ {2} + y ^ {2} + z ^ {2}} \\hat {\\mathbf {r}}\n$$\n\n- Electric field from charge  $q$ moving in  $x$-direction:\n\n$$\n\\mathbf {E} ^ {\\prime} = \\frac {\\gamma q r ^ {\\prime}}{\\left(\\gamma^ {2} x ^ {\\prime 2} + y ^ {\\prime 2} + z ^ {\\prime 2}\\right) ^ {3 / 2}} \\hat {\\mathbf {r}} ^ {\\prime}\n$$", "source": "mit-ocw", "source_doc_id": "0007-electromagnetism-handout-30b7ee57-99b26a7c", "source_title": "Key formula summary: How relativity and electricity implies magnetism", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Key formula summary: How relativity and electricity implies magnetism", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-electromagnetism-handout-30b7ee57-99b26a7c:page-0002", "text": "# How relativity and electricity implies magnetism\n\n- We know that the force  $\\mathbf{F}$ on charged particle of charge  $q$ in an electric field  $\\mathbf{E}$ is\n\n$$\n\\mathbf {F} = q \\mathbf {E},\n$$\n\nindependent of the velocity u of the particle.\n\n- We can rewrite this equation in a mathematically equivalent way using 4-vectors:\n\n$$\n\\mathbb {F} = \\frac {q}{c} \\mathbf {M U}, \\tag {1}\n$$\n\nwhere F is the force 4-vector, U is the velocity 4-vector and M is the   $4 \\times 4$  matrix\n\n$$\n\\mathbf {M} = \\left( \\begin{array}{c c c c} 0 & 0 & 0 & E _ {x} \\\\ 0 & 0 & 0 & E _ {y} \\\\ 0 & 0 & 0 & E _ {z} \\\\ E _ {x} & E _ {y} & E _ {z} & 0 \\end{array} \\right).\n$$\n\nThe 4th component of this equation reads   $P = qE \\cdot u$ , so   $P = F \\cdot u$  as should be.\n\n- Let's Lorentz transform to a frame  $S'$ moving with velocity  $v = \\beta c$ in the  $x$-direction:\n\n$$\n\\mathbb {F} ^ {\\prime} \\equiv \\boldsymbol {\\Lambda} \\mathbb {F} = \\frac {q}{c} \\boldsymbol {\\Lambda} \\mathbf {M} \\mathbf {U} = \\frac {q}{c} \\boldsymbol {\\Lambda} \\mathbf {M} \\boldsymbol {\\Lambda} ^ {- 1} \\boldsymbol {\\Lambda} \\mathbf {U} = \\frac {q}{c} \\mathbf {M} ^ {\\prime} \\mathbf {U} ^ {\\prime},\n$$\n\nwhere the transformed matrix is\n\n$$\n\\mathbf {M} ^ {\\prime} \\equiv \\boldsymbol {\\Lambda} \\mathbf {M} \\boldsymbol {\\Lambda} ^ {- 1}. \\tag {2}\n$$\n\n- Plugging in our M-matrix above, this gives\n\n$$\n\\left( \\begin{array}{c c c c} 0 & - \\beta \\gamma E _ {y} & - \\beta \\gamma E _ {z} & E _ {x} \\\\ \\beta \\gamma E _ {y} & 0 & 0 & \\gamma E _ {y} \\\\ \\beta \\gamma E _ {z} & 0 & 0 & \\gamma E _ {z} \\\\ E _ {x} & \\gamma E _ {y} & \\gamma E _ {z} & 0 \\end{array} \\right).\n$$\n\n- This shows two things. First we see that, hardly surprisingly by now, the E-field is picks up some   $\\gamma$ -factors — specifically,   $E_{x}^{\\prime}=E_{x}$  whereas   $E_{y}^{\\prime}=\\gamma E_{y}$  and   $E_{z}^{\\prime}=\\gamma E_{z}$ . Second, we see that new terms appear in the matrix that don't correspond to an E-field! The component   $M_{23}$  would also become non-zero if we transformed to a frame moving in a different direction. So to be able to describe the general case, we need to introduce more field components in M. It's easy to show that the upper left   $3\\times3$  block of the matrix is antisymmetric regardless of how we Lorentz transform, i.e.,   $M_{11}=M_{22}=M_{33}=0$  and   $M_{32}=-M_{23}$ ,   $M_{13}=-M_{31}$ ,   $M_{21}=-M_{12}$ , so we simply need to keep track of the three quantities   $M_{23}$ ,   $M_{31}$  and   $M_{12}$ . We could denote these three numbers by whatever symbols", "source": "mit-ocw", "source_doc_id": "0007-electromagnetism-handout-30b7ee57-99b26a7c", "source_title": "Key formula summary: How relativity and electricity implies magnetism", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Key formula summary: How relativity and electricity implies magnetism", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-electromagnetism-handout-30b7ee57-99b26a7c:page-0003", "text": "we want — let's call them   $B_{x}$ ,   $B_{y}$  and   $B_{z}$ . This means that, by definition, the M-matrix takes the form\n\n$$\n\\mathbf {M} \\equiv \\left( \\begin{array}{c c c c} 0 & B _ {z} & - B _ {y} & E _ {x} \\\\ - B _ {z} & 0 & B _ {x} & E _ {y} \\\\ B _ {y} & - B _ {x} & 0 & E _ {z} \\\\ E _ {x} & E _ {y} & E _ {z} & 0 \\end{array} \\right).\n$$\n\n- Plugging this back into equation (1) now gives\n\n$$\n\\mathbf {F} = q (\\mathbf {E} + \\frac {1}{c} \\mathbf {u} \\times \\mathbf {B}), \\tag {3}\n$$\n\nfor the first three components, i.e., the famous Lorentz force law from 8.02! The 4th component gives   $P = qE \\cdot u = F \\cdot u$  as should be.\n\n- For simplicity, we've used c.g.s. units here, where  $B$ has the same units as  $E$. To switch to m.k.s. units, replace  $\\mathbf{B}$ by  $c\\mathbf{B}$.\n- In conclusion, starting with a pure electric field in  $S$, we found that in  $S'$, the force on our particle will also depend on its velocity according to equation (3), i.e., there is a magnetic field!", "source": "mit-ocw", "source_doc_id": "0007-electromagnetism-handout-30b7ee57-99b26a7c", "source_title": "Key formula summary: How relativity and electricity implies magnetism", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Key formula summary: How relativity and electricity implies magnetism", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-electromagnetism-handout-30b7ee57-99b26a7c:page-0004", "text": "# Transforming the electromagnetic field\n\n- Having figured out that these three new components correspond to a B-field, let us now use equation (2) to derive the transformation properties of an arbitrary electromagnetic field:\n\n$$\n\\begin{array}{l} \\left( \\begin{array}{c c c c} 0 & B _ {z} ^ {\\prime} & - B _ {y} ^ {\\prime} & E _ {x} ^ {\\prime} \\\\ - B _ {z} ^ {\\prime} & 0 & B _ {x} ^ {\\prime} & E _ {y} ^ {\\prime} \\\\ B _ {y} ^ {\\prime} & - B _ {x} ^ {\\prime} & 0 & E _ {z} ^ {\\prime} \\\\ E _ {x} ^ {\\prime} & E _ {y} ^ {\\prime} & E _ {z} ^ {\\prime} & 0 \\end{array} \\right) = \\\\ = \\boldsymbol {\\Lambda} \\left( \\begin{array}{c c c c} 0 & B _ {z} & - B _ {y} & E _ {x} \\\\ - B _ {z} & 0 & B _ {x} & E _ {y} \\\\ B _ {y} & - B _ {x} & 0 & E _ {z} \\\\ E _ {x} & E _ {y} & E _ {z} & 0 \\end{array} \\right) \\boldsymbol {\\Lambda} ^ {- 1} = \\\\ = \\left( \\begin{array}{c c c c} \\gamma & 0 & 0 & - \\gamma \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ - \\gamma \\beta & 0 & 0 & \\gamma \\end{array} \\right) \\left( \\begin{array}{c c c c} 0 & B _ {z} & - B _ {y} & E _ {x} \\\\ - B _ {z} & 0 & B _ {x} & E _ {y} \\\\ B _ {y} & - B _ {x} & 0 & E _ {z} \\\\ E _ {x} & E _ {y} & E _ {z} & 0 \\end{array} \\right) \\left( \\begin{array}{c c c c} \\gamma & 0 & 0 & \\gamma \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ \\gamma \\beta & 0 & 0 & \\gamma \\end{array} \\right) \\\\ = \\left( \\begin{array}{c c c c} 0 & \\gamma (B _ {z} - \\beta E _ {y}) & - \\gamma (B _ {y} + \\beta E _ {z}) & E _ {x} \\\\ - \\gamma (B _ {z} - \\beta E _ {y}) & 0 & B _ {x} & \\gamma (E _ {y} - \\beta B _ {z}) \\\\ \\gamma (B _ {y} + \\beta E _ {z}) & - B _ {x} & 0 & \\gamma (E _ {z} + \\beta B _ {y}) \\\\ E _ {x} & \\gamma (E _ {y} - \\beta B _ {z}) & \\gamma (E _ {z} + \\beta B _ {y}) & 0 \\end{array} \\right), \\\\ \\end{array}\n$$\n\nSO\n\n$$\n\\begin{array}{l} E _ {x} ^ {\\prime} = E _ {x} \\\\ {E _ {y} ^ {\\prime}} = {\\gamma (E _ {y} - \\beta B _ {z})} \\\\ E _ {z} ^ {\\prime} = \\gamma (E _ {z} + \\beta B _ {y}) \\\\ B _ {x} ^ {\\prime} = B _ {x} \\\\ B _ {y} ^ {\\prime} = \\gamma (B _ {y} + \\beta E _ {z}) \\\\ {B _ {z} ^ {\\prime}} = {\\gamma (B _ {z} - \\beta E _ {y}).} \\\\ \\end{array}\n$$\n\n- In case you're familiar with tensor notation, the Einstein summation convention and raising/ lowering indices, the matrix denoted  $\\mathbf{M}$ above is the electromagnetic field tensor  $F_{\\mu}^{\\nu}$ which, unlike  $F_{\\mu \\nu} \\equiv A_{\\nu, \\mu} - A_{\\mu, \\nu}$, is not antisymmetric.", "source": "mit-ocw", "source_doc_id": "0007-electromagnetism-handout-30b7ee57-99b26a7c", "source_title": "Key formula summary: How relativity and electricity implies magnetism", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Key formula summary: How relativity and electricity implies magnetism", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-electromagnetism-handout-30b7ee57-99b26a7c:page-0005", "text": "# Transforming charge and current densities\n\n- The theory of electromagnetism consists of two parts: how matter affects fields and how fields affect matter. Above we studied the latter — let us now study the former.\n- Analogy: the theory of gravity consists of two parts: how matter affects fields (the gravitational field) and how fields affect matter. In general relativity, the role of the gravitational field is played by the metric, and we will find that both parts of the theory get a geometric interpretation: the former that matter moves along geodesics through spacetime and the latter that matter curves spacetime.\n- The source of electromagnetic fields is matter carrying electric charge, characterized at each spacetime eventy by a charge density   $\\rho(\\mathbf{r},t)$  and a current density   $\\mathbf{J}(\\mathbf{r},t)$ .\n- These can be combined into the current 4-vector (or “4-current”)\n\n$$\n\\mathbb {J} \\equiv \\left( \\begin{array}{c} J _ {x} \\\\ J _ {y} \\\\ J _ {z} \\\\ \\rho c \\end{array} \\right).\n$$\n\nFor a blob of charge of uniform density   $\\rho_{0}$  in its rest frame that moves with velocity 4-vector U, the 4-current is simply\n\n$$\n\\mathbb {J} \\equiv \\rho_ {0} \\mathbf {U},\n$$\n\nand the total 4-current from many sources (say electrons and ions moving in opposite directions) is simply the sum of all the individual 4-currents.\n\n-  $\\rho_0$ is called the proper charge density.", "source": "mit-ocw", "source_doc_id": "0007-electromagnetism-handout-30b7ee57-99b26a7c", "source_title": "Key formula summary: How relativity and electricity implies magnetism", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Key formula summary: How relativity and electricity implies magnetism", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-electromagnetism-handout-30b7ee57-99b26a7c:page-0006", "text": "- The first part of the theory (how fields affect matter) is given by the Lorentz force law that we derived,\n\n$$\n\\mathbf {F} = q (\\mathbf {E} + \\frac {1}{c} \\mathbf {u} \\times \\mathbf {B}). \\tag {4}\n$$\n\n- The second part of the theory (how matter determines the fields) is given by Maxwell's equations:\n\n$$\n\\nabla \\cdot \\mathbf {E} = 4 \\pi \\rho , \\tag {5}\n$$\n\n$$\n\\nabla \\times \\mathbf {B} - \\frac {1}{c} \\dot {\\mathbf {E}} = \\frac {4 \\pi}{c} \\mathbf {J}, \\tag {6}\n$$\n\n$$\n\\nabla \\cdot \\mathbf {B} = 0, \\tag {7}\n$$\n\n$$\n\\nabla \\times \\mathbf {E} + \\frac {1}{c} \\dot {\\mathbf {B}} = \\mathbf {0}. \\tag {8}\n$$\n\n- We derived magnetism from electricity by assuming that the first part of the theory was Lorentz invariant. Let's now show that the second part is Lorentz invariant too, so that everything is consistent. We've already shown that the wave equation (which gives solutions to Maxwell's equations in vacuum, i.e., with  $\\mathbb{J} = \\mathbf{0}$) is Lorentz invariant, but we need to show more: that the full Maxwell equations are Lorentz invariant in general, even in the presence of charges and currents.\n- You won't be responsible for the material below in this course — I'm just presenting it here so that you can admire the full elegance of electromagnetism, which only becomes manifest in relativistic 4-vector notation.\n- A standard vector calculus result is that a vector field with no curl can be written as a gradient of some scalar field, say  $\\phi$, and a vector field with no divergence can be written as a curl of some vector field, A. Maxwell's last two equations above therefore imply that we can write\n\n$$\n\\mathbf {E} = - \\nabla \\phi - \\frac {1}{c} \\dot {\\mathbf {A}}, \\tag {9}\n$$\n\n$$\n\\mathbf {B} = \\nabla \\times \\mathbf {A}, \\tag {10}\n$$\n\nwhere   $\\phi$  and A are referred to as the scalar potential and the vector potential, respectively. Proof:   $\\nabla \\cdot B = 0$  gives   $B = \\nabla \\times A$ , after which the 4th Maxwell equation shows that   $\\nabla \\times \\left( E + \\frac{1}{c} \\dot{A} \\right) = 0$  so that we can write   $E + \\frac{1}{c} \\dot{A} = -\\nabla \\phi$ .\n\n• These are conveniently combined into a 4-vector\n\n$$\n\\mathbb {A} \\equiv \\left( \\begin{array}{c} A _ {x} \\\\ A _ {y} \\\\ A _ {z} \\\\ \\phi \\end{array} \\right).\n$$\n\n- The differential operator\n\n$$\n\\Box \\equiv \\left(\\frac {\\partial}{c \\partial t}\\right) ^ {2} - \\nabla^ {2}\n$$", "source": "mit-ocw", "source_doc_id": "0007-electromagnetism-handout-30b7ee57-99b26a7c", "source_title": "Key formula summary: How relativity and electricity implies magnetism", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 5, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Key formula summary: How relativity and electricity implies magnetism", "Page 6"], "page_start": 6, "page_end": 6, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-electromagnetism-handout-30b7ee57-99b26a7c:page-0007", "text": "is called the d'Alembertian, and is a spacetime generalization of the Laplace operator   $\\nabla^{2}$ . It is easy to show that it is Lorentz invariant.\n\n- In terms of this operator, the wave equation for some scalar field  $\\psi$ can be written in the extremely compact form\n\n$$\n\\Box \\psi = 0.\n$$\n\n- It turns out that there is some slop (known as gauge freedom) involved in the choices of  $\\phi$ and  $\\mathbf{A}$: for any scalar field  $\\psi$ satisfying the wave equation, you can replace  $\\mathbf{A}$ by  $\\mathbf{A} + \\nabla \\psi$ and  $\\phi$ by  $\\phi + \\frac{\\partial \\psi}{c\\partial t}$ without changing the fields  $\\mathbf{E}$ and  $\\mathbf{B}$. Without loss of generality, we can use this freedom to make  $\\mathbb{A}$ satisfy the so-called Lorentz gauge condition\n\n$$\n\\nabla \\cdot \\mathbf {A} - \\frac {1}{c} \\dot {\\phi} = 0. \\tag {11}\n$$\n\n- Plugging equations (9), (10) and (11) into Maxwell's first two equations and doing some vector algebra now gives the beautiful result\n\n$$\n\\Box \\mathbb {A} = - \\frac {4 \\pi}{c} \\mathbb {J}.\n$$\n\nWe have solved Maxwell's equations. This equation shows how matter determines the fields. For the special case  $\\mathbb{J} = \\mathbf{0}$, we see that it simply reduces to the wave equation  $\\square \\mathbb{A} = \\mathbf{0}$, i.e., each of the four components of  $\\mathbb{A}$ must separately satisfy the wave equation.\n\n- We set out to prove that the second half of the theory was Lorentz invariant. The last equation shows this explicitly, since  $\\square$ is Lorentz invariant and  $\\mathbb{A}$ and  $\\mathbb{J}$ are both 4-vectors.\n- Here's some doubly optional material in case you're interested. If you're familiar with the tensor notation, raising and lowering of indices and the Einstein summation convention (certainly not neccessary for this course!), here's an electromagnetism synopsis:\n\n$$\nA _ {, \\mu} ^ {\\mu} = 0 (\\mathrm{Lorentzgaugecondition}),\n$$\n\n$$\nJ ^ {\\mu} _ {, \\mu} = 0 (\\text { charge   conservation }),\n$$\n\n$$\nF _ {\\mu \\nu} = A _ {\\nu , \\mu} - A _ {\\mu , \\nu} (\\text { definition   of } F),\n$$\n\n$$\nF _ {\\mu} = F _ {\\mu} ^ {\\nu} U _ {\\nu} \\quad (\\text { how   fields   affect   matter }),\n$$\n\n$$\n\\Box A _ {\\mu} = - \\frac {4 \\pi}{c} J _ {\\mu} \\text {(how matter affects fields),}\n$$\n\n$$\nF ^ {\\mu \\nu} _ {, \\mu} = 4 \\pi J _ {\\nu} \\quad (\\text { Maxwell's   1st   2   equations }),\n$$\n\n$$\nF _ {\\mu \\nu , \\lambda} + F _ {\\nu \\lambda , \\mu} + F _ {\\lambda \\mu , \\nu} = 0 (\\text { Maxwell's   2nd   2   equations }).\n$$", "source": "mit-ocw", "source_doc_id": "0007-electromagnetism-handout-30b7ee57-99b26a7c", "source_title": "Key formula summary: How relativity and electricity implies magnetism", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 6, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Key formula summary: How relativity and electricity implies magnetism", "Page 7"], "page_start": 7, "page_end": 7, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-electromagnetism-handout-30b7ee57-99b26a7c:page-0008", "text": "# Retarded positions\n\n- As above, Maxwell's equations determine the field from arbitrary collections of moving charges from their 4-current density. To boost our intuition, let us look at the special case of a single charge.\n\n- The electric and magnetic fields from a stationary charge  $q$ at  $\\mathbf{r} = 0$ are (in c.g.s. units)\n\n$$\n{\\bf E} = q \\frac {\\hat {\\bf r}}{r ^ {2}} = q \\frac {\\bf r}{r ^ {3}},\n$$\n\n$$\n\\mathbf {B} = \\mathbf {0}.\n$$\n\n- What are the fields created by a charge moving with velocity  $v$ in the  $x$-direction? Since the last two equations give the answer in the rest frame  $S$ of the charge, all we need to do is Lorentz transform them into the frame  $S'$ where the charge is moving. Doing this gives the new electric field\n\n$$\nE _ {x} ^ {\\prime} = E _ {x} = \\frac {q}{r ^ {3}} x,\n$$\n\n$$\nE _ {y} ^ {\\prime} = \\gamma E _ {y} = \\frac {q}{r ^ {3}} \\gamma y,\n$$\n\n$$\nE _ {z} ^ {\\prime} = \\gamma E _ {z} = \\frac {q}{r ^ {3}} \\gamma z.\n$$\n\n- All that remains is to reexpress this result in terms of the new coordinates   $(x', y', z')$ . In   $S'$ , the charge is moving, so   $E'$  will depend on the new time   $t'$ . Let us calculate the field at the time   $t' = 0$  in   $S'$  (this is when the charge is at the origin of the frame   $S'$ ). At this instant,   $x = \\gamma x'$ , since more generally   $x = \\gamma x' + \\gamma vt'$ . Since   $y' = y$  and   $z' = z$  at all time, this gives\n\n$$\nE _ {x} ^ {\\prime} = \\frac {q}{r ^ {3}} \\gamma x ^ {\\prime},\n$$\n\n$$\n{E _ {y} ^ {\\prime}} = {\\frac {q}{r ^ {3}} \\gamma y ^ {\\prime},}\n$$\n\n$$\n{E _ {z} ^ {\\prime}} = {\\frac {q}{r ^ {3}} \\gamma z ^ {\\prime},}\n$$\n\ni.e.\n\n$$\n\\mathbf {E} ^ {\\prime} = \\gamma \\frac {q}{r ^ {3}} \\mathbf {r} ^ {\\prime} = \\frac {\\gamma q}{\\left(\\gamma^ {2} x ^ {\\prime 2} + y ^ {\\prime 2} + z ^ {\\prime 2}\\right) ^ {3 / 2}} \\mathbf {r} ^ {\\prime} = \\gamma q \\frac {\\left(x ^ {\\prime 2} + y ^ {\\prime 2} + z ^ {\\prime 2}\\right) ^ {1 / 2}}{\\left(\\gamma^ {2} x ^ {\\prime 2} + y ^ {\\prime 2} + z ^ {\\prime 2}\\right) ^ {3 / 2}} \\hat {\\mathbf {r}} ^ {\\prime}\n$$\n\n- Conclusion 1: The magnitude  $E = |\\mathbf{E}|$ of the field becomes anisotropic: decreased by a factor  $\\gamma^2$ in front of and behind the moving charge and increased by a factor  $\\gamma$ in the perpendicular direction.\n\n- Conclusion 2: The direction of the field (  $\\hat{r}'$ ) still points straight away from the instantaneous position of the charge. This is remarkable, since the electromagnetic field can only propagate at the speed of light, so the charge must have caused this field at a time when it was in a different position (the so-called retarded position). Sure enough, this remarkable property no longer holds if the charge accelerates, leading to the Abraham-Lorentz force fiasco.", "source": "mit-ocw", "source_doc_id": "0007-electromagnetism-handout-30b7ee57-99b26a7c", "source_title": "Key formula summary: How relativity and electricity implies magnetism", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 7, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Key formula summary: How relativity and electricity implies magnetism", "Page 8"], "page_start": 8, "page_end": 8, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-electromagnetism-handout-30b7ee57-99b26a7c:page-0009", "text": "# A fly in the ointment\n\n- Finally, you should know that as beautiful as it is, this whole theory has a lethal flaw (resolved by quantum mechanics). There is an instability whereby the electric field created by an accelerating electron acts back on the the electron causing it to undergo runaway linear acceleration! Aside from the fact that  $E$ is formally infinite at the location of a point charge (a problem avoided if the electron somehow were to have a finite size), one might think that there should be no net force by symmetry. This is clearly the case for a stationary charge and, as we saw on the previous page, it is even true for a charge in uniform motion since all field lines point towards the current rather than retarded position. However, it breaks down for accelerated motion: in c.g.s. units and for speeds  $v \\ll c$, this so-called Abraham-Lorentz force on a particle of charge  $q$ is\n\n$$\n\\mathbf {F} = \\frac {2}{3} \\frac {q ^ {2}}{c ^ {3}} \\dot {\\mathbf {a}},\n$$\n\ni.e., it depends on the time-derivative of the acceleration, the third derivative of the position with respect to time. For   $v \\ll c$  we have F = ma and hence\n\n$$\n\\dot {\\mathbf {a}} = \\omega \\mathbf {a},\n$$\n\nwhere the frequency\n\n$$\n\\omega = \\frac {3}{2} \\frac {c ^ {3} m}{q ^ {2}} \\approx (6. 2 6 6 \\times 1 0 ^ {- 2 4} s) ^ {- 1}\n$$\n\nfor the case of an electron. The solution to this equation is\n\n$$\n\\mathbf {a} = \\mathbf {a} _ {0} e ^ {\\omega t}, \\quad \\mathbf {u} = \\mathbf {u} _ {0} + \\frac {1}{\\omega} \\mathbf {a} _ {0} e ^ {\\omega t},\n$$\n\ni.e., the electron will all on its own increase its velocity exponentially, doubling on a timescale around   $10^{-23}$  seconds, in stark contrast to what we actually observe!\n\n- Note: This problem is alleviated in quantum field theory, but remains important — it is intimately linked with the so-called self-energy of the electron and the issue known as renormalization.", "source": "mit-ocw", "source_doc_id": "0007-electromagnetism-handout-30b7ee57-99b26a7c", "source_title": "Key formula summary: How relativity and electricity implies magnetism", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 8, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Key formula summary: How relativity and electricity implies magnetism", "Page 9"], "page_start": 9, "page_end": 9, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0008-general-relativity-handout-35307630-39b76ca4:page-0001", "text": "# Topics\n\n- Key concept summary\n- Summary of useful metrics\n\n# Special relativity concept summary\n\n- Space and time unified into 4D spacetime.\n- Analogous unification for other 4-vectors (momentum+energy, etc.).\n- Lorentz transform relates 4-vectors in different inertial frames. Example: fast moving clocks are slower, shorter and heavier.\n- $E = mc^{2}$. Example: nuclear power.\n\n# General relativity concept summary\n\n- Spacetime is not static but dynamic, globally expanding and locally curving and contracting to form black holes etc.\n- Matter curves spacetime so that things moving “straight” (along geodesics) through curved spacetime appear deflected/accelerated (gravity).", "source": "mit-ocw", "source_doc_id": "0008-general-relativity-handout-35307630-39b76ca4", "source_title": "Topics: Special relativity concept summary", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Topics: Special relativity concept summary", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0008-general-relativity-handout-35307630-39b76ca4:page-0002", "text": "# Summary of useful metrics\n\n- Minkowski metric:\n\n$$\nd \\tau^ {2} = d (c t) ^ {2} - d x ^ {2} - d y ^ {2} - d z ^ {2}\n$$\n\n- Newtonian metric:\n\n$$\nd \\tau^ {2} = \\left(1 + \\frac {2 \\phi}{c ^ {2}}\\right) d (c t) ^ {2} - d x ^ {2} - d y ^ {2} - d z ^ {2}\n$$\n\n- Minkowski metric in polar coordinates:\n\n$$\nd \\tau^ {2} = d (c t) ^ {2} - d r ^ {2} - r ^ {2} d \\theta^ {2} - r ^ {2} \\sin^ {2} \\theta d \\varphi^ {2}\n$$\n\n- Friedman-Robertson-Walker (FRW) metric:\n\n$$\nd \\tau^ {2} = d (c t) ^ {2} - a (t) ^ {2} \\left(\\frac {d r ^ {2}}{1 - k r ^ {2}} + r ^ {2} d \\theta^ {2} + r ^ {2} \\sin^ {2} \\theta d \\varphi^ {2}\\right)\n$$\n\n• Schwartzschild metric:\n\n$$\nd \\tau^ {2} = \\left(1 - \\frac {r _ {s}}{r}\\right) d (c t) ^ {2} - \\left(1 - \\frac {r _ {s}}{r}\\right) ^ {- 1} d r ^ {2} - r ^ {2} d \\theta^ {2} - r ^ {2} \\sin^ {2} \\theta d \\varphi^ {2},\n$$\n\nwhere the Schwartzschild radius is defined as\n\n$$\nr _ {s} \\equiv \\frac {2 M G}{c ^ {2}}.\n$$\n\n- In GR, it's convenient to use units where  $c = G = 1$, simplifying these metrics:\n\n- Minkowski metric:\n\n$$\nd \\tau^ {2} = d t ^ {2} - d x ^ {2} - d y ^ {2} - d z ^ {2}\n$$\n\n- Newtonian metric:\n\n$$\nd \\tau^ {2} = (1 + 2 \\phi) d t ^ {2} - d x ^ {2} - d y ^ {2} - d z ^ {2}\n$$\n\n- Minkowski metric in polar coordinates:\n\n$$\nd \\tau^ {2} = d t ^ {2} - d r ^ {2} - r ^ {2} d \\theta^ {2} - r ^ {2} \\sin^ {2} \\theta d \\varphi^ {2}\n$$\n\n- Friedman-Robertson-Walker (FRW) metric:\n\n$$\nd \\tau^ {2} = d t ^ {2} - a (t) ^ {2} \\left(\\frac {d r ^ {2}}{1 - k r ^ {2}} + r ^ {2} d \\theta^ {2} + r ^ {2} \\sin^ {2} \\theta d \\varphi^ {2}\\right)\n$$\n\n- Schwartzschild metric ( $r_s = 2M$):\n\n$$\nd \\tau^ {2} = \\left(1 - \\frac {2 M}{r}\\right) d t ^ {2} - \\left(1 - \\frac {2 M}{r}\\right) ^ {- 1} d r ^ {2} - r ^ {2} d \\theta^ {2} - r ^ {2} \\sin^ {2} \\theta d \\varphi^ {2},\n$$", "source": "mit-ocw", "source_doc_id": "0008-general-relativity-handout-35307630-39b76ca4", "source_title": "Topics: Special relativity concept summary", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Topics: Special relativity concept summary", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0008-general-relativity-handout-35307630-39b76ca4:page-0003", "text": "# Summary of how to work with metrics\n\n- Computing the ageing  $\\Delta\\tau$ along a curve  $\\mathbf{r}(t)$ though spacetime:\n\n$$\n\\Delta \\tau = \\int d \\tau .\n$$\n\n• Example: for Minkowski metric,\n\n$$\n\\begin{array}{l} {\\Delta \\tau} = {\\int d \\tau = \\int \\sqrt {d t ^ {2} - d x ^ {2} - d y ^ {2} - d z ^ {2}} = \\int \\sqrt {1 - \\dot {x} ^ {2} - \\dot {y} ^ {2} - \\dot {z} ^ {2}} d t} \\\\ = \\int \\sqrt {1 - u ^ {2}} d t = \\int \\frac {d t}{\\gamma}. \\\\ \\end{array}\n$$\n\n• Example: for Newtonian metric in limit   $|\\phi| \\ll 1$  and   $u \\ll 1$ ,\n\n$$\n\\begin{array}{l} \\Delta \\tau = \\int d \\tau = \\int \\sqrt {(1 + 2 \\phi) d t ^ {2} - d x ^ {2} - d y ^ {2} - d z ^ {2}} = \\int \\sqrt {1 + 2 \\phi - u ^ {2}} d t \\\\ \\approx \\int \\left(1 + \\phi - \\frac {1}{2} u ^ {2}\\right) d t. \\\\ \\end{array}\n$$\n\nInterpretation: you age slower if you go faster but age faster if you go higher up in the gravitational potential.\n\n- Computing the trajectory of a massive particle from event A to event B: find the geodesic between A and B. This is the path of maximal aging, i.e., the path that maximizes   $\\Delta\\tau$  from above. This is a variational calculus problem and can be zapped with the Euler-Lagrange equation.\n\n- Computing the trajectory of a photon from event A to event B: photons move along geodesics that have  $\\Delta = \\tau = 0$, so-called null geodesics. Many of our applications will involve only one space-dimension (say, motion in the  $r$-direction, in which case you don't need to use variational calculus and can simply solve the equation  $d\\tau = 0$.", "source": "mit-ocw", "source_doc_id": "0008-general-relativity-handout-35307630-39b76ca4", "source_title": "Topics: Special relativity concept summary", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Topics: Special relativity concept summary", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0008-general-relativity-handout-35307630-39b76ca4:page-0004", "text": "# Newtonian gravity\n\nThe “gravitational field” g is minus the gradient   $\\nabla\\phi$  of the Newtonian gravitational potential   $\\phi$ . Units:   $\\phi/c^{2}$  is dimensionless.\n\n- How matter affects the gravitational field:\n\n$$\n\\nabla^ {2} \\phi = 4 \\pi G \\rho\n$$\n\nImplication: the gravitational potential from a single point mass  $M$ at the origin is\n\n$$\n\\phi = - \\frac {G M}{r},\n$$\n\nand fields from different masses simply add.\n\n- How the gravitational field affects matter:\n\n$$\nF = m \\mathbf {g} = - m \\nabla \\phi .\n$$", "source": "mit-ocw", "source_doc_id": "0008-general-relativity-handout-35307630-39b76ca4", "source_title": "Topics: Special relativity concept summary", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Topics: Special relativity concept summary", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0008-general-relativity-handout-35307630-39b76ca4:page-0005", "text": "# Equivalence principle (1911)\n\n- General relativity (GR) consists of two parts: how matter (particles, electromagnetic fields, etc.) affects spacetime and how spacetime affect matter. The second part is specified by the strong equivalence principle.\n- Weak equivalence principle: No local experiment can distinguish between a uniform gravitational field  $\\mathbf{g}$ and a frame accelerated with  $\\mathbf{a} = \\mathbf{g}$.\n- Strong equivalence principle: The laws of physics take on their special-relativistic form in any locally inertial frame frame.\n- A freely falling elevator is a locally inertial frame (if the elevator is small enough and our experiment short enough), so the strong version says that special relativity applies in all such elevators anywhere and anytime in the universe, i.e., independently of the spacetime position and velocity of the elevator.\n- Where did this idea come from? Combining\n\n$$\nF = m a\n$$\n\nwith\n\n$$\nF = \\frac {G m M}{r ^ {2}}\n$$\n\nshows that the gravitational acceleration\n\n$$\na = \\frac {G M}{r ^ {2}}\n$$\n\nis mass-independent as long as\n\n“inertial mass” = “gravitational mass”.\n\nIs it?\n\n- Galileo's Pisa experiment showed it with low precision.\n- Eötvös (1890) and later others showed with high precision that  $a$ independent of both mass and composition (density, atomic element, matter/antimatter, etc). Coincidence? Einstein thought that no, it was telling us something.\n- In other words, if you know the direction of the worldline of an object freely floating through a spacetime event (i.e., the direction of the velocity 4-vector), then the continuation of the worldline under the influence of gravity is the same regardless of the mass and composition of the object. This suggested to Einstein that gravity was a purely geometric effect.", "source": "mit-ocw", "source_doc_id": "0008-general-relativity-handout-35307630-39b76ca4", "source_title": "Topics: Special relativity concept summary", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Topics: Special relativity concept summary", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0008-general-relativity-handout-35307630-39b76ca4:page-0006", "text": "# Gravitational redshift\n\n- Implied by equivalence principle. When light travels a distance h from the floor to the ceiling of an elevator free-falling downward in a uniform gravitational field g, it will have no redshift according to an observer in the elevator. If the elevator was at rest in the lab frame when the light was emitted, then when it reaches the ceiling, the ceiling (and the locally inertial frame where the light is not redshifted) is moving downward with\n\n$$\nu \\approx g t \\approx \\frac {g h}{c}.\n$$\n\nLorentz transforming to the lab frame and using the Doppler shift formula thus gives a gravitational redshift\n\n$$\n\\frac {\\nu^ {\\prime}}{\\nu} \\approx 1 - \\frac {v}{c} \\approx 1 - \\frac {g h}{c ^ {2}}.\n$$\n\n- Also implied by energy conservation and   $E = mc^{2}$ . If a photon travels upward in a gravitational field, the increase in potential energy   $mgh = \\frac{E}{c^{2}}gh$  must be offset by a reduction in the photon energy   $h\\nu$ , i.e., be a lowering of the frequency. This again gives the result\n\n$$\n\\frac {\\nu^ {\\prime}}{\\nu} \\approx 1 - \\frac {g h}{c ^ {2}}.\n$$\n\n- Both of these calculations are only approximate, correct to first order (in the small quantities  $v / c$ and  $\\phi / c^2$).\n\n- The gravitational redshift is simply given by the Newtonian gravitational potential   $\\phi$ : Adding up many infinitesimal contributions from the above formula gives\n\n$$\n\\frac {\\Delta \\nu}{\\nu} \\int \\frac {1}{c ^ {2}} \\mathbf {g} \\cdot d \\mathbf {r} = - \\frac {1}{c ^ {2}} \\int \\nabla \\phi \\cdot d r = - \\frac {\\phi}{c ^ {2}}.\n$$\n\n- Implication: Time runs slower further down in the gravitational potential:\n\n$$\n\\frac {d \\tau}{d t} = 1 + \\frac {\\phi}{c ^ {2}}.\n$$\n\n- Implication: Taking  $c = 1$, this means that we need a factor  $(1 + \\phi)^2 \\approx 1 + 2\\phi$ multiplying the  $dt^2$-factor in the metric:\n\n$$\nd \\tau^ {2} = (1 + 2 \\phi) d t ^ {2} - d x ^ {2} - d y ^ {2} - d z ^ {2}\n$$\n\n- This simple metric reproduces all of Newtonian gravity! The Euler-Lagrange equation shows that time-like geodesics in this metric obey the Newtonian result\n\n$$\n\\ddot {r} = - \\nabla \\phi .\n$$\n\n- Tidbit for the curious: Adding terms multiplying elsewhere in the metric (say, for the  $dy^2$-term or the off-diagonal  $dtdx$-term) has no effect in the Newtonian limit, since they would get suppressed by a factor  $v/c$ or  $(v/c)^2$ in the Euler-Lagrange equation.", "source": "mit-ocw", "source_doc_id": "0008-general-relativity-handout-35307630-39b76ca4", "source_title": "Topics: Special relativity concept summary", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 5, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Topics: Special relativity concept summary", "Page 6"], "page_start": 6, "page_end": 6, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0008-general-relativity-handout-35307630-39b76ca4:page-0007", "text": "# Spherical coordinates\n\n- Spherical coordinates  $(r, \\theta, \\varphi)$ are defined by\n\n$$\nx = r \\sin \\theta \\cos \\varphi ,\n$$\n\n$$\ny = r \\sin \\theta \\sin \\varphi ,\n$$\n\n$$\nz = r \\cos \\theta .\n$$\n\n- This implies\n\n$$\n{ d x } { = } { \\sin \\theta \\cos \\varphi d r + r \\cos \\theta \\cos \\varphi d \\theta - r \\sin \\theta \\sin \\varphi d \\varphi , }\n$$\n\n$$\nd y = \\sin \\theta \\sin \\varphi d r + r \\cos \\theta \\sin \\varphi d \\theta + r \\sin \\theta \\cos \\varphi d \\varphi ,\n$$\n\n$$\n{ d z } { = } { \\cos \\theta d r - r \\sin \\theta d \\theta . }\n$$\n\n- This let's us reexpress the Minkowski metric in spherical coordinates:\n\n$$\n\\begin{array}{l} d \\tau^ {2} = d t ^ {2} - d x ^ {2} - d y ^ {2} - d z ^ {2} \\\\ { = } { d t ^ { 2 } - d r ^ { 2 } - r ^ { 2 } d \\theta ^ { 2 } - r ^ { 2 } \\sin ^ { 2 } \\theta d \\varphi ^ { 2 } . } \\\\ \\end{array}\n$$\n\n(To get the second line, we simply plugged in the expressions for dx, dy and dz and simplified the result.)", "source": "mit-ocw", "source_doc_id": "0008-general-relativity-handout-35307630-39b76ca4", "source_title": "Topics: Special relativity concept summary", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 6, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Topics: Special relativity concept summary", "Page 7"], "page_start": 7, "page_end": 7, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0008-general-relativity-handout-35307630-39b76ca4:page-0008", "text": "# General covariance\n\n- The analogous procedure is used to transform any metric into any coordinate system.\n\n- Key concept: this means that we can do our calculations with metrics and geodesics in any system of space and time coordinates we like. In Minkowski space, inertial frames are just a special class of coordinate systems (the standard spacetime coordinates   $(x,y,z,ct)$  and Lorentz transforms thereof), so we're not limited to working in inertial frames in GR.\n\n- Einstein insisted that not only the metric but indeed all laws of physics should be expressible using any coordinate system. This requirement is called general covariance.\n\n- This is why GR is called General relativity, special relativity being merely the special case where you were allowed to start with an inertial frame and make a Lorentz transformation (a particular linear coordinate transformation).\n\n- If you think of Lorentz transformations as coordinate transformations, they are simply the ones that have the property\n\n$$\nd \\tau^ {2} = d (c t) ^ {2} - d x ^ {2} - d y ^ {2} - d z ^ {2} = d (c t ^ {\\prime}) ^ {2} - d x ^ {\\prime 2} - d y ^ {\\prime 2} - d z ^ {\\prime 2},\n$$\n\nsince we previously proved that   $d\\tau$  is Lorentz invariant.\n\n- General covariance defines the second part of General relativity (how the metric affects matter): to compute the evolution of something near a certain spacetime event (the electromagnetic field, the position of a particle, etc.), simply change to coordinates that correspond to a free-falling frame locally (in the spacetime region surrounding that event) and apply the equations special relativity. In particular, a particle not subjected to any non-gravitational forces moves along a geodesic.\n\n- Note that it's not at all obvious just from staring at a metric that someone writes down whether it's really just Minkowski space in disguise, expressed in some funny coordinates.", "source": "mit-ocw", "source_doc_id": "0008-general-relativity-handout-35307630-39b76ca4", "source_title": "Topics: Special relativity concept summary", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 7, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Topics: Special relativity concept summary", "Page 8"], "page_start": 8, "page_end": 8, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0009-einstein-s-field-equations-handout-13798c16-f720c60a:page-0001", "text": "# MIT Course 8.033, Fall 2006, Formula Sheet\n\n# (Dated: August 24, 2006)\n\n- 4-vectors:\n\n$$\n\\mathbf {X} \\equiv \\left( \\begin{array}{c} x \\\\ y \\\\ z \\\\ c t \\end{array} \\right), \\quad \\mathbf {U} \\equiv \\frac {d \\mathbf {X}}{d \\tau} = \\gamma_ {u} \\left( \\begin{array}{c} u _ {x} \\\\ u _ {y} \\\\ u _ {z} \\\\ c \\end{array} \\right),\n$$\n\n$$\n\\mathbf {K} \\equiv \\gamma_ {u} \\left( \\begin{array}{c} k _ {x} \\\\ k _ {y} \\\\ k _ {z} \\\\ w / c \\end{array} \\right), \\quad \\mathbf {P} \\equiv m _ {0} \\mathbf {U} = \\left( \\begin{array}{c} p _ {x} \\\\ p _ {y} \\\\ p _ {z} \\\\ E / c \\end{array} \\right)\n$$\n\n$$\n\\mathcal {F} \\equiv \\frac {d}{d \\tau} \\mathbf {P} = \\gamma_ {u} \\left( \\begin{array}{l} \\mathbf {F} \\\\ P / c \\end{array} \\right), \\quad \\mathbb {J} \\equiv \\rho_ {0} \\mathbf {U} = \\left( \\begin{array}{l} J _ {x} \\\\ J _ {y} \\\\ J _ {z} \\\\ \\rho c \\end{array} \\right)\n$$\n\n- Lorentz transformation:\n\n$$\n\\boldsymbol {\\Lambda} (\\hat {\\mathbf {x}} v) = \\left( \\begin{array}{c c c c} \\gamma & 0 & 0 & - \\gamma \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ - \\gamma \\beta & 0 & 0 & \\gamma \\end{array} \\right), \\quad \\left( \\begin{array}{c} x ^ {\\prime} \\\\ y ^ {\\prime} \\\\ z ^ {\\prime} \\\\ c t ^ {\\prime} \\end{array} \\right) = \\left( \\begin{array}{c} \\gamma (x - \\beta c t) \\\\ y \\\\ z \\\\ \\gamma (c t - \\beta x) \\end{array} \\right)\n$$\n\n- Parallel velocity addition:\n\n$$\nv ^ {\\prime \\prime} = \\frac {v + v ^ {\\prime}}{1 + \\frac {v v ^ {\\prime}}{c ^ {2}}}\n$$\n\n• Aberration & Doppler effect:\n\n$$\n\\cos \\theta^ {\\prime} = \\frac {\\cos \\theta - \\beta}{1 - \\beta \\cos \\theta}, \\quad \\omega^ {\\prime} = \\omega \\gamma (1 - \\beta \\cos \\theta)\n$$\n\n- Energy:\n\n$$\nE = m _ {0} \\gamma c ^ {2} = \\sqrt {(m _ {0} c ^ {2}) ^ {2} + (c p) ^ {2}}\n$$\n\n- Electromagnetism:\n\n$$\n\\mathbf {F} = q (\\mathbf {E} + \\frac {1}{c} \\mathbf {u} \\times \\mathbf {B})\n$$\n\n$$\nE _ {x} ^ {\\prime} = E _ {x}\n$$\n\n$$\nE _ {y} ^ {\\prime} = \\gamma (E _ {y} - \\beta B _ {z})\n$$\n\n$$\nE _ {z} ^ {\\prime} = \\gamma (E _ {z} + \\beta B _ {y})\n$$\n\n$$\nB _ {x} ^ {\\prime} = B _ {x}\n$$\n\n$$\nB _ {y} ^ {\\prime} = \\gamma (B _ {y} + \\beta E _ {z})\n$$\n\n$$\nB _ {z} ^ {\\prime} = \\gamma (B _ {z} - \\beta E _ {y}).\n$$\n\n- Euler-Lagrange equation:\n\n$$\n\\frac {\\partial f}{\\partial x} - \\frac {d}{d t} \\frac {\\partial f}{\\partial \\dot {x}} = 0\n$$\n\n- Metrics (Minkowski, Newtonian, FRW, Schwartzschild),  $c = G = 1$,  $d\\Omega^2 \\equiv d\\theta^2 + \\sin^2 \\theta d\\varphi^2$:\n\n$$\nd \\tau^ {2} = d t ^ {2} - d x ^ {2} - d y ^ {2} - d z ^ {2}\n$$\n\n$$\n= d t ^ {2} - d r ^ {2} - r ^ {2} d \\Omega^ {2},\n$$\n\n$$\nd \\tau^ {2} = (1 + 2 \\phi) d t ^ {2} - r ^ {2} d \\Omega^ {2},\n$$\n\n$$\nd \\tau^ {2} = d t ^ {2} - a (t) ^ {2} \\left(\\frac {d r ^ {2}}{1 - k r ^ {2}} + r ^ {2} d \\Omega^ {2}\\right),\n$$\n\n$$\nd \\tau^ {2} = \\left(1 - \\frac {2 M}{r}\\right) d t ^ {2} - \\left(1 - \\frac {2 M}{r}\\right) ^ {- 1} d r ^ {2} - r ^ {2} d \\Omega^ {2}\n$$\n\n- Friedmann equation:\n\n$$\nH ^ {2} = \\frac {8 \\pi G}{3} \\rho - \\frac {k c ^ {2}}{a ^ {2}}, H \\equiv \\frac {\\dot {a}}{a}\n$$\n\n- Shell coordinates:\n\n$$\nd t _ {\\text { shell }} = \\gamma_ {r} ^ {- 1} d t, \\quad d r _ {\\text { shell }} = \\gamma_ {r} d r\n$$\n\n$$\n\\gamma_ {r} \\equiv \\frac {1}{\\sqrt {1 - \\beta_ {r} ^ {2}}}, \\quad \\beta_ {r} \\equiv \\left(\\frac {2 M}{r}\\right) ^ {1 / 2}\n$$\n\n• Schwarzschild orbits:\n\n$$\n\\left(\\frac {d r}{d \\tau}\\right) ^ {2} = \\tilde {E} ^ {2} - \\tilde {V} (\\tilde {L}, r) ^ {2},\n$$\n\n$$\n\\frac {d \\varphi}{d \\tau} = \\frac {\\tilde {L}}{r ^ {2}},\n$$\n\n$$\n\\frac {d t}{d \\tau} = \\gamma_ {r} ^ {2} \\tilde {E},\n$$\n\n$$\n\\tilde {V} (\\tilde {L}, r) ^ {2} \\equiv \\left(1 - \\frac {2 M}{r}\\right) \\left(1 + \\frac {\\tilde {L} ^ {2}}{r ^ {2}}\\right)\n$$\n\n- Just kidding:\n\n$$\nR _ {\\mu \\nu} - \\frac {1}{2} R g _ {\\mu \\nu} = 8 \\pi G T _ {\\mu \\nu},\n$$\n\n$$\nR \\equiv g ^ {\\mu \\nu} R _ {\\mu \\nu},\n$$\n\n$$\nR _ {\\mu \\nu} \\equiv R _ {\\mu \\alpha \\nu} ^ {\\alpha},\n$$\n\n$$\nR _ {\\mu \\nu \\beta} ^ {\\alpha} \\equiv \\Gamma_ {\\nu \\beta , \\mu} ^ {\\alpha} - \\Gamma_ {\\mu \\beta , \\nu} ^ {\\alpha} + \\Gamma_ {\\mu \\beta} ^ {\\gamma} \\Gamma_ {\\nu \\gamma} ^ {\\alpha} - \\Gamma_ {\\nu \\beta} ^ {\\gamma} \\Gamma_ {\\mu \\gamma} ^ {\\alpha},\n$$\n\n$$\n\\Gamma_ {\\mu \\nu} ^ {\\alpha} \\equiv \\frac {1}{2} g ^ {\\alpha \\sigma} \\left(g _ {\\sigma \\mu , \\nu} + g _ {\\sigma \\nu , \\mu} - g _ {\\mu \\nu , \\sigma}\\right) Ⓥ\n$$", "source": "mit-ocw", "source_doc_id": "0009-einstein-s-field-equations-handout-13798c16-f720c60a", "source_title": "MIT Course 8.033, Fall 2006, Formula Sheet: (Dated: August 24, 2006)", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MIT Course 8.033, Fall 2006, Formula Sheet: (Dated: August 24, 2006)", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-problem-set-1-264e6cf3-4a983b79:page-0001", "text": "# MASSACHUSETTS INSTITUTE OF TECHNOLOGY\n\n# DEPARTMENT OF PHYSICS\n\nPhysics 8.033\n\nSeptember 8, 2006\n\n# Problem Set 1\n\nDue: Friday September 15 4:00PM. Please deposit the problem set in the appropriate 8.033 bin, labeled by recitation number.\n\nReading: Resnick Chapter 1 & beginning of Chapter 2. Parallel reading: Chapters 1 & 2 in French.\n\n- Problem 0(3 points): Write your name, section number and staple your problem set!\n\n- Problem 1(3 points): “First Measurement of Speed of Light”\n\nIn 1675, the Danish astronomer Olaus Roemer showed for the first time that the speed of light is finite. He did this using measurements by French the astronomer Giovanni Cassini of the orbital period of Jupiter's moon Io. The time between two successive appearances of Io from behind Jupiter was observed to be slightly longer when Earth was moving away from Jupiter rather than approaching. Relative to the orbital period when Earth was closest to Jupiter (around A), all periods were thus measured to be slightly longer during the six months when Earth moved from B to C to the point farthest from Jupiter (around D). The sum of these six months of period excesses was measured to be 22 minutes. How would you interpret this result and estimate the speed of light? (Neglect the motion of Jupiter, which takes 12 years to orbit the Sun.)\n\nIO\nJUPITER\nEARTH\nA\nF\nB\nC\nD\nE\nEARTH SIX MONTHS LATER\n\nHint: This effect has nothing to do with doppler shifting of light! The difference in observed period values is related to the finite speed of light i.e, the time taken for light to reach Earth varies with its orbital position.\n\n- Problem 2(6 points): “This Goodly Frame, the Earth...” (Hamlet)\n\nFor some purposes the Earth cannot be taken as a totally “goodly” inertial frame because its motion is accelerated. Calculate the accelerations associated with\n\na) the Earth's rotation about its axis (assume an equatorial point),\n\nb) the Earth's orbital motion about the Sun\n\nc) the orbital motion of the solar system about the Galactic Center and\n\nd) motion around Earth-Moon center of mass.\n\nThe Earth-Sun distance is about   $1.5 \\times 10^{11}$ m and the Earth-Moon distance is about   $3.8 \\times 10^{8}$ m. Mass ratio of Earth and the Moon is about 80. Assume that the Sun is about 10 kpc from the Galactic center and orbits around it at about 300 km/s. One parsec =   $10^{-3}$  kpc   $\\approx 3.09 \\times 10^{16}$  m.", "source": "mit-ocw", "source_doc_id": "0010-problem-set-1-264e6cf3-4a983b79", "source_title": "MASSACHUSETTS INSTITUTE OF TECHNOLOGY: DEPARTMENT OF PHYSICS", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MASSACHUSETTS INSTITUTE OF TECHNOLOGY: DEPARTMENT OF PHYSICS", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-problem-set-1-264e6cf3-4a983b79:page-0002", "text": "- Problem 3(6 points): “The Galilean Transformation Generalized”\n\n- a) Resnick, Chapter 1, problem number 7, page 46. In other words, generalize equations 1.1a and 1.1b in the book.\n- b) Write down the equivalent matrix equation.\n\nNote: We will return to this problem in Special Relativity where it becomes substantially more complicated.\n\n- Problem 4(9 points): “Momentum conservation”\n\nAn observer on the ground watches a collision between two particles whose masses are   $m_{1}$  and   $m_{2}$  and finds, by measurement, that momentum is conserved. Use the classical velocity addition theorem to show that an observer on a moving train will also find that momentum is conserved in this collision.\n\nRepeat this calculation under the assumption that a transfer of mass from one particle to the other takes place during the collision, the initial masses being   $m_{1}$  and   $m_{2}$  and the final masses being   $m_{1}^{\\prime}$  and   $m_{2}^{\\prime}$ . Again, assume that the ground observer finds, by measurement, that momentum is conserved. Show that the train observer will also find that momentum is conserved only if mass is also conserved, that is, if   $m_{1}^{\\prime} + m_{2}^{\\prime} = m_{1} + m_{2}$ .\n\nNote: In this course we generally reserve u for velocities of objects within a given frame, and v for velocities of transformation.\n\n- Problem 5(9 points): “The Invariance of Elastic”\n\nA collision between two particles in which energy is conserved is described as elastic. Show, using the Galilean velocity transformation equations, that if a collision is found to be elastic in one inertial reference frame, then it will also be found to be elastic in all other such frames.\n\nCould this result have been predicted from the conservation of energy principle?\n\nNote: You may restrict your analysis to the special case where all velocities are along the x-axis.\n\n- Problem 6(6 points): “The work-energy theorem holds in all inertial frames”\n\nObserver G is on the ground and observer T is on a train moving with uniform velocity v with respect to the ground. Each observes that a particle of mass m, initially at rest with respect the train, is acted on by a constant force F applied to it in the forward direction for a time t.\n\na) Show that the two observers will find, for the work done on the particle by the force  $\\mathbf{F}$,\n\n$$\nW _ {T} = \\frac {1}{2} m a ^ {2} t ^ {2} \\quad \\text { and } \\quad W _ {G} = \\frac {1}{2} m a ^ {2} t ^ {2} + m v a t,\n$$\n\nrespectively, where a is the common acceleration of the particle.\n\nb) Show that  $\\Delta K_{T}$ and  $\\Delta K_{G}$, the changes in kinetic energy calculated by each observer, are also given by these same two expressions.\n\nThus the work-energy theorem  $W = \\Delta K$ is valid in all inertial reference frames.\n\n- Problem 7(6 points): “Initial Conditions Depend on the Frame”\n\nA person standing on a train moving with speed v (with respect to the ground) releases a ball from rest at a height h above the floor. The ball falls straight down and hits the floor at a time   $t = (2h/g)^{1/2}$ , where g is the gravitational acceleration. Sketch the trajectory that an observer at rest on the ground measures. (Assume that one side of the train is made of transparent material.) How do you reconcile, quantitatively, the different trajectories with the fact that Newton's second law, F = ma, is supposed to work equally well in both of these inertial frames?\n\n- Problem A: (Optional) “Vacuum Speed of Light is Independent of Frequency”\n\nX-ray pulses, visible-light pulses, and radio pulses (the latter corrected for dispersion in the interstellar plasma) emitted by an astronomical object called a “pulsar” are all observed to arrive simultaneously at the Earth — with an uncertainty of only 200 microseconds. The particular pulsar in question is located at a distance from the Earth of 6000 light years. Use this information to make a quantitative estimate of how much the speed of electromagnetic radiation can vary with frequency (or wavelength). Express your answer as a limit on the fractional difference in speed over this wide range of electromagnetic frequencies.\n\n- Feedback: Roughly how much time did you spend on this problem set?", "source": "mit-ocw", "source_doc_id": "0010-problem-set-1-264e6cf3-4a983b79", "source_title": "MASSACHUSETTS INSTITUTE OF TECHNOLOGY: DEPARTMENT OF PHYSICS", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MASSACHUSETTS INSTITUTE OF TECHNOLOGY: DEPARTMENT OF PHYSICS", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-problem-set-2-dd187cc5-d9d10882:page-0001", "text": "# Problem Set 2\n\nDue: Friday 22 September 2006 at 4:00PM. Please Write your name deposit the problem set in the appropriate 8.033 bin, labeled by recitation section number and stapled as needed (3 points).\n\nReadings: Chapter II of Resnick. Parallel reading: Chapters 3 & 4 in French.\n\nProblem 1: (3 points) Consider a general wave equation of the form:\n\n$$\n\\frac {\\partial^ {2} y}{\\partial x ^ {2}} = \\frac {1}{V ^ {2}} \\frac {\\partial^ {2} y}{\\partial t ^ {2}}. \\tag {1.1}\n$$\n\n(a) Show that any function of the quantity  $(x\\pm Vt)$ will satisfy this equation.\n\n(b) Make an argument to show why V represents the propagation speed of waves or other disturbances described by this equation (x is the space and t is time).\n\nProblem 2: “Maxwell versus Galileo” (6 points)\n\nMaxwell's wave equation for the  $z$-component of an electric field propagating in the  $x$-direction is\n\n$$\n\\frac {\\partial^ {2} E _ {z}}{\\partial x ^ {2}} = \\frac {1}{c ^ {2}} \\frac {\\partial^ {2} E _ {z}}{\\partial t ^ {2}}, \\tag {2.1}\n$$\n\nwhere   $E(x,t)$  is the amplitude of the electric field. Show that this equation is not invariant under a Galilean transformation to a reference frame moving with relative speed v in the x direction.\n\nHint: Use the chain rule in which, if  $f = f(x,t)$, then\n\n$$\n\\frac {\\partial f}{\\partial x} = \\frac {\\partial f}{\\partial x ^ {\\prime}} \\left(\\frac {\\partial x ^ {\\prime}}{\\partial x}\\right) + \\frac {\\partial f}{\\partial t ^ {\\prime}} \\left(\\frac {\\partial t ^ {\\prime}}{\\partial x}\\right). \\tag {2.2}\n$$\n\nOptional: Show that any function  $E_{z}(x + Vt)$ is a solution to the transformed wave equation, where  $V = v \\pm c$.\n\nProblem 3: “Michelson-Morley Experiment With a Real Wind” (3+3+3 points)\n\nA pilot plans to fly due east from A to B and back again. If u is his airspeed and if l is the distance between A and B, it is clear that his round-trip timr   $t_{0}$ —if there is no wind—will be 2l/u.\n\n(a). Suppose, however, that a steady headwind of speed v blows from the east (or from the west). Show that the round-trip time will now be\n\n$$\nt _ {1} = \\frac {t _ {0}}{1 - (v / u) ^ {2}}. \\tag {3.1}\n$$\n\n(b). If the wind is from the north (or from the south), show that the expected round-trip travel time is\n\n$$\nt _ {2} = \\frac {t _ {0}}{\\sqrt {1 - (v / u) ^ {2}}}. \\tag {3.2}\n$$\n\n(c). Note that these two travel times are not equal. In the Michelson-Morley experiment, however, the experiment seems to show that (for arms of equal length), the travel times for light are equal; otherwise these experimenters would have found a fringe shift when they rotated the interferometer. What is the essential difference between the two situations?", "source": "mit-ocw", "source_doc_id": "0011-problem-set-2-dd187cc5-d9d10882", "source_title": "Problem Set 2", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Problem Set 2", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-problem-set-2-dd187cc5-d9d10882:page-0002", "text": "Problem 4: “Not Much of a Contraction” (3 points)\n\nIn the Michelson-Morley experiment, by about how many atomic diameters would the appropriate arm of the interferometer have to shrink with respect to the other arm according to the Lorentz-Fitzgerald contraction hypothesis? Assume that the Earth is moving along the direction of the first arm. The actual arm length of the interferometer was 2.8m. Take 0.1 nm as a typical atomic diameter.\n\nProblem 5: (3 points) In Special Relativity we will often want to evaluate the relativistic factor   $\\gamma = (1 - v^{2}/c^{2})^{-1/2}$  for: (1) small values of speed v, (2) for values of v close to the speed of light c. While this can be done with a calculator, it is often more instructive to derive an approximate analytic result explicitly in terms of powers of   $\\beta = v/c$  for case (1), and powers of   $1/\\gamma$  for case (2).\n\n(a). Expand  $\\gamma^2$ and  $\\gamma$ in a Taylor series in powers of  $\\beta^2$, keeping the first 3 terms (constant plus next two terms). If you have not yet learned about Taylor series, simply adopt the expression below:\n\n$$\n(1 + x) ^ {n} = 1 + n x + \\frac {n (n - 1)}{2 !} x ^ {2} + \\frac {n (n - 1) (n - 2)}{3 !} x ^ {3} + \\dots \\tag {5.1}\n$$\n\nCheck your answers:\n\n$$\n\\gamma^ {2} = \\frac {1}{1 - \\beta^ {2}} = 1 + \\beta^ {2} + \\beta^ {4} + \\beta^ {6} + \\dots , \\tag {5.2}\n$$\n\n$$\n\\gamma = \\frac {1}{\\sqrt {1 - \\beta^ {2}}} = 1 + \\frac {1}{2} \\beta^ {2} + \\frac {3}{8} \\beta^ {4} + \\frac {5}{1 6} \\beta^ {6} + \\dots . \\tag {5.3}\n$$\n\nEvaluate your expression for with values of   $\\beta\\in\\{0.1,0.001,10^{-5}\\}$ . Compare to the values you get by evaluating the full expression for   $\\gamma$  with a calculator.\n\n(b). Show that for large  $\\gamma$,  $1 - \\beta \\approx 1 / (2\\gamma^2)$. Use this expression to find the values of  $\\beta$ when  $\\gamma \\in \\{2, 10, 10^3\\}$. Compare the approximate results with those obtained from the full expression using a calculator.\n\nProblem 6: “Aberration and Relativity” (3+1.5+3 points)\n\nConsider the aberration arrangement of Fig. 1-9(Resnick page 29).\n\n- (a). At what speed does light pass along the telescope axis according to the ether hypothesis?\n- (b). According to the special theory of relativity?\n- (c). Show that, according to relativity theory, the classical aberration equation (see Eq. 1-11)\n\n$$\n\\tan \\alpha = \\frac {v}{c} \\quad (\\text { classical   theory }) \\tag {6.1}\n$$\n\nmust be replaced by\n\n$$\n\\sin \\alpha = \\frac {v}{c} \\quad (\\text { relativity   theory }). \\tag {6.2}\n$$\n\nThus, the ether theory and the theory of relativity make different predictions for the aberration of starlight.\n\nOptional: If  $\\alpha_{c}$ and  $\\alpha_{r}$ are the predictions of these two theories, find their fractional difference  $(\\alpha_{r} - \\alpha_{c}) / \\alpha_{r}$. Assume that  $v = 30\\mathrm{km / s}$ (the earth's orbital speed).\n\nHint: The predictions of the two theories are so close that their difference may be beyond the scope of your hand calculator. Use the series expansions of   $\\sin^{-1}$  and   $\\tan^{-1}$ .", "source": "mit-ocw", "source_doc_id": "0011-problem-set-2-dd187cc5-d9d10882", "source_title": "Problem Set 2", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Problem Set 2", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-problem-set-2-dd187cc5-d9d10882:page-0003", "text": "Problem 7: “The Speed of Light Really is the Same in All Frames” (3+3 points)\n\n(a). Start with the equation:\n\n$$\nx ^ {2} + y ^ {2} + z ^ {2} - c ^ {2} t ^ {2} = 0, \\tag {7.1}\n$$\n\nand apply the Lorentz-Einstein transformations to yield equation:\n\n$$\nx ^ {\\prime 2} + y ^ {\\prime 2} + z ^ {\\prime 2} - c ^ {2} t ^ {\\prime 2} = 0. \\tag {7.2}\n$$\n\n(b). The corresponding matrix equation is\n\n$$\nX ^ {t} \\eta X = 0 \\tag {7.3}\n$$\n\nAlso we saw in class that  $\\Lambda^t\\eta \\Lambda = \\eta$ for any Lorentz matrix  $\\Lambda$. Start here and show that\n\n$$\nX ^ {\\prime t} \\eta X ^ {\\prime} = 0 \\tag {7.4}\n$$\n\nwhere  $X' = \\Lambda X$. You can see that the matrix formalism is easier.\n\nProblem 8: “A Moving Clock” (3 points)\n\nA clock moves along the x-axis at a speed of 0.6c and reads zero as it passes the origin. What time does it read(to an observer stationary w.r.t the clock) as it passes the 180 m mark on the axis? What is the time difference with a stationary clock synchronized at the origin?\n\nProblem 9: “A Moving Rod” (1.5 points)\n\nA rod lies parallel to the x-axis of reference frame S, moving along this at a speed of 0.6c. Its rest length is 1.0 m. What will its measured length be in frame S?\n\nProblem 10: “Inverse Lorentz transformation” (3 points)\n\nAs you know the Lorentz transformation along x-axis may be written in matrix form as\n\n$$\n\\left( \\begin{array}{c} c t ^ {\\prime} \\\\ x ^ {\\prime} \\\\ y ^ {\\prime} \\\\ z ^ {\\prime} \\end{array} \\right) = \\left( \\begin{array}{c c c c} \\gamma & - \\beta \\gamma & 0 & 0 \\\\ - \\beta \\gamma & \\gamma & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{array} \\right) \\left( \\begin{array}{c} c t \\\\ x \\\\ y \\\\ z \\end{array} \\right). \\tag {10.1}\n$$\n\nFind the inverse transformation, that is, find the coordinates   $(ct, x, y, z)$  in terms of   $(ct', x', y', z')$ , by the physical expectation of the inverse being given by changing the sign of the relative velocity v. Show by explicit matrix multiplication, that the transformation found in this way is indeed inverse to the matrix in (10.1).\n\nOptional Problem 11: “Fascinating Thought Experiment”\n\nEhrenfest (1880 - 1933) proposed the following thought experiment to illustrate the different behavior expected for light under the ether-wind hypothesis and under Einstein's second postulate.\n\nImagine yourself seated at the center of a spherical shell of radius   $3 \\times 10^{8}$  m, the inner surface being diffusely reflecting. A source at the center of the sphere emits a sharp pulse of light, which travels outward through the darkness with uniform intensity in all directions. What would you see during the 3-s interval following the pulse under the assumptions that (a) there is a steady ether wind blowing through the sphere at 100 km/s and (b) that there is no ether and Einstein's second postulate holds. (c) Discuss the relationship of this thought experiment to the Michelson-Morley experiment.\n\nSource: Problems 2, 3, 4, 6, 7, 8, 9, 10, 11 are taken from Basic Concepts in Relativity by Robert Resnick, David Halliday.\n\nFeedback: Roughly how much time did you spend on this problem set?", "source": "mit-ocw", "source_doc_id": "0011-problem-set-2-dd187cc5-d9d10882", "source_title": "Problem Set 2", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Problem Set 2", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0012-problem-set-3-f2bce924-27a294de:page-0001", "text": "# Massachusetts Institute of Technology\n\n# Department of Physics\n\nPhysics 8.033\n\nOut: Friday 22 September 2006\n\nDue: Friday 29 September 2006\n\n# Problem Set 3\n\nDue: Friday 29 September 2006 at 4:00PM. Please Write your name deposit the problem set in the appropriate 8.033 bin, labeled by recitation section number and stapled as needed (3 points).\n\nReadings: No new reading assignment in Resnick. Parallel reading: Chapter 5 in French.\n\nProblem 1: (2004 Exam Question) (1+1+1+1 pts)\n\nChoose one answer for each of the following multiple choice problems. Circle your answer.\n\n(a). A sphere is moving at  $90\\%$ of the speed of light perpendicularly to your line of sight. In your rest frame, it is\n\n- (a) a sphere.\n- (b) a prolate ellipsoid (like a watermelon: One principal axis is longer than the other two.)\n- (c) an oblate ellipsoid (like an M&M: One principal axis is shorter than the other two.)\n\n(b). To you, it looks like\n\n- (a) a sphere.\n- (b) a prolate ellipsoid.\n- (c) an oblate ellipsoid.\n\n(c). A lamp is moving around you in a circle (you are at the center) at 50% of the speed of light. You observe it\n\n- (a) redshifted.\n- (b) blueshifted.\n- (c) neither.\n\n(d). An airplane is moving around you in a circle (you are at the center) at  $50\\%$ of the speed of sound. You hear it\n\n- (a) redshifted (lower frequency).\n- (b) blueshifted (higher frequency).\n- (c) neither.\n\nProblem 2: “A Moving, Slanting Rod” (3+3 pts)\n\nA thin rod of length  $L'$, at rest in the  $S'$ frame, makes an angle of  $\\theta'$ with the  $x'$ axis, as in Figure 1.\n\n(a) What is its length L as measured by an observer in the S frame, for whom the rod is moving at a speed of   $\\beta c$  in the direction of increasing x? (b) What angle   $\\theta$  does this moving rod make with the x axis? (b) Evaluate these quantities for   $L' = 1.00 \\, m$ ,   $\\theta' = 30^\\circ$ , and   $\\beta = 0.40$ .\n\nProblem 3: “Decay in Flight” (3 pts)\n\nAn unstable high-energy particle enters a detector and leaves a track 1.05 mm long before it decays. Its speed relative to the detector was 0.992c. What is its proper lifetime? That is, how long would it have lasted before decay had it been at rest with respect to the detector?", "source": "mit-ocw", "source_doc_id": "0012-problem-set-3-f2bce924-27a294de", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0012-problem-set-3-f2bce924-27a294de:page-0002", "text": "y'\nS'\nS\nL'\nθ'\nx'\nx\ny'\n\n# Problem 4: “Pole Vaulter Problem”:   $(3+3+3+3$  pts)\n\nA pole oriented in the x direction is at rest in the   $S'$  frame, which moves to the right with respect to the S frame at a speed   $v = (3^{1/2}/2)c$ , (i.e.,   $\\gamma = 2$ ). The pole is 10 meters long in its own rest frame. It passes into a barn which is at rest in S with the right door closed and the left door open. The barn is 8m long in its own rest frame. At time t = 0, the left barn door closes with the left end of the rod (  $x' = 0$ ) just inside the door. At that same time in S the right end of the rod (  $x' = 10m$ ) is well inside the barn at x = 5m. Call the event (x = 0, t = 0) “A”, and the event (x = 5, t = 0) “B”. Thus, the barn doors are closed with the pole contained completely inside the barn. A short while later, the right door opens and the pole passes harmlessly out of the barn.\n\n- (a). Find  $t_B'$ in the vaulter's frame (assume that  $t_A' = 0$).\n- (b). At  $t_B'$ in the vaulter's frame locate both ends of the rod in  $S$.\n- (c). At  $t_A' = 0$ in the vaulter's frame locate both ends of the rod in  $S$.\n- (d). Sketch the rod and the barn in the   $S'$  frame at   $t_{A}'$  and   $t_{B}'$ , showing qualitatively how the pole vaulter understands the sequence of events.\n\n# Problem 5: “Lorentz-Einstein Transformation Along an Angle to the X-Axis” (6 pts):\n\nA frame   $S'$  moves at constant speed v with respect to the S frame, but in a direction that makes an angle   $\\theta$  with respect to the x axis (see the diagram on the last page). Find the Lorentz-Einstein transformation between these two frames. Start with the Lorentz-Einstein transformations as we have derived them, i.e., with   $\\theta = 0$ , and find the appropriate transformation when the relative motion of the two frames makes an arbitrary angle   $\\theta$  with respect to the x axis.\n\nThe easiest way to approach this problem is to carry out a rotation (rotate the velocity vector by an angle   $\\theta$ ) in the   $S'$  frame until the direction of motion of the   $S'$  frame (relative to S) is along the new   $x'$  axis. Then the usual Lorentz-Einstein transformation can be applied to find x, y, and t. However, the S frame needs to be rotated by an angle   $-\\theta$  to get back to the original orientation shown in the sketch. Thus, the operations consist of a rotation, followed by a Lorentz-Einstein transformation, followed by a counter rotation. This can easily be carried out in linear algebra by using a product of three   $3 \\times 3$  transformation matrices. The rotation matrix is given by:\n\n$$\nR (\\theta) = \\left( \\begin{array}{c c c} \\cos \\theta & \\sin \\theta & 0 \\\\ - \\sin \\theta & \\cos \\theta & 0 \\\\ 0 & 0 & 1 \\end{array} \\right), \\tag {5.1}\n$$\n\nand the Lorentz-Einstein transformation matrix for the variables x, y, and t is:\n\n$$\n\\Lambda_ {x} = \\left( \\begin{array}{c c c} \\gamma & 0 & \\gamma \\beta \\\\ 0 & 1 & 0 \\\\ \\gamma \\beta & 0 & \\gamma \\end{array} \\right). \\tag {5.2}\n$$", "source": "mit-ocw", "source_doc_id": "0012-problem-set-3-f2bce924-27a294de", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0012-problem-set-3-f2bce924-27a294de:page-0003", "text": "To reverse the direction of rotation, simply change  $\\theta$ to  $-\\theta$. Therefore the operation described above is\n\n$$\n\\Lambda = R (- \\theta) \\Lambda_ {x} R (\\theta) \\tag {5.3}\n$$\n\nProblem 6: “Transforming Maxwells Wave Equation” (3+6 pts)\n\na) Using chain rule, derive how time and space derivatives   $\\frac{\\partial f}{\\partial t}$ ,   $\\frac{\\partial f}{\\partial x}$  of a function f transform under a Lorentz transformation to a reference frame moving with relative speed v along the x-axis.\n\nb) Show that Maxwell's wave equation:\n\n$$\n\\frac {\\partial^ {2} E}{\\partial x ^ {2}} = \\frac {1}{c ^ {2}} \\frac {\\partial^ {2} E}{\\partial t ^ {2}} \\tag {6.1}\n$$\n\nis invariant under a Lorentz transformation to a reference frame moving with relative speed  $v$ along the  $x$-axis.\n\nProblem 7: “Two Events - Same Time Difference, Same Separation, Different Sequence” (3+3 pts)\n\nObserver S notes that two colored flashes of light, separated by 2400 m, occur along the positive branch of the x axis of his reference frame. A blue flash occurs first, followed after 5.00   $\\mu$ s by a red flash, the latter being the most distant from the origin of his reference frame. A second observer   $S'$  obtains exactly the same numerical values for the time difference and the absolute spatial separation between the two events but declares that the red flash occurs first. (a) What is the relative speed of   $S'$  with respect to S? (b) Which flash will   $S'$  find to be the more distant from the origin of her reference frame?\n\nProblem 8: “An Event Pair - Timelike or Spacelike” (3+3+3+1 pts)\n\nTwo events occur on the x axis of reference frame S, their spacetime coordinates being:\n\n| Event | x | t |\n| --- | --- | --- |\n| 1 | 200 m | 5.0 μs |\n| 2 | 1200 m | 2.0 μs |\n\n(a) What is the square of the spacetime interval   $(\\Delta s)^{2}$  for these two events? (b) What is the proper distance interval   $\\Delta\\sigma$  between them? (c) If two events possess a (mathematically real) proper distance interval, it should be possible to find a frame   $S'$  in which these events would be seen to occur simultaneously. Find this frame. (d) Can you calculate a (mathematically real) proper time interval   $\\Delta\\tau$  for this pair of events? (e) Would you describe this pair of events as timelike? Spacelike? Lightlike?\n\nProblem 9: “Transformation of Speeds”: (3 pts)\n\nA particle in the  $S'$ frame has velocity components  $u_x'$ and  $u_y'$. In turn,  $S'$ is moving to the right with speed  $v$. Find the speed of the particle in frame  $S$. Express your answer as  $u(u_x', u_y', v)$, without any explicit  $\\gamma$'s being included. Show—in the limit of either  $u_x'$,  $u_y'$, or  $v$ approaching  $c$—that  $u$ remains less than  $c$. Also, see problem #62, chapter 2, page 87 (Resnick & Halliday) for an interesting form of the answer which makes the limiting speed  $u$ more obvious.\n\nProblem 10: “Quasar, Quasar Burning Bright” (3+3 pts)\n\nIn the spectrum of the quasar 3C9, some of the familiar hydrogen lines appear but they are shifted so far forward toward the red that their wavelengths are observed to be three times as large as that observed in the light from hydrogen atoms at the rest in the laboratory. (a) Show that the classical Doppler equation gives a velocity of recession greater than c. (b) Assuming that the relative motion of 3C9 and the earth is entirely one of recession, find the recession speed predicted by the relativistic Doppler equation.\n\nProblem 11: (2004 Exam Question)(3 pts)\n\nThe Starship Enterprise fires two identical laser guns in opposite directions. How fast and in what direction (draw a picture) is the Enterprise moving in your frame if in your frame, the two rays make an angle of 120 degrees and are observed to have the same frequency?", "source": "mit-ocw", "source_doc_id": "0012-problem-set-3-f2bce924-27a294de", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0012-problem-set-3-f2bce924-27a294de:page-0004", "text": "Problem 12: “The Ives-Stillwell Experiment” (9 pts):\n\nNeutral hydrogen atoms are moving along the axis of an evacuated tube with a speed of   $2.0 \\times 10^{6}$  m/s. A spectrometer is arranged to receive light emitted by these atoms in the direction of their forward motion. This light, if emitted from resting hydrogen atoms, would have a measured (proper) wavelength of 486.133 nm. (a) Calculate the expected wavelength for light emitted from the forward-moving (approaching) atoms, using the exact relativistic formula. (b) By use of a mirror this same spectrometer can also measure the wavelength of light emitted by these moving atoms in the direction opposite to their motion. What wavelength is expected under this arrangement, in which the light source and the observer are -effectively- separating? (c) Calculate the difference between the average of the two wavelengths found in (a) and (b) and the unshifted (proper) wavelength. By this technique, Ives and Stillwell were able to distinguish between the predictions of the classical and the relativistic Doppler formulas.\n\nOptional Problem 13: \"To the Galactic Center!\"\n\n- (a) Can a person, in principle, travel from earth to the galactic center (which is about 28,000 ly distant) in a normal lifetime? Explain, using either time-dilation or length contraction arguments.\n- (b) What constant velocity would be needed to make the trip 30 y (proper time)?\n\nOptional Problem 14: \"What Time is it Anyway?\":\n\nObservers S and   $S'$  stand at the origins of their respective frames, which are moving relative to each other with a speed of 0.60c. Each has a standard clock, which, as usual, they set to zero when the two origins coincide. Observers S keeps the   $S'$  clock visually in sight. (a) What time will the   $S'$  clock record when the S clock records 5.00   $\\mu$ s? (b) What time will observer S actually read on the   $S'$  clock when his own clock reads 5.00   $\\mu$ s?\n\nOptional Problem 15: “The Unreachable Goal”:\n\nA spaceship, at rest in a certain reference frame S, is given a speed increment of 0.50c. It is then given a further 0.50c increment in this new frame, and this process is continued until its speed with respect to its original frame S exceeds 0.999c. How many increments does it require?\n\nOptional Problem 16: “A Philosophical Difficulty”:\n\nSuppose that A causes event B, the effect now being propagated from A to B with a speed greater than c. Show, using the relativistic velocity transformation equation, that there exists an inertial frame   $S'$ , which moves relative to S with a velocity less than c, in which the order of these events would be reversed. Hence, if concepts of cause and effect are to be preserved, it is impossible to send signals with a speed greater than that of light.\n\nSource: Problems 2, 3, 7, 8, 9, 10, 12, 13, 14, 15, 16 are taken from Basic Concepts in Relativity by Robert Resnick, David Halliday.\n\nFeedback: Roughly how much time did you spend on this problem set?", "source": "mit-ocw", "source_doc_id": "0012-problem-set-3-f2bce924-27a294de", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0012-problem-set-3-f2bce924-27a294de:page-0005", "text": "S\ny\nS'\ny'\nv\nθ\nx'\nx", "source": "mit-ocw", "source_doc_id": "0012-problem-set-3-f2bce924-27a294de", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0013-problem-set-4-5f886682-1f927e99:page-0001", "text": "# Massachusetts Institute of Technology\n\nDepartment of Physics\n\nPhysics 8.033\n\nOut: Friday 29 September 2006\n\nDue: Friday 6 October 2006\n\n# Problem Set 4\n\nDue: Friday 6 October 2006 at 4:00PM. Please deposit the problem set in the appropriate 8.033 bin, labeled with name and recitation section number and stapled as needed (3 points).\n\nReadings: Supplementary Topics A & B in Resnick (pages 188–209). No new parallel reading assignments in French.\n\nProblem 1: “Identifying the Lines and Determining the Redshift of a Quasar” (3 pts):\n\nFigure 1 is the visible spectrum of a distant quasar. Five prominent spectral lines can be seen in the spectrum. Also provided is a table of the laboratory wavelengths of the most observed emission lines in the spectra of quasars. Identify the lines and calculate the redshift   $z = (\\lambda - \\lambda_{0}) / \\lambda_{0}$  of this quasar. (  $\\lambda$  is the wavelength measured in the laboratory).\n\nHint: First measure   $\\lambda$  for the five lines in the spectrum. Then look for five lines in the table such that   $\\lambda/\\lambda_{0}$  has a nearly constant value.", "source": "mit-ocw", "source_doc_id": "0013-problem-set-4-5f886682-1f927e99", "source_title": "Massachusetts Institute of Technology: Problem Set 4", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Problem Set 4", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0013-problem-set-4-5f886682-1f927e99:page-0002", "text": "| λ    | F(λ_obs) |\n| ---- | -------- |\n| 3400 | ~2.0     |\n| 4000 | ~1.5     |\n| 4600 | ~1.2     |\n| 5200 | ~0.8     |\n| 5800 | ~0.6     |\n| 6400 | ~0.4     |\n| 7000 | ~0.2     |\n\nTABLE I\nLaboratory wavelengths of the most observed emission lines in the spectra of quasars\n| $\\lambda_{\\text{lab}}$ | ion |\n| --- | --- |\n| 6563 Å | H alp |\n| 5007 | [O III] |\n| 4956 | [O III] |\n| 4861 | H bet |\n| 4363 | [O III] |\n| 4340 | H gam |\n| 4102 | H del |\n| 3969 | H eps |\n|  | [Ne III] |\n| 3869 | [Ne III] |\n| 3728 | [O II] |\n| 2799 | Mg II |\n| 1909 | C III] |\n| 1549 | C IV |\n| 1402 | Si IV |\n|  | O IV] |\n| 1241 | N v |\n| 1216 | Ly alp |", "source": "mit-ocw", "source_doc_id": "0013-problem-set-4-5f886682-1f927e99", "source_title": "Massachusetts Institute of Technology: Problem Set 4", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Problem Set 4", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0013-problem-set-4-5f886682-1f927e99:page-0003", "text": "Problem 2: “Relativistic Jets from the Astrophysical Object SS433” (3+3+3+3):\n\nThe astronomical object SS433 is a binary system consisting of a collapsed star orbiting a normal companion star from which it accretes matter. This process results in the expulsion of two relativistic beams (“jets”) of matter which leave the system in opposite directions (180° apart). The atoms in these beams emit characteristic spectral lines which can be used to measure the Doppler shifts of these beams as a function of time. The time varying nature of the Doppler curves indicates that the two beams are periodically precessing so that we are viewing them from different angles at different times. The two intersecting Doppler curves correspond to the two oppositely directed beams.\n\n- (a). Determine the Doppler shift   $(\\lambda - \\lambda_{0})/\\lambda_{0}$  at the times when both beams exhibit the same wavelength.\n- (b). Show from the relativistic Doppler equation  $\\lambda / \\lambda_0 = \\gamma (1 - \\beta \\cos \\theta)$ that this will occur only when  $\\theta = 90^{\\circ}$ (i.e., when viewing the beams perpendicular to their direction). Note that  $\\lambda$ and  $\\theta$ are measured in the Earth's frame, while  $\\lambda_0$ is measured in the frame of the moving beam.\n- (c). Use the results of (a) and (b) to find  $\\beta$ for the beam.\n- (d). Measure the maximum Doppler shift for the precessing beam. Use the relativistic Doppler equation and the value of   $\\beta$  from part (c) to find the maximum angle between the beam and the direction to the Earth.\n\n| Julian day | 1979 | 1980 | 1981 | 1982 | 1983 |\n| ---------- | ---- | ---- | ---- | ---- | ---- |\n| 4600       | -0.10 | 0.00 | 0.05 | -0.10 | 0.00 |\n| 4800       | 0.15 | 0.15 | 0.15 | -0.10 | 0.05 |\n| 5000       | -0.10 | 0.00 | 0.05 | -0.10 | 0.15 |\n| 5200       | 0.15 | 0.15 | 0.15 | -0.10 | 0.05 |\n| 5400       | -0.10 | 0.00 | 0.05 | -0.10 | 0.15 |\n| 5600       | 0.15 | 0.15 | 0.15 | -0.10 | 0.05 |", "source": "mit-ocw", "source_doc_id": "0013-problem-set-4-5f886682-1f927e99", "source_title": "Massachusetts Institute of Technology: Problem Set 4", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Problem Set 4", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0013-problem-set-4-5f886682-1f927e99:page-0004", "text": "Problem 3: “The Headlight Effect—A High-Speed Limit” (3+3+3 pts)\n\nA source of light, at rest in the   $S'$  frame, emits uniformly in all directions. The source is viewed from frame S, the relative speed parameter relating the two frames being   $\\beta$ . (a) Show that at high speeds (that is as   $\\beta \\rightarrow 1$ ), the forward-pointing cone into which the source emits half of its radiation has a half-angle   $\\theta_{0.5}$  given closely, in radian measure by   $\\theta_{0.5} = \\sqrt{2(1 - \\beta)}$ . (b) What value of   $\\theta_{0.5}$  is predicted for the gamma radiation emitted by a beam of energetic neutral pions, for which   $\\beta = 0.993?$  (c) At what speed would light source have to move toward an observer to have half of its radiation concentrated into a narrow forward cone of half-angle   $5.0^{\\circ}$ ?\n\nHint: Recall the approximation  $(1 - \\beta) \\approx 1 / (2\\gamma^2)$.\n\nProblem 4: (2004 Exam Question (3 pts))\n\nA lamp moves past you in a straight line. At what angle relative to its direction of motion (draw a picture and label this angle!) do you see it when it is neither blueshifted nor redshifted?\n\nProblem 5: (2004 Final Question (3+4+1+1+3 pts))\n\nConsider three intertial frames, S,   $S'$ , and   $S''$ .   $S'$  moves with speed   $v' = \\frac{3}{5}c$  along the x-axis of frame S and   $S''$  moves with speed   $v'' = \\frac{4}{5}c$  along the   $y'$ -axis of frame   $S'$ . The origins and orientations of the frames are such that they coincide at time   $t = t' = t'' = 0$  and the origin   $x = x' = x'' = y = y' = y'' = z = z' = z'' = 0$ .\n\n(a). Write down the Lorentz transformations for  $S \\to S'$ and  $S' \\to S''$, i.e., express  $(x', y', z', ct')$ as functions of  $(x, y, z, ct)$ and  $(x'', y'', z'', ct'')$ as functions of  $(x', y', z', ct')$. You may find it convenient to express your two transformations in matrix form.)\n\n(b). Complete the following table of coordinates for the event  $A = (1,2,3,4)$:\n\n| Frame | x | y | z | ct |\n| --- | --- | --- | --- | --- |\n| S | 1 | 2 | 3 | 4 |\n| S' |  |  |  |  |\n| S'' |  |  |  |  |\n\n(c). A rocket of proper length  $L_0$ moves along the  $x$-axis of frame  $S$ with speed  $u = \\frac{3}{5} c$. What is its velocity in frame  $S'$?\n\n(d). How long is it in frame  $S'$?\n\n(e). List three pieces of observational evidence supporting special relativity.\n\nProblem 6: “Bob Is Older Than Dave This Time” (3+3+3+3 pts)\n\nHere is a different version of the twin paradox, where the twins are Bob and Dave:\n\nBob, once started on his outward journey from Dave, keeps on going at his original uniform speed of 0.8c. Dave, knowing that Bob was planning to do this, decides, after waiting for three years, to catch up with Bob and to do so in three additional years. (a) To what speed must Dave accelerate to do so? (b) What will be the elapsed time by Bob's clock when they meet? (c) How far will they each have traveled when they meet, measured in Dave's original reference frame? (d) Who is older and by how much when they meet?\n\nProblem 7: “Bob And Dave Are Twins Again” (3 pts)\n\nBob travels away from his brother Dave with speed 0.8c and Dave stays at rest. After three years, each counting the years by his own onboard clock, Bob come to rest and Dave will accelerate to 0.8c and eventually catch up with Bob. What will be the total elapsed times on each of their clocks when they meet?\n\nProblem 8: (3+3+3 pts) “More General Version of the Twin Paradox”: Two clocks start at rest at the same point in space and are set to the same time in frame S. Clock A remains at rest in the S reference frame. Clock B takes an accelerated trip along the x-axis of total duration T as recorded by Clock A, and ends up at rest in S, at the location of Clock A. The trip can be broken up into 6 segments which are described below. All times, velocities, and accelerations are measured in S. At the end of the trip, Clock B shows an elapsed time   $T'$ .", "source": "mit-ocw", "source_doc_id": "0013-problem-set-4-5f886682-1f927e99", "source_title": "Massachusetts Institute of Technology: Problem Set 4", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Problem Set 4", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0013-problem-set-4-5f886682-1f927e99:page-0005", "text": "# Outbound portion of trip (segments 1 through 3):\n\n- 1. Clock B undergoes a constant acceleration  $a_0$ for time  $fT / 4$, where  $f$ is the fraction of the entire trip which is spent accelerating.\n- 2. Clock B then coasts at constant speed   $\\beta$  for time   $(1 - f)T/2$ .\n- 3. Clock B decelerates at   $-a_{0}$  for time fT/4, and comes to rest.\n\nReturn portion of trip (segments 4 through 6): Clock B undergoes all the same motions as in the outbound trip, but in reverse order, and ends up at rest in S, at the location of Clock A. You can make use of the symmetry in the problem by noting that steps 4-6 take a time equal to steps 1-3.\n\n- (a). Compute   $T'/T$  as a function of   $a_{0}$ ,   $\\beta$ , abd f.\n- (b). Show that in the limit of  $f \\to 1$ (i.e., no coasting period), and  $\\beta$ approaches unity, the ratio  $T' / T \\approx \\pi / 4$.\n- (c). Show that in the limit of   $f \\rightarrow 0$  (i.e., no significant time spent during the acceleration), and   $\\beta$  still large (i.e., large accelerations at the turning points), the “standard” result for the twin paradox is recovered.\n\nNote: the 8.01 relations   $v = v_{0} + at$  and   $x = x_{0} + v_{0}t + at^{2}/2$  still hold in S.\n\nProblem 9: “Geodesic on a Sphere”   $(3+3+3+3$  pts): Use the calculus of variations to find the curve representing the shortest path between any two points on a unit sphere (of radius 1).\n\n(a). Show that the infinitesimal distance between two neighboring points is given by:\n\n$$\n\\mathrm{d} s = (\\mathrm{d} \\theta^ {2} + \\sin^ {2} \\theta \\mathrm{d} \\phi^ {2}) ^ {1 / 2}. \\tag {9.1}\n$$\n\n(b). Take the independent variable to be  $\\theta$. Use Euler's equation (of the first form) to find the following differential equation:\n\n$$\n\\frac {\\mathrm{d} \\phi}{\\mathrm{d} \\theta} = \\frac {a \\csc^ {2} \\theta}{(1 - a ^ {2} \\csc^ {2} \\theta) ^ {1 / 2}}, \\tag {9.2}\n$$\n\nwhere a is the first constant of integration.\n\n(c). Integrate this equation to find:\n\n$$\n\\cot \\theta = - \\frac {\\sqrt {1 - a ^ {2}}}{a} \\sin (\\phi - \\alpha), \\tag {9.3}\n$$\n\nwhere   $\\alpha$  is the second constant of integration.\n\n(d). Show that the result of part (c) can be written in the form:\n\n$$\nA y - B x = z, \\tag {9.4}\n$$\n\nin Cartesian coordinates. Note that this is a plane passing through the center of the sphere and the two chosen points on the surface. Use this to deduce the nature of a geodesic curve on a sphere.\n\nUseful relations:\n\n$$\nx = \\sin \\theta \\cos \\phi , \\quad y = \\sin \\theta \\sin \\phi , \\quad z = \\cos \\theta . \\tag {9.5}\n$$\n\nOptional Problem 10: “Fresnel Drag Coefficient”: Use the relativistic addition of velocities to compute the “Fresnel drag coefficient”. Start with the fact that the speed of light in a medium with index of refraction n is c/n. Compute how fast the light appears to travel with respect to a frame in which the medium is moving with speed v (in the same direction as the light is propagating). Show that to the lowest order in v/c this speed is given by   $u = c/n + v(1 - 1/n^{2})$ . The term in brackets on the right hand side is called the “Fresnel drag coefficient”, although there was never any need to assume a “drag” mechanism in the first place. Typical indices of refraction for water and glass are   $n \\sim 1.5$ .\n\nThe Fresnel prediction (1817) of light “drag” by a moving medium, and the experimental “verification” of this effect by Fizeau (1851) are discussed on pages 30–33 of Resnick.", "source": "mit-ocw", "source_doc_id": "0013-problem-set-4-5f886682-1f927e99", "source_title": "Massachusetts Institute of Technology: Problem Set 4", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Problem Set 4", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0013-problem-set-4-5f886682-1f927e99:page-0006", "text": "Optional Problem 11: “Calibration by Hyperbolas”\n\n(2003 Final Question)\n\nMinkowski Diagrams: The quadrant of the Minkowski diagram shown on the drawing below contains the ct and x axes of the S frame, as well as the   $ct'$  and   $x'$  axes of an   $S'$  frame moving with speed   $\\beta = 4/5$ , (  $\\gamma = 5/3$ ). Point A on the   $x'$  axis has a coordinate (  $x', ct'$ ) = (1,0). The figure is drawn accurately to scale.\n\n| x    | ct     |\n| ---- | ------ |\n| 0    | 0      |\n| 1    | 1.3    |\n| 2    | 2.5    |\n| 3    | 3.7    |\n| 4    | 4.9    |\n| 5    | 6.1    |\n\n- (i) Line segment AD is parallel to the ct axis. Use the Lorentz transformations to find the coordinate x of point D in S. Confirm your answer by reading the value from the graph.\n- (ii) Line segment AC is parallel to the  $ct'$ axis. Use the Lorentz transformations to find the coordinate  $x$ of point C in  $S$. Confirm your answer by reading the value from the graph.\n- (iii) The heavy curve that connects points B and A and its extrapolation is designed so as to intersect the  $x$ axis at a value of 1.0. Furthermore, it would intersect any other  $x''$ axis drawn on the figure (corresponding to  $\\beta''$) at a value of  $x'' = 1$. Find the equation,  $x(t)$, of this curve.\n\nSource: Problems 3, 6, 7, 11 are taken from Basic Concepts in Relativity by Robert Resnick, David Halliday.\n\nFeedback: Roughly how much time did you spend on this problem set?", "source": "mit-ocw", "source_doc_id": "0013-problem-set-4-5f886682-1f927e99", "source_title": "Massachusetts Institute of Technology: Problem Set 4", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 5, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Problem Set 4", "Page 6"], "page_start": 6, "page_end": 6, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0014-problem-set-5-fa912778-235d3d76:page-0001", "text": "# Problem Set 5\n\nDue: Friday 20 October 2006 at 4:00PM. Please deposit the problem set in the appropriate 8.033 bin, labeled with name and recitation section number and stapled as needed (3 points).\n\nReadings: Relativistic Dynamics Chapters in Resnick and French.\n\nProblem 1: “The Great Neutron Shootout” (3+3+3+3+3 points)\n\nSue and Jim are two experimenters at rest with respect to one another at different points in space. They “fire” neutrons at each other, each neutron leaving its “gun” with a relative speed of 0.60c. Jim makes five observations about what is going on: (a) “My separation from Sue is 10 km.” (b) “The speed of ‘my’ neutron is 0.60c.” (c) “Two of our neutrons have collided; relativistic momentum and kinetic energy are conserved.” (d) “After this collision, one of the neutrons was scattered through an angle of   $30^{\\circ}$ .” (e) “I am firing neutrons at the rate of   $10,000 s^{-1}$ .”\n\nFor each of these observations, state the corresponding observation that would be reported by a third person (Sam, say), who is in a frame  $S$ chosen so that Sue's neutrons are at rest in it.\n\nProblem 2: “A Ton of Sunlight” (3 points)\n\nThe Sun radiates energy at the rate of   $4.0 \\times 10^{26}$  W. How many “tons of sunlight” (mass equivalent) does the Earth intercept each day?\n\nProblem 3: “An Enormous Source of Energy” (3 points)\n\nQuasars are thought to be the nuclei of active galaxies in the early stages of their formation. A typical quasar radiates energy at the rate of   $10^{41}$  W. At what rate is the mass of this quasar being reduced to supply this energy? Express your answer in solar mass units per year, where one solar mass unit, (smu=2 ×   $10^{30}$  kg), is the mass of our sun.\n\nProblem 4: (9 points) “Relativistic Transformation of Acceleration”: Starting from the definition of acceleration,   $a = du/dt$ , derive the x and y components of acceleration in an arbitrary inertial frame   $S'$  moving with speed v along the x-axis of S. The answers are:\n\n$$\na _ {x} = \\frac {a _ {x} ^ {\\prime}}{\\gamma^ {3} (1 + v u _ {x} ^ {\\prime} / c ^ {2}) ^ {3}}, \\tag {4.1}\n$$\n\n$$\na _ {y} = \\frac {a _ {y} ^ {\\prime}}{\\gamma^ {2} (1 + v u _ {x} ^ {\\prime} / c ^ {2}) ^ {2}}, \\tag {4.2}\n$$\n\nwhere the expression for   $a_{y}$  is valid only for the case where either   $u_{y}^{\\prime}=0$  or   $a_{x}^{\\prime}=0$ . In general, the transformation of   $a_{y}$  is rather messy.\n\nProblem 5: “Constant Acceleration”: (6 points) French, problem #7-8, Chapter 7, page 226. Hint: You should end up solving   $d(\\gamma u)/dt = g$  to find   $u(t)$ ; then integrate   $u dt = dx$  to find   $x(t)$ .\n\nProblem 6: “Relativistic Rocket Problem”:   $(3+3+6+6+3$  points)\n\n(a). Show that in the rest frame of the rocket, for non-relativistic dynamics:\n\n$$\n\\Delta M _ {0} V _ {\\mathrm{ex}} = - M _ {0} \\Delta u ^ {\\prime} \\tag {6.1}\n$$", "source": "mit-ocw", "source_doc_id": "0014-problem-set-5-fa912778-235d3d76", "source_title": "Problem Set 5", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Problem Set 5", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0014-problem-set-5-fa912778-235d3d76:page-0002", "text": "or equivalently\n\n$$\n\\frac {\\mathrm{d} M _ {0}}{\\mathrm{d} t ^ {\\prime}} V _ {\\mathrm{ex}} = - M _ {0} \\frac {\\mathrm{d} u 0}{\\mathrm{d} t ^ {\\prime}}, \\tag {6.2}\n$$\n\nwhere   $M_{0}$  is the rest mass of the rocket,   $\\Delta M_{0}$  is the differential mass loss in the exhaust,   $V_{ex}$  is the speed of the rocket exhaust (in the rocket's frame,   $S'$ ), and   $\\Delta u'$  is the change in the rocket's speed after ejecting   $\\Delta M_{0}$ . This is correct unless   $V_{ex}$  is comparable with the speed of light.\n\n(b). Consider first non-relativistic motion of the rocket in a fixed (stationary frame). Note that in the non-relativistic case  $a = a'$ and  $F = F'$. Thus, the rocket equation in  $S$ is:\n\n$$\n\\frac {\\mathrm{d} M _ {0}}{\\mathrm{d} t} V _ {\\mathrm{ex}} = - M _ {0} \\frac {\\mathrm{d} u}{\\mathrm{d} t}. \\tag {6.3}\n$$\n\nIf the rate of mass loss in the exhaust is known as a function of time, then the above expression can be directly integrated to find   $u(t)$  since   $M_{0}(t)$  is known. Whether or not the rate of exhaust ejection is known, the dt's can be canceled on both sides and the equation integrated to find u as a function of the initial and final masses of the rocket. Show that this leads to:\n\n$$\nu = V _ {\\mathrm{ex}} \\ln \\left(M _ {\\text { initial }} / M _ {\\text { final }}\\right). \\tag {6.4}\n$$\n\n(c). Suppose that the rocket in part (b) is firing its engine vertically in a gravitational field. Further assume that the atmospheric drag is negligible (not the case in the Earth's atmosphere, but okay on the Moon), and that the rocket motor is on while the rocket covers a distance that is relatively short compared with the radius of the gravitating body—such that the acceleration due to gravity, g, can be taken to be approximately constant. Show that if the rocket starts from rest, and ejects its exhaust uniformly in time (i.e.,   $dM_{0}/dt = \\dot{M}_{0} = \\text{constant}$ , and   $M_{0} = M_{0,\\text{initial}} + \\dot{M}_{0}t$ ), the rocket equation becomes:\n\n$$\n\\frac {\\mathrm{d} u}{\\mathrm{d} t} = - \\frac {V _ {\\mathrm{ex}} \\dot {M}}{M _ {\\text {initial}} + \\dot {M} t} - g. \\tag {6.5}\n$$\n\nShow that if the rocket is fired for a total time  $T$, the above equation is easily integrated to yield  $u(t)$:\n\n$$\nu (T) = V _ {\\mathrm{ex}} \\ln \\left[ \\frac {M _ {\\text {initial}}}{M _ {\\text {initial}} + \\dot {M} T} \\right] - g T = V _ {\\mathrm{ex}} \\ln \\left[ \\frac {M _ {\\text {initial}}}{M _ {\\text {final}}} \\right] - g T. \\tag {6.6}\n$$\n\n(d). Next, consider the case where the rocket motion is at least somewhat relativistic, but the exhaust velocity is not. Then, the equation in part (a) tells us that in  $S'$ the rocket acceleration is:\n\n$$\na ^ {\\prime} = \\frac {\\mathrm{d} u ^ {\\prime}}{\\mathrm{d} t ^ {\\prime}} = - \\frac {V _ {\\mathrm{ex}}}{M _ {0}} \\frac {\\mathrm{d} M _ {0}}{\\mathrm{d} t ^ {\\prime}}. \\tag {6.7}\n$$\n\nNow, transform the acceleration to the S frame to find:\n\n$$\na = \\frac {\\mathrm{d} u}{\\mathrm{d} t} = - \\frac {1}{\\gamma^ {3}} \\frac {V _ {\\mathrm{ex}}}{M _ {0}} \\frac {\\mathrm{d} M _ {0}}{\\mathrm{d} t ^ {\\prime}}. \\tag {6.8}\n$$\n\nFrom time dilation show that:\n\n$$\n\\frac {\\mathrm{d} u}{\\mathrm{d} t} = - \\frac {1}{\\gamma^ {2}} \\frac {V _ {\\mathrm{ex}}}{M _ {0}} \\frac {\\mathrm{d} M _ {0}}{\\mathrm{d} t}. \\tag {6.9}\n$$\n\nFinally, cancel the dt's and integrate to find:\n\n$$\n\\frac {u}{c} = \\frac {\\left(\\frac {M _ {i}}{M _ {f}}\\right) ^ {2 V / c} - 1}{\\left(\\frac {M _ {i}}{M _ {f}}\\right) ^ {2 V / c} + 1}, \\tag {6.10}\n$$\n\nwhere V,   $M_{i}$ , and   $M_{f}$  are shorthand for   $V_{ex}$ ,   $M_{initial}$ , and   $M_{final}$ , respectively.\n\n(e). Relating the answers from parts (b) and (d): Show that for exhaust speeds much lower than c, the answer from part (d) matches the non-relativistic expression in part (b). Show that unless   $V_{ex}$  is a substantial fraction of the speed of light, the rocket will never attain truly relativistic speeds.", "source": "mit-ocw", "source_doc_id": "0014-problem-set-5-fa912778-235d3d76", "source_title": "Problem Set 5", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Problem Set 5", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0014-problem-set-5-fa912778-235d3d76:page-0003", "text": "Optional Problem 7: “Human Power Output Per Unit Mass vs. That of the Sun”: Suppose that a “typical” person of mass   $\\sim$  60 kg, puts out 60 Watts of power while at rest. By contrast, the Sun has a luminosity (power output) of   $4 \\times 10^{26}$  Watts, and mass of   $2 \\times 10^{30}$  kg. Compare the power generated per unit mass of the Sun and a person. How do you make sense of this result?\n\nOptional Problem 8: “Powering the Sun by Gravitational Potential Energy”: Suppose that the luminosity of the Sun were powered by the release of gravitational potential energy only. Its power output is   $4 \\times 10^{26}$  Watts. If the Sun contracted from its current radius of   $R_{\\odot} = 7 \\times 10^{8}$  m, to half that value, how long would the contraction have to last in order to supply the observed power output? Assume that the gravitational potential energy of the Sun is   $\\sim GM_{\\odot}^{2}/R$ , where   $M_{\\odot} = 2 \\times 10^{30}$  kg, and R is the radius at any given time.\n\nOptional Problem 9: “Snell’s Law Derived From Fermat’s Principle”: Fermat’s Principle states that the path of a light ray from one point to another is that which requires the least time. Suppose that light is emitted by a source at   $(x = 0, y = y_{0})$  in air, heads in the direction of a semi-infinite slab of glass located at y = 0, and enters the glass such as to reach the point   $(x = x0, y = -y_{0})$ . The geometry is shown in the diagram. Take the speed of light to be c in air, and c/n in glass, where n is called the “index of refraction”, and is typically a number in the range of 1.4-1.6.\n\nAssume, from Fermat's Principle, the obvious result that within any given medium (i.e., of fixed  $n$) light travels in a straight line. Then use Fermat's Principle to prove Snell's law:  $\\sin \\theta_i = n \\sin \\theta_t$. You need not bother with the calculus of variations for this problem.\n\nHint: take the distances traveled to be   $y_{0}/\\cos\\theta_{i}$  and   $y_{0}/\\cos\\theta_{t}$  in the air and glass, respectively. Then note that   $x_{0}=y_{0}(\\tan\\theta_{i}+\\tan\\theta_{t})$ , which defines the constraint between   $\\theta_{i}$  and   $\\theta_{t}$ .\n\nOptional Problem 10: “Lagrange’s Equations of Motion”: Marion & Thornton’s book on “Classical Dynamics” introduces Lagrangian mechanics with the following two instructive statements:\n\n- 1. “Minimal principles in physics have a long and interesting history. The search for such principles is predicated on the notion that nature always minimizes certain important quantities when a physical process takes place.”\n- 2. In two papers in 1834 and 1835, Hamilton announced the dynamical principle on which it is possible to base all of mechanics and, indeed, most of classical physics. Hamilton's Principle may be stated as follows: \"Of all the possible paths along which a dynamical system may move from one point to another within a specified time interval (consistent with any constraints), the actual path followed is that which minimizes the time integral of the difference between the kinetic and potential energies.\"\n\nLet's use this principle and the Euler equations for extremizing path integrals to derive Lagrange's equations of motion and solve a couple of simple mechanics problems. First, identify the “Lagrangian” as the difference between the kinetic and potential energies:\n\n$$\n\\mathcal {L} \\equiv K - V, \\tag {10.1}\n$$\n\nwhere  $K$ and  $V$ are the kinetic and potential energies, respectively. Show that if  $K$ and  $V$ depend on coordinates  $q_k$, then the equations that satisfy Hamilton's principle are:\n\n$$\n\\frac {\\partial \\mathcal {L}}{\\partial q _ {k}} - \\frac {\\mathrm{d}}{\\mathrm{d} t} \\frac {\\partial \\mathcal {L}}{\\partial \\dot {q} _ {k}} = 0. \\tag {10.2}\n$$\n\n- (a). As a first illustrative mechanics problem, consider a mass  $M$ suspended (in gravity) on a spring with a spring constant  $k$. Call the vertical direction  $z$. Take the position of the mass to be  $z = 0$ when the spring is neither stretched nor compressed. What is the potential energy as a function of  $z$. Write down the kinetic energy in terms of  $\\dot{z}$. Use Lagrange's equation to find the equation of motion of the mass. How does the equation compare with that found by starting with  $F = ma$?\n- (b). As a second illustrative problem, consider a pendulum consisting of a mass,  $M$, suspended on a string of length  $l$. Dispace the pendulum by an angle  $\\phi$ with respect to the vertical and release the mass. Use Lagrange's equation to describe the subsequent motion of the mass. The problem is effectively one-dimensional in that a complete description of the motion is given in terms of a single dependent variable  $(\\phi)$ and its derivative  $(\\dot{\\phi})$. Simple analytic solutions to this equation are found by making a small-angle approximation, i.e.,  $\\sin \\phi \\approx \\phi$. You also made the same approximation in the 8.01 version of this problem.", "source": "mit-ocw", "source_doc_id": "0014-problem-set-5-fa912778-235d3d76", "source_title": "Problem Set 5", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Problem Set 5", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0014-problem-set-5-fa912778-235d3d76:page-0004", "text": "Feedback: Roughly how much time did you spend on this problem set?", "source": "mit-ocw", "source_doc_id": "0014-problem-set-5-fa912778-235d3d76", "source_title": "Problem Set 5", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Problem Set 5", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0015-problem-set-6-e0639d49-c6764ef3:page-0001", "text": "# Massachusetts Institute of Technology\n\n# Department of Physics\n\nPhysics 8.033\n\nOut: Friday 20 October 2006\n\nDue: Friday 27 October 2006\n\n# Problem Set 6\n\nDue: Friday 27 October 2006 at 4:00PM. Please deposit the problem set in the appropriate 8.033 bin, labeled with name and recitation section number and stapled as needed (3 points).\n\nReadings: Chapter 3 in Resnick and Chapter 6 and 7 in French.\n\nProblem 1: (2004 Exam Question)  $(5 + 3 + 1 + 1 + 1$ points)\n\n- (a). A particle of rest mass  $m_0$ moving with Lorentz factor  $\\gamma$ collides with an identical particle that is initially at rest. Compute the total (rest+kinetic) energy  $E_{\\mathrm{tot}}$ of the two-particle system both in the lab frame  $S$ and in the CM frame  $S'$. Compute the ratio  $E_{\\mathrm{tot}} / E_{\\mathrm{tot}}'$.\n- (b). CERN's Large Hadron Collider will use colliding particle beams rather than beams hitting fixed targets to make new forms of matter — in one sentence, why is this a good idea based on your result in (a)?\n- (c). In the Hindenburg blimp disaster, the fire was dominated by the chemical reaction   $2H_{2}+O_{2}\\rightarrow2H_{2}O$ . Based on this fact, the combined rest masses of two hydrogen molecules and one oxygen molecule is (circle one) SMALLER/LARGER than that of a two water molecules.\n- (d). Order the following particles by increasing rest mass: proton, neutron, photon, Z-boson, electron.\n- (e). You consist of quarks called ____ and ____ and leptons called ____.\n\nProblem 2: (2003 Final Question) (4 points)\n\nConsider the following reactions:\n\n- (a). ____ An isolated 1 GeV photon decays into an electron-positron pair.\n- (b). ____ A neutron decays into a proton and an electron\n- (c). ____ A neutron decays into an electron-positron pair\n- (d). ____ A neutron decays into a proton and a photon\n\nFor each one, write one of the letters from the option list below.\n\n- L violates lepton number conservation\n- B violates baryon number conservation\n- P violates parity conservation\n- E violates energy-momentum conservation\n- Q violates charge conservation\n- N violates none of the above conservation laws\n\nProblem 3: Concept questions (6 points): For each of these pairs, circle the one with the largest rest mass.\n\n- (a). (A) a proton\n- (B) a neutron\n\n(b). (A) a proton\n\n(B) 1000 electrons at rest far from each other", "source": "mit-ocw", "source_doc_id": "0015-problem-set-6-e0639d49-c6764ef3", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0015-problem-set-6-e0639d49-c6764ef3:page-0002", "text": "- (c). (A) a 10 GeV muon\n- (B) a 100 GeV electron\n\n(d). two neutrons and two protons in a fixed arrangement where all pairs are separated by of order\n\n- (A) 1 m\n- (B)   $10^{-10}$ m\n\n(e). two neutrons and two protons arranged so that all pairs are separated by of order\n\n- (A) 1 m\n- (B)  $10^{-15}\\mathrm{m}$ (bound as an  $\\alpha$-particle, i.e., a Helium nucleus)\n\n(f). (A) two hydrogen molecules  $(H_{2})$ at rest and one oxygen molecule  $(O_{2})$ at rest\n\n(B) two water molecules at rest\n\nFor the systems involving more than one particle, interpret the rest mass of the system as its total relativistic mass   $E/c^{2}$  in the frame where the total momentum is zero. When answering these questions, you can ignore any kinetic energy due to internal motions (from the particles in the system moving relative to each other).\n\nProblem 4: “Transforming a photon” (3+3+3+3+3 points):\n\n- (a). Write down the momentum 4-vector  $\\mathbf{P}$ for a photon of angular frequency  $\\omega$ traveling in an arbitrary direction specified by angles  $\\theta$ and  $\\phi$ in polar coordinates.\n- (b). Write down an equation relating  $\\mathbf{P}$ to the wave 4-vector  $\\mathbf{K}$ for light of the same frequency traveling in this same direction.\n- (c). Compute the momentum 4-vector  $\\mathbf{P}'$ in a frame  $S'$ that is moving with velocity  $v = \\beta c$ in the  $z$-direction.\n- (d). Derive the energy ratio  $E' / E$ and the new direction of motion ( $\\phi'$ and  $\\theta'$) in terms of  $\\phi$,  $\\theta$ and  $\\beta$.\n- (e). Comment on any similarities between your results and the Doppler effect and aberration formulas that we derived in class.\n\nProblem 5: This problem appeared on the second 8.033 quiz in 2001. (6+3 points)\n\nA particle with rest mass   $M_{0}$  is kept moving in a circular orbit of fixed radius R, by a magnetic field, B, whose field lines are perpendicular to the orbit. A force of constant amplitude F is applied to the particle in a direction that is always along its velocity vector (i.e., in the   $\\theta$  direction). As the particle is accelerated to higher   $(\\gamma,\\beta)$ , the magnetic field strength is adjusted so that the particle continues to move in its orbit with constant radius R.\n\n- (a). Starting with  $\\vec{F} = \\mathrm{d}\\vec{p} / \\mathrm{d}t$, derive an equation for the increase in particle speed,  $u$, with time. Express your answer in terms of  $\\mathrm{du} / \\mathrm{dt}$ equals a function of  $\\gamma$,  $M_0$, and  $F$. You need not solve the equation.\n- (b). Find how the magnetic field must change as a function of  $\\gamma$. Note, this part of the problem can be done independently of part (a).\n\nProblem 6: Photoelectric effect: “Calculating the Planck Constant and More”:   $(3+3+3$  points)\n\nIn a photoelectric experiment in which a sodium surface is used, one finds a stopping potential of 1.85 V for a wavelength of 300 nm and a stopping potential of 0.82 V for a wavelength of 400 nm. From these data find (a) a value for the Planck constant, (b) the work function for sodium and (c) the cutoff wavelength for sodium.\n\nProblem 7: “Photon Rocket” (9 points): Review and discuss French’s Photon Rocket problem on pages 183-184. Derive a quadratic equation for f, the payload’s mass as a fraction of the rest mass of the rocket. How feasible is the photon rocket? Note that he seems to have written this in a hurry and gets into lots of totally unnecessary algebra towards the end.", "source": "mit-ocw", "source_doc_id": "0015-problem-set-6-e0639d49-c6764ef3", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0015-problem-set-6-e0639d49-c6764ef3:page-0003", "text": "Optional Problem 8: “More on the Calculus of Variations”: Consider two points in the x - y plane as shown in the sketch:   $(x_{1}, y_{1})$  and   $(x_{2}, y_{2})$ . The two points are connected by a curve specified by   $y(x)$ . This curve is then rotated about the y axis to form a surface. Find the shape of the curve   $y(x)$  that minimizes the area of the resultant figure of rotation.\n\ny\n(x2,y2)\ny(x)\n(x1,y1)\nx\n\n- (a). Show that the infinitesimal arc length along the path is   $ds = dx \\sqrt{1 + (dy/dx)^2}$ .\n- (b). Show that the integral for the surface area of the figure of rotation is:\n\n$$\nA = 2 \\pi \\int_ {x _ {2}} ^ {x _ {1}} x \\sqrt {1 + y ^ {\\prime 2}} \\mathrm{d} x, \\tag {8.1}\n$$\n\nwhere   $y' = dy/dx$ .\n\n(c). Apply the appropriate Euler equation:\n\n$$\n\\frac {\\partial f}{\\partial y} - \\frac {\\mathrm{d}}{\\mathrm{d} x} \\frac {\\partial f}{\\partial y ^ {\\prime}} = 0 \\tag {8.2}\n$$\n\nto find the differential equation for the shape of the curve  $y(x)$. Show that\n\n$$\n\\frac {x y ^ {\\prime}}{\\sqrt {1 + y ^ {\\prime 2}}} = a = \\text { constant }. \\tag {8.3}\n$$\n\n(d). Solve for  $\\mathrm{dy} / \\mathrm{dx}$ and show that the result is:\n\n$$\ny = \\int \\frac {a}{\\sqrt {x ^ {2} - a ^ {2}}} \\mathrm{d} x. \\tag {8.4}\n$$\n\n(e). Look up the integral and show that it can be written as:\n\n$$\ny = a \\cosh^ {- 1} \\left(\\frac {x}{a}\\right) + b, \\tag {8.5}\n$$", "source": "mit-ocw", "source_doc_id": "0015-problem-set-6-e0639d49-c6764ef3", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0015-problem-set-6-e0639d49-c6764ef3:page-0004", "text": "or\n\n$$\nx = a \\cosh \\left(\\frac {y - b}{a}\\right), \\tag {8.6}\n$$\n\nwhere b is a second constant of integration. This function is called a “catenary”. It has the same shape as a flexible chain hanging freely between two points of support.\n\nFeedback: Roughly how much time did you spend on this problem set?", "source": "mit-ocw", "source_doc_id": "0015-problem-set-6-e0639d49-c6764ef3", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0016-problem-set-7-330fe5ae-04e389d4:page-0001", "text": "# Massachusetts Institute of Technology\n\n# Department of Physics\n\nPhysics 8.033\n\nOut: Friday 27 October 2006\n\nDue: Friday 3 November 2006\n\n# Problem Set 7\n\nDue: Friday 3 November 2006 at 4:00PM. Please deposit the problem set in the appropriate 8.033 bin, labeled with name and recitation section number and stapled as needed (3 points).\n\nReadings: Chapter IV in Resnick and Chapter 8 in French. (As stressed in lecture, please don't get bogged down with their excessive E&M algebra.)\n\nProblem 1: Key concepts (3+3+3 points):\n\n- (a). Professor D. Ubious tells you that he has observed a 1 GeV photon decay into an electron-positron pair (there was no other particle nearby and no external electromagnetic field). Is this consistent with relativity and standard particle physics? If not, explain which conservation law it violates and give an explicit calculation if needed. (Hint: if you find yourself making a nontrivial calculation, try working in an frame where things are simpler.)\n- (b). Consider a neutron undergoing  $\\beta$-decay. For each of the three decay products, list at least one conservation law that would be violated if it weren't produced.\n- (c). Give an inventory of a water molecule in terms of elementary particles (specify how many quarks and leptons it contains of various types).\n\nProblem 2: “Transforming the electromagnetic field” (6+3+3+3 points):\n\n(a). Resnick spends pages and pages working out how  $\\mathbf{E}$ and  $\\mathbf{B}$ are Lorentz transformed. French spends 37 pages on the subject without even obtaining this key result. Now let's see if you can derive it more elegantly. In class, we showed that Lorentz transforming the Force law gave (in c.g.s. units)\n\n$$\n\\left( \\begin{array}{c c c c} 0 & B _ {z} ^ {\\prime} & - B _ {y} ^ {\\prime} & E _ {x} ^ {\\prime} \\\\ - B _ {z} ^ {\\prime} & 0 & B _ {x} ^ {\\prime} & E _ {y} ^ {\\prime} \\\\ B _ {y} ^ {\\prime} & - B _ {x} ^ {\\prime} & 0 & E _ {z} ^ {\\prime} \\\\ E _ {x} ^ {\\prime} & E _ {y} ^ {\\prime} & E _ {z} ^ {\\prime} & 0 \\end{array} \\right) = \\boldsymbol {\\Lambda} \\left( \\begin{array}{c c c c} 0 & B _ {z} & - B _ {y} & E _ {x} \\\\ - B _ {z} & 0 & B _ {x} & E _ {y} \\\\ B _ {y} & - B _ {x} & 0 & E _ {z} \\\\ E _ {x} & E _ {y} & E _ {z} & 0 \\end{array} \\right) \\boldsymbol {\\Lambda} ^ {- 1},\n$$\n\nwhere\n\n$$\n\\boldsymbol {\\Lambda} = \\left( \\begin{array}{c c c c} \\gamma & 0 & 0 & - \\gamma \\beta \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ - \\gamma \\beta & 0 & 0 & \\gamma \\end{array} \\right)\n$$\n\nis the usual matrix corresponding to a Lorentz transformation in the  $x$-direction, and its inverse  $\\mathbf{\\Lambda}^{-1}$ is the same matrix with  $-\\beta$ replaced by  $\\beta$. Perform the above matrix multiplications explicitly and solve for the new field quantities  $E_x'$,  $E_y'$,  $E_z'$,  $B_x'$,  $B_y'$ &  $B_z'$ in terms of the old ones  $E_x$,  $E_y$,  $E_z$,  $B_x$,  $B_y$ &  $B_z$. Does your result agree with Resnick's? Which derivation was shorter?\n\n(b). To get some intuition for what your result means, describe what happens in the following special cases:\n\n- (A) Both E and B point in the x-direction in S (parallel to direction in which   $S'$  is moving).\n- (B)  $S$ has a pure  $\\mathbf{E}$-field in the  $y$-direction (perpendicular to the direction in which  $S'$ is moving), i.e.,  $E_x = E_z = B_x = B_y = B_z = 0$.\n\n(c). In what sense does your last result imply that one cannot consider electric and magnetic fields as fundamentally separate entities?", "source": "mit-ocw", "source_doc_id": "0016-problem-set-7-330fe5ae-04e389d4", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0016-problem-set-7-330fe5ae-04e389d4:page-0002", "text": "(d). If B vanishes in S, then in what other frames does it vanish? (In other words, in which directions can the new frame   $S'$  move and still have no magnetic field?\n\nProblem 3: “Center of Mass Collisions” (6 points):\n\n(a) In modern experimental physics, energetic particles are made to circulate in opposite directions in so-called storage rings and permitted to collide head-on. In this arrangement each particle has the same kinetic energy K in the laboratory. The collisions may be viewed as totally inelastic, in that the rest energy K in the laboratory. The collisions may be viewed as totally inelastic, in that the rest energy of the two colliding protons, plus all available kinetic energy, can be used to generate new particles and to endow them with kinetic energy. Show that the available energy in this arrangement can be written in the form.\n\n$$\n\\epsilon = 2 m _ {0} c ^ {2} \\big (1 + \\frac {k}{m _ {0} c ^ {2}} \\big)\n$$\n\n(b) How much energy is made available when 100 GeV protons are used in fashion? (c) What proton energy would be required to make 100 GeV available?\n\nProblem 4: “Atomic Decay With Recoil”(9 points): French, problem #6-10, Chapter 6, page 201.\n\nProblem 5: “Compton Scattering” (9 points): French, problem #6-17, Chapter 6, page 202.\n\nNote: The block of matter serves only as a source of “free” electrons; it does not absorb any momentum as a collective unit.\n\nProblem 6: “Creation of Pions”(6 points): A   $10^{15}$  eV cosmic-ray (proton) collides with a proton (at rest) in the Earth’s atmosphere. Compute the maximum number of pions (rest mass = 140 MeV) that can be created in such a collision. [Hint: Either work in the center of mass reference frame or make use of the fact that   $E_{TOT}^{2} - p_{TOT}^{2}c^{2}$  is Lorentz invariant.\n\nProblem 7: “Pair Creation”(6 points): Two protons of mass   $\\sim$  1 GeV collide to produce a particle of rest mass 300 GeV. The two protons remain after the collision. Find the threshold energy for this particle production to occur if: (i) one of the protons is initially at rest, and, (ii) both protons have the same energy and collide head on. Comment on the relative efficiency of colliding beam accelerators versus those using fixed targets.\n\nOptional Problem 8: “Proton-Antiproton Collision”: French, problem #6-12, Chapter 6, page 202.\n\nOptional Problem 9: “Scattering Angles in Elastic Collisions”: French, problem #7-5, Chapter 7, page 226.\n\nOptional Problem 10: “Cyclotron”: French, problem #7-9, Chapter 7, page 226.\n\nOptional Problem 11: “Relativistic Elastic Collision”: A relativistic particle of rest mass   $M_{0}$  moves along the x-axis of S with motion characterized by   $\\beta_{0}$  and   $\\gamma_{0}$ . It collides with an identical particle that is at rest. After the collision it is observed that both particles are moving at angles   $\\pm\\theta$  with respect to the x-axis.\n\n(a). Show that the velocity of the CM frame ( $S'$ frame) with respect to  $S$ is given by\n\n$$\n\\beta_ {\\mathrm{CM}} = \\frac {\\gamma_ {0} \\beta_ {0}}{\\gamma_ {0} + 1}, \\quad \\gamma_ {\\mathrm{CM}} = \\sqrt {\\frac {\\gamma_ {0} + 1}{2}}. \\tag {11.1}\n$$\n\nHint: one way to proceed is to use the relation:   $\\beta_{CM} = p_{TOT}c / E_{TOT}$ .\n\n- (b). Argue that before the collision, both particles have the same energy and magnitude of momentum in the CM frame, and that these quantities are characterized by   $\\beta_{CM}$  and   $\\gamma_{CM}$ .\n- (c). Argue that after the collision both particles are moving at right angles to the  $x'$-axis in the CM frame.\n- (d). For each particle, after the collision, find  $p_x$ and  $p_y$ back in the  $S$ frame; express your answers in terms of  $\\beta_{\\mathrm{CM}}$ and  $\\gamma_{\\mathrm{CM}}$.\n- (e). Compute   $\\tan(\\theta)=p_{y}/p_{x}$  and show that   $\\tan(\\theta)=1/\\gamma_{\\mathrm{CM}}=\\sqrt{2/(\\gamma_{0}+1)}$ .\n\nSource: Problem 3 is taken from Basic Concepts in Relativity by Robert Resnick, David Halliday.\n\nFeedback: Roughly how much time did you spend on this problem set?", "source": "mit-ocw", "source_doc_id": "0016-problem-set-7-330fe5ae-04e389d4", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0017-problem-set-8-d468add1-17168b91:page-0001", "text": "# Problem Set 8 (Cosmology)\n\nDue: Friday 17 November 2006 at 4:00PM. Please deposit the problem set in the appropriate 8.033 bin, labeled with name and recitation section number and stapled as needed (3 points).\n\nReadings: Taylor & Wheeler until page 2-18, also project G. Also the three items linked to under “handouts” on the course web site:\n\n- (a). the popular article on cosmology\n- (b). the spacetime review article\n- (c). Ned Wrights cosmology Tutorial\n\nThe lecture notes state a number of properties of the expanding Universe (the FRW metric). To strengthen your intuition and get used to dealing with metrics, you will now derive some of them.\n\nProblem 1: “Geodesics in the expanding universe:(6+3+3 points)” For the special case of flat space (k=0), the FRW metric can be written in Cartesian coordinates as (using units where c=1)\n\n$$\n\\begin{array}{l} d \\tau^ {2} = d t ^ {2} - a (t) ^ {2} \\left(d x ^ {2} + d y ^ {2} + d z ^ {2}\\right) \\\\ = \\left[ 1 - a (t) ^ {2} \\left(\\dot {x} ^ {2} + \\dot {y} ^ {2} + \\dot {z} ^ {2}\\right) \\right] d t ^ {2} \\\\ = \\left[ 1 - a (t) ^ {2} | \\dot {\\mathbf {r}} | ^ {2} \\right] d t ^ {2}, \\\\ \\end{array}\n$$\n\nwhere dots denote d/dt and   $\\mathbf{r} \\equiv (x, y, z)$  is called the comoving position.\n\n- (a). Find the geodesic between the two spacetime events   $(t,\\mathbf{r})=(t_{0},\\mathbf{r}_{0})$  and   $(t,\\mathbf{r})=(t_{1},\\mathbf{r}_{0})$ . Hint: Find the curve of maximal aging. The result is very simple and a straightforward mathematical argument suffices to show this — there is no need to use calculus of variations for this.\n- (b). What is the total time that elapses on a clock moving on this geodesic curve between these two events?\n- (c). An object is said to be comoving if   $\\dot{r}=0$ . Consider a galaxy in the this FRW spacetime that is comoving early on. Based on your result in (a), describe in one sentence its future motion.\n\nProblem 2: “Cosmic expansion”(6+3+3 points): Consider two comoving galaxies in a flat FRW universe (same metric as above), one at   $\\mathbf{r} = (x, y, z) = (0, 0, 0)$  and one at   $\\mathbf{r} = (x, y, z) = (x_0, 0, 0)$ .\n\n(a). Calculate the distance  $\\sigma$ between them at a fixed time  $t = t_0$, i.e., the length of the spacelike geodesics between the event A with  $(x,y,z,t) = (0,0,0,t_0)$ and the event B with  $(x,y,z,t) = (x_0,0,0,t_0)$.\n\nHint: This geodesic is simply the straight line  $(x,y,z,t) = (x,0,0,t_0)$ where  $x$ goes from 0 to  $x_0$ (you don't need to prove this), so you merely need to integrate along this curve:\n\n$$\n\\sigma = \\int d \\sigma = \\int_ {0} ^ {x _ {0}} \\frac {d \\sigma}{d x} d x,\n$$\n\nwhere  $d\\sigma$ is the proper space interval defined by  $d\\sigma^2 = -\\mathrm{d}\\tau^2$.\n\n(b). Give the ratio of this separation  $\\sigma$ at two different times,  $\\frac{\\sigma(t_2)}{\\sigma(t_1)}$. You would have obtained this exact same result for any pair of comoving galaxies, regardless their positions, so as the Universe expands, all distances increase by the same factor.", "source": "mit-ocw", "source_doc_id": "0017-problem-set-8-d468add1-17168b91", "source_title": "Problem Set 8 (Cosmology)", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Problem Set 8 (Cosmology)", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0017-problem-set-8-d468add1-17168b91:page-0002", "text": "(c). Using your result from part (a), compute the Hubble parameter defined as recession velocity over distance, i.e.,\n\n$$\nH \\equiv \\frac {\\dot {\\sigma}}{\\sigma}.\n$$\n\nProblem 3: “Deriving the Friedmann equation”: Although the rigorous way to do this is using Einstein’s theory of General Relativity, it turns out that you can obtain exactly the same result with classical mechanics as you will now show. This is helpful for intuitively understanding the Friedmann equation, arguably the most important equation in all of cosmology. N.B. Make sure to use classical mechanics throughout this problem, not special relativity.\n\nI   $(3+3+3+3+3$  points) Consider a large mass M at rest and a negligibly small mass m a distance a away, moving straight away from M with a velocity v.\n\n- (a) Illustrate this with a picture. Write down the total energy  $E$ of the system, including both the kinetic energy and the gravitational potential energy.\n- (b) Calculate the escape velocity  $v$, defined as the smallest velocity that would let  $m$ escape to infinity eventually. (Hint: This corresponds to the case  $E = 0$.)\n- (c) For the special case  $E = 0$, solve for the position  $a$ as a function of time  $t$. Make the assumption that  $m \\ll M$, so that the large mass remains at rest throughout. Define  $t = 0$ as the time when  $a = 0$ (this fixes your integration constant).\n- Hint: If you have no experience with differential equations, try plugging in a solution of the form  $a(t) = At^B$ and solve for the constants  $A$ and  $B$.\n- (d) Using your solution from part (c), how long ago were the two masses at the same place if the current separation is  $a_0$?\n- (e) In words, describe what would happen eventually for the two cases  $E < 0$ and  $E > 0$.\n\nII  $(6 + 3 + 3 + 3 + 3$ points) Perhaps without knowing it, you have just described the expanding Universe. All that remains to be done is to reinterpret your equations in terms of a different problem.\n\n(a) In problem 2, you showed that the a comoving object moves radially away from the origin with a distance  $\\sigma(t) \\propto a(t)$ and speed  $v = H\\sigma$. This means that no matter overtakes any other matter, i.e., that the total mass contained within an expanding sphere of radius  $a(t)$ remains constant. Assuming that the Universe is full of uniformly distributed comoving matter of density  $\\rho(t)$, this constant enclosed mass is\n\n$$\nM = \\frac {4}{3} \\pi a ^ {3} \\rho .\n$$\n\nYou will determine the function  $a(t)$ by studying the motion of a tiny blob of stuff of mass  $m$ at the edge of this sphere. Illustrate this with a picture.\n\nSince the mass distribution is spherically symmetric about the origin, the blob will feel a gravitational force equivalent to that of single mass  $M = \\frac{4}{3}\\pi a^3\\rho$ at the origin. (This is a famous theorem by Newton. The result that we can ignore all the matter outside of the sphere holds in general relativity as well, and is known as Birkoff's theorem.) Use this to eliminate  $M$ from your answer in part I (a) and write down the combined kinetic and gravitational energy  $E$ for the blob.\n\n(b) Plug in the relation  $v = Ha$ to eliminate  $v$ from this result, and solve for  $H^2$. This is the famous Friedmann equation. Tidy it up by replacing the integration constant  $E$ by the dimensionless curvature constant  $k$ defined as\n\n$$\nk \\equiv - \\frac {2 E}{m c ^ {2}}.\n$$\n\n- (c) Using your answer in part I (c) (for the  $k = 0$ case), write down the solution to the Friedmann equation as  $a(t) \\propto t^B$ for some constant  $B$.\n- (d) How much time has elapsed since the Big Bang, i.e., since the time when  $\\rho = \\infty$? Express your answer in terms of  $H$. Hint: You simply need to relate  $H$ to  $t$. A simple way to do this is to write  $a = At^{B}$ as in part I (c) and compute  $H = \\dot{a} / a$.\n- (e) In words, describe what would happen eventually for the two cases k < 0 and k > 0.", "source": "mit-ocw", "source_doc_id": "0017-problem-set-8-d468add1-17168b91", "source_title": "Problem Set 8 (Cosmology)", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Problem Set 8 (Cosmology)", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0017-problem-set-8-d468add1-17168b91:page-0003", "text": "Problem 4: “Solving the Friedmann equation:”(3+3+3 points) In the last problem, you derived the Friedmann equation\n\n$$\nH ^ {2} = \\frac {8 \\pi G}{3} \\rho - \\frac {k c ^ {2}}{a ^ {2}}\n$$\n\nand solved it for the special case of flat space   $(k=0)$  and ordinary matter   $(\\rho\\propto a^{-3})$ . Now you will solve it for more interesting cases. Different types of matter dilute differently as space expands:\n\n- $\\rho_{\\gamma} \\propto a^{-4}$  (photons)\n- $\\rho_{m} \\propto a^{-3}$ (ordinary matter, dark matter)\n- $\\rho_{k} \\propto a^{-2}$ (spatial curvature, i.e., the  $k$-term)\n- $\\rho_{\\Lambda} \\propto a^{0}$ (vacuum energy, i.e., cosmological constant)\n\n(a). Compute  $a(t)$ up to a proportionality constant for a Universe containing only photons ( $H^2 \\propto a^{-4}$).\n\nHint: For the matter-dominated case   $H^{2} \\propto a^{-3}$  that you solved above, the key steps would be   $(a^{-1}da/dt)^{2} \\propto a^{-3}$ ,   $a^{1/2}da \\propto dt$ ,   $a^{3/2} \\propto t$ ,   $a \\propto t^{2/3}$ .\n\n(b). Compute  $a(t)$ up to a proportionality constant for an empty Universe with curvature only  $(H^2 \\propto a^{-2})$.\n\n(c). Compute  $a(t)$ up to a proportionality constant for a Universe with only vacuum energy ( $H = H_0 = \\text{constant}$). (Express your answer in terms of  $H_0$.)\n\nNote: In the theory of cosmological inflation, this solution applies approximately in the very early Universe, creating a vast volume in a very short time. This solution may also apply to our distant future.\n\nIf you want something harder, one of the optional problems below is to show that the case with both matter and curvature makes   $a(t)$  a cycloid.\n\nProblem 5: “Age of the Universe”(9 × 3 points): The Friedmann equation implies that we can write\n\n$$\nH (a) = H _ {0} \\left[ \\Omega_ {\\gamma} a ^ {- 4} + \\Omega_ {m} a ^ {- 3} + \\Omega_ {k} a ^ {- 2} + \\Omega_ {\\Lambda} \\right] ^ {1 / 2},\n$$\n\nwhere a is normalized so that a = 1 at the present time and where the parameters   $H_{0}$ ,   $\\Omega_{\\gamma}$ ,   $\\Omega_{m}$ ,   $\\Omega_{k}$  and   $\\Omega_{\\Lambda}$  are all constants.   $H = a^{-1}da/dt$  implies dt = da/aH, so the age of the Universe is\n\n$$\nt _ {0} = \\int_ {0} ^ {1} {\\frac {d a}{a H}} = \\int {\\frac {H _ {0} ^ {- 1} d a}{a [ \\Omega_ {\\gamma} a ^ {- 4} + \\Omega_ {m} a ^ {- 3} + \\Omega_ {k} a ^ {- 2} + \\Omega_ {\\Lambda} ] ^ {1 / 2}}}.\n$$\n\n- (a). Compute the dimensionless age of the Universe  $H_0t_0$ for the matter-dominated case ( $\\Omega_m = 1$,  $\\Omega_\\gamma = \\Omega_k = \\Omega_\\Lambda = 0$).\n- (b). Compute the dimensionless age of the Universe  $H_0t_0$ for the photon-dominated case ( $\\Omega_{\\gamma} = 1$,  $\\Omega_m = \\Omega_k = \\Omega_\\Lambda = 0$).\n- (c). Compute the dimensionless age of the Universe  $H_0t_0$ for the empty Universe case ( $\\Omega_k = 1$,  $\\Omega_\\gamma = \\Omega_m = \\Omega_\\Lambda = 0$).\n- (d). Compute the dimensionless age of the Universe   $H_{0}t_{0}$  for the currently favored case (  $\\Omega_{k}=0$ ,   $\\Omega_{\\gamma}\\approx0$ ,   $\\Omega_{m}=0.3$ ,   $\\Omega_{\\Lambda}=0.7$ ). Hint: This integral can be done analytically, so feel free to use the result below.\n\n$$\n\\begin{array}{l} H _ {0} t _ {0} = \\int_ {0} ^ {1} \\frac {d a}{(\\Omega_ {\\mathrm{m}} / a + \\Omega_ {\\Lambda} a ^ {2}) ^ {1 / 2}} = \\frac {2}{3} (1 - \\Omega_ {\\mathrm{M}}) ^ {- 1 / 2} \\sinh^ {- 1} \\left[ \\left(\\frac {1 - \\Omega_ {\\mathrm{m}}}{\\Omega_ {\\mathrm{m}}}\\right) ^ {1 / 2} \\right] \\\\ { = } { \\frac { 2 } { 3 } ( 1 - \\Omega _ { \\mathrm{m} } ) ^ { - 1 / 2 } \\ln \\left[ \\Omega _ { \\mathrm{m} } ^ { - 1 / 2 } + \\left( \\frac { 1 - \\Omega _ { \\mathrm{m} } } { \\Omega _ { \\mathrm{m} } } \\right) ^ { 1 / 2 } \\right] , } \\\\ \\end{array}\n$$\n\nwhere the second of the expressions on the right-hand-side is the more useful form. (Optionally: prove this formula — it is helpful to make a change of variables, where   $x = a^{3/2}$ .)", "source": "mit-ocw", "source_doc_id": "0017-problem-set-8-d468add1-17168b91", "source_title": "Problem Set 8 (Cosmology)", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Problem Set 8 (Cosmology)", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0017-problem-set-8-d468add1-17168b91:page-0004", "text": "(e). For the above cases, give the age of the Universe   $t_{0}$  in Gigayears, using the fact that   $H_{0}^{-1} \\approx h^{-1} \\times 9.7846$  Gyr. Assume a dimensionless Hubble parameter h = 0.7 (  $H_{0} = h \\times 100km/s/Mpc$ .)\n\n(f). Check your answers against Ned Wright's Javascript calculator at\n\n(make sure to enter   $H_{0} = 70km/s/Mpc$  and z = 0) Does it agree with your results? (Note that it can't handle the photon-dominated case).\n\nUse it to calculate the age of a closed Universe with  $\\Omega_{m} = 0.3$,  $\\Omega_{\\Lambda} = 0.8$.\n\n(g). Using either analytic arguments or this calculator, answer the following qualitative questions:\n\n- Increasing  $H_0$ makes the Universe YOUNGER / OLDER (circle one).\n- Increasing  $\\Omega_{m}$ makes the Universe YOUNGER / OLDER (circle one).\n- Increasing  $\\Omega_{\\Lambda}$ makes the Universe YOUNGER / OLDER (circle one).\n\n(h). The age of the Universe at redshift z (at the time of emission of photons that are just now arriving at the Earth with a redshift of z) is given by changing variables from   $a = (1 + z)^{-1}$  to z in the integral above:\n\n$$\nt (z) = \\int_ {z} ^ {\\infty} \\frac {d z ^ {\\prime}}{(1 + z ^ {\\prime}) H} = \\int_ {z} ^ {\\infty} \\frac {H _ {0} ^ {- 1} d z ^ {\\prime}}{(1 + z ^ {\\prime}) [ \\Omega_ {\\gamma} (1 + z ^ {\\prime}) ^ {4} + \\Omega_ {m} (1 + z ^ {\\prime}) ^ {3} + \\Omega_ {k} (1 + z ^ {\\prime}) ^ {2} + \\Omega_ {\\Lambda} ] ^ {1 / 2}}.\n$$\n\nCompute  $H_0t(z)$ by doing this integral for the matter-dominated case ( $\\Omega_m = 1$,  $\\Omega_\\gamma = \\Omega_k = \\Omega_\\Lambda = 0$).\n\n(i). Using the Javascript calculator, compute the age  $t(z)$ for for the currently favored cosmology ( $\\Omega_k = 0$,  $\\Omega_\\gamma \\approx 0$,  $\\Omega_m = 0.3$,  $\\Omega_\\Lambda = 0.7$) when the cosmic microwave background radiation was released ( $z = 10^3$) and corresponding to one of the most distant quasars ever observed ( $z = 6$).\n\n# Problem 6: Key concepts  $(3 + 5 + 3 + 4)$:\n\n- (a). Order the following epochs chronologically and give the approximate age of the Universe corresponding to each one: emission of the Cosmic Microwave Background, formation of typical galaxies, primordial nucleosynthesis, Planck time, today, death of Sun, inflation, formation of first stars. (Hint: see the Time Magazine handout.)\n- (b). Give each of the following quantities to the nearest power of 10 (don't show calculations, being off by one power of 10 is OK):\n\n- Number of stars in our Galaxy\n- Light travel time to closest star (Sun!) in minutes\n- Light travel time to Pluto in hours\n- Light travel time to 2nd closest star in years\n- Distance to Andromeda galaxy (M31) in lightyears\n\n(c). List (no explanations needed) three pieces of evidence supporting the Big Bang model.\n\n(d). Give rough current estimates of the Hubble parameter h, the dark energy density parameter   $\\Omega_{\\Lambda}$ , the baryon density parameter   $\\Omega_{b}$  and the dark matter density parameter   $\\Omega_{dm}$ .\n\nOptional Problem 7: “Accelerating Universe”: Consider a currently fashionable Universe with   $\\Omega_{m}=0.3$  and   $\\Omega_{\\Lambda}=0.7$ . Ignoring the photon density, the squared expansion velocity is\n\n$$\n\\dot {a} ^ {2} = a ^ {2} H ^ {2} \\propto \\frac {\\Omega_ {m}}{a} + \\Omega_ {\\Lambda} a ^ {3}.\n$$\n\nGraph the right hand side as a function of the scale factor  $a$ for this cosmology. Find the scale factor at the time of minimum expansion velocity (minimum  $\\dot{a}$). Is the Universe currently accelerating or not, i.e., is  $\\dot{a}$ currently increasing or decreasing?\n\nOptional Problem 8: “Evolution of the Scale Factor for a Universe with   $\\Omega_{\\Lambda}=0$ , and   $\\Omega_{m}>1$ ”: Show that the evolution of the scale factor of a closed Universe (only matter and curvature) can", "source": "mit-ocw", "source_doc_id": "0017-problem-set-8-d468add1-17168b91", "source_title": "Problem Set 8 (Cosmology)", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Problem Set 8 (Cosmology)", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0017-problem-set-8-d468add1-17168b91:page-0005", "text": "be worked out analytically to yield the following parametric expressions:\n\n$$\na = \\frac {1}{2} \\left(\\frac {\\Omega_ {m}}{\\Omega_ {m} - 1}\\right) (1 - \\cos \\alpha), \\tag {8.1}\n$$\n\n$$\nt = \\frac {1}{2 H _ {0}} \\frac {\\Omega_ {m}}{(\\Omega_ {m} - 1) ^ {3 / 2}} (\\alpha - \\sin \\alpha). \\tag {8.2}\n$$\n\nThe curve   $a(t)$  is a cycloid, i.e., the same curve that solved the brachistochrone problem. Hint: To carry out the integral over a, make a substitution of variables where\n\n$$\na = \\frac {\\Omega_ {m}}{\\Omega_ {m} - 1} \\sin^ {2} \\alpha / 2. \\tag {8.3}\n$$\n\nAlternatively, simply plug in the above expressions and show that they satisfy the Friedmann equation, using the identity\n\n$$\n{\\frac {d a}{d t}} = {\\frac {d a / d \\alpha}{d t / d \\alpha}}.\n$$\n\nOptional Problem 9: For the above closed universe cycloid solution, show that a photon leaving the origin at the Big Bang arrives back at the same place at the Big Crunch, i.e., just barely has time to go circumnavigate the Universe once. Hint: The photon trajectory is defined by   $d\\tau = 0$  which gives an expression for dr/dt that you can integrate.\n\nOptional Problem 10: “Minkowski space in disguise” (hard!): Show by a clever choice of coordinates that the FRW metric with   $\\Omega_{\\Lambda} = \\Omega_{m} = \\Omega_{\\gamma} = 0$ ,   $\\Omega_{k} = 1$  (this is the special case with   $a(t) = t$ , k = -1, corresponding to an empty and maximally open universe) is simply the Minkowski metric in disguise.\n\nOptional Problem 11: “Ant Universe”: The following problem is taken from Shu’s book, “The Physical Universe”, Problem 15.5, page 371.\n\nThe premise: Imagine a world of ants on the surface of an expanding balloon of instantaneous radius   $R(t)$ . These ants have no concept of the existence of a third dimension perpendicular to the surface of the balloon. They are doomed to crawl forever only in the two dimensions parallel to the surface of the balloon. Because their world is a big place compared to any ant city, these ants have always imagined that they live on a flat surface. Moreover, being two-dimensional creatures, they, unlike us, cannot visualize their geometrically round world. (This is a slight rephrasing of Shu's setup for the problem that follows.)\n\nThe problem: “Consider the great circle which corresponds to a line of sight from any given ant city. Let the radius of curvature of this great circle be   $R(t)$ , the instantaneous radius of the world relative to a center displaced in the third, unobservable dimension. Suppose that there are N ant cities at any time, distributed more or less evenly along this great circle. Show the average distances   $s(t)$  between two ant cities along this (or any other) line of ant sight is given by   $s(t) = \\theta R(t)$ , where   $\\theta = 2\\pi/N$  radians. Argue that, in general, the distance between two ant cities which are initially separated by an angle   $\\theta$  (relative to the unobservable center of the world) is given at any later time t by the formula   $S(t) = \\theta R(t)$ . The recessional velocity of one city with respect to the other is given by   $v = \\dot{s}(t)$  where the dot denotes differentiation with respect to time. Show that this recessional velocity satisfies Hubble’s law:   $v = H(t)s$ , where   $H(t) = \\dot{R}(t)/R(t)$  depends only on the history of the radius of curvature R. Notice, in particular, that all reference to the unobservable angle   $\\theta$  drops out.”", "source": "mit-ocw", "source_doc_id": "0017-problem-set-8-d468add1-17168b91", "source_title": "Problem Set 8 (Cosmology)", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Problem Set 8 (Cosmology)", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0017-problem-set-8-d468add1-17168b91:page-0006", "text": "Optional Problem 12: “  $\\Omega_{\\Lambda}$  vs.   $\\Omega_{m}$ ”: Consider this plot showing various quantities graphed in the parameter space   $\\Omega_{\\Lambda}$  vs.   $\\Omega_{m}$ :\n\n| Region | Percentage |\n|--------|----------|\n| No Big Bang | 99.7% |\n| Accelerating | 68.3% |\n| Decelerating | 95.4% |\n| Expands to Infinity | 99.7% |\n| Recóttapses | 68.3% |\n| Closed Open | 0% |\n| Formation Ring + Maxima | 0% |\n| Transition | 0% |\n\nThere are three curves delineating regions (1) of expansion to infinity vs. recollapsing\", (2) \"acceleration vs. deceleration\", and (3) \"no big bang\" vs. \"big bang\". Try to either explain these curves qualitatively, or write a simple program to compute them quantitatively. The curve separating \"acceleration vs. deceleration\" is a simple analytic curve (i.e., a straight line of slope  $1/2$).\n\nFeedback: Roughly how much time did you spend on this problem set?", "source": "mit-ocw", "source_doc_id": "0017-problem-set-8-d468add1-17168b91", "source_title": "Problem Set 8 (Cosmology)", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 5, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Problem Set 8 (Cosmology)", "Page 6"], "page_start": 6, "page_end": 6, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0018-problem-set-9-b803ac10-500e2c8c:page-0001", "text": "# MASSACHUSETTS INSTITUTE OF TECHNOLOGY\n\n# Physics Department\n\nPhysics 8.033\n\nNovember 17, 2006\n\n# Problem Set 9\n\nDue: December 8, at 4:00PM. Please deposit the problem set in the appropriate 8.033 bin, labeled with name and recitation section number and stapled as needed (3 points).\n\nReading: Chapters 2, 3, 4, 5, and Project D in the Taylor & Wheeler book - \"Exploring Black Holes, Introduction to General Relativity\". You will be responsible only for the corresponding material that was actually covered in the lectures. Project E should also be understandable, but this topic will be mentioned only very briefly in lecture.\n\n# Problem 1\n\nConcept questions\n\n- 1. According to the no hair theorem, which three physical quantities uniquely characterize a black hole? (2 points)\n- 2. To explain why most astrophysicists now believe that black holes really exist, briefly give one piece of evidence for the existence of supermassive black holes and one piece of evidence for the existence of stellar-mass black holes. (Hint: see the black hole section in the Science handout.) (3 points)\n- 3. Due to a miscalculation, your friend falls into the supermassive black hole at the center of our Galaxy. Assuming that this black hole is neither charged nor rotating, indicate whether each of the following statements is true or false.(10 points)\n- (a) He will die just as he enters the event horizon.\n- (b) He will die only after he has entered the event horizon.\n- (c) He will be killed by tidal forces.\n- (d) He will be killed by the singularity.\n- (e) He will emerge unscathed from a while hole in another Universe.\n- (f) You will see him disappear through the event horizon.\n- (g) You will see him forever, seemingly frozen on the event horizon.\n- (h) You will see him seemingly frozen on the event horizon until he redshifts out of sight.\n- (i) Most physicists now believe that this black hole will last forever, since nothing can come out from it.", "source": "mit-ocw", "source_doc_id": "0018-problem-set-9-b803ac10-500e2c8c", "source_title": "MASSACHUSETTS INSTITUTE OF TECHNOLOGY: Physics Department", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MASSACHUSETTS INSTITUTE OF TECHNOLOGY: Physics Department", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0018-problem-set-9-b803ac10-500e2c8c:page-0002", "text": "(j) The concepts of space and time as we know them are no longer valid inside the event horizon.\n\n# Problem 2 (9 points)\n\nShow that the Gullstrand-Painlevé (GP) metric\n\n$$\nd \\tau^ {2} = d t _ {\\mathrm{ff}} ^ {2} - (d r + \\beta_ {r} d t _ {\\mathrm{ff}}) ^ {2} - r ^ {2} \\left(d \\theta^ {2} + \\sin^ {2} \\theta d \\varphi^ {2}\\right)\n$$\n\nand the standard Schwarzschild metric\n\n$$\nd \\tau^ {2} = \\left(1 - \\frac {2 M}{r}\\right) d t ^ {2} - \\left(1 - \\frac {2 M}{r}\\right) ^ {- 1} d r ^ {2} - r ^ {2} \\left(d \\theta^ {2} + \\sin^ {2} \\theta d \\varphi^ {2}\\right),\n$$\n\nare equivalent. Here\n\n$$\n\\beta_ {r} \\equiv \\left(\\frac {2 M}{r}\\right) ^ {1 / 2}\n$$\n\nis the escape velocity.\n\n1. first rewrite the Schwarzschild metric as\n\n$$\nd \\tau^ {2} = \\gamma_ {r} ^ {- 2} d t ^ {2} - \\gamma_ {r} ^ {2} d r ^ {2} - r ^ {2} \\left(d \\theta^ {2} + \\sin^ {2} \\theta d \\varphi^ {2}\\right),\n$$\n\nwhere\n\n$$\n\\gamma_ {r} \\equiv \\frac {1}{\\sqrt {1 - \\beta_ {r} ^ {2}}}.\n$$\n\n2. Start with the relation\n\n$$\nd t _ {\\mathrm{ff}} = d t + \\beta_ {r} \\gamma_ {r} ^ {2} d r\n$$\n\nfrom the lecture notes which defines the time coordinate   $t_{ff}$  and show that\n\n$$\nd r + \\beta_ {r} d t _ {\\mathrm{ff}} = \\beta_ {r} d t + \\gamma_ {r} ^ {2} d r.\n$$\n\n3. Show that when you plug these two relations into the GP metric, you get the Schwarzschild metric.", "source": "mit-ocw", "source_doc_id": "0018-problem-set-9-b803ac10-500e2c8c", "source_title": "MASSACHUSETTS INSTITUTE OF TECHNOLOGY: Physics Department", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MASSACHUSETTS INSTITUTE OF TECHNOLOGY: Physics Department", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0018-problem-set-9-b803ac10-500e2c8c:page-0003", "text": "# Problem 3(6 points)\n\n“Why tidal forces make you tall and slim”\n\nFor simplicity, use classical mechanics and Newtonian gravity for this problem.\n\nConsider a closed capsule that is freely falling radially in the gravitational field outside a spherically symmetric body of mass M. At a given time, the center of the capsule is at a distance r from the center of the mass, and the capsule size is very small compared with r. Define a local Cartesian coordinate system with an origin fixed at the center of the capsule. The coordinate y is then the distance from the origin along the radial direction (r), while x is the distance from the origin in a direction perpendicular to the radial direction.\n\nShow that for   $|x| \\ll r$  and   $|y| \\ll r$ , the accelerations of free test particles, relative to an observer at   $(0,0)$ , are given by:\n\n$$\n\\pmb {a} = \\frac {G M}{r ^ {3}} (- x \\hat {x} + 2 y \\hat {y})\n$$\n\nConsider 4 test particles at locations   $(0,+y)$ ,   $(0,-y)$ ,   $(x,0)$ , and   $(-x,0)$ . Compute the initial motions (before the particle has moved very far) for each particle relative to   $(0,0)$ , and sketch their trajectories.\n\nIf fall feet first into a Schwarzschild black hole, why would you become tall and slim before you die?\n\n# Problem 4(6 points)\n\n“Comparison of   $r_{shell}$  and   $r''$ ”\n\nCompute and plot the shell radius,   $r_{shell}$ , vs. the coordinate radius r. Follow the integration steps outlined in Taylor & Wheeler, Sample Problem 2, page 2-28. Integrate   $r_{shell}$  from   $r_{0} \\rightarrow r_{s}$ , where   $r_{0}$  is an arbitrary starting value. Make a plot of   $(r_{s} - r_{0})$  vs. r, starting from r = 2M to a sufficiently large value of r to be able to discern the asymptotic behavior. [Note: we use units where G = 1 and c = 1.]\n\n# Problem 5\n\n\"Gravitational Redlift\" (3+3+3 points)\n\nA radioactive Fe source emits a 6 keV X-ray line in its rest frame. Suppose such a source is located on the surface of a neutron star of mass   $M = 2.8 \\times 10^{30}$  kg, and radius 10 km. Assume that the atomic transition which gives rise to the X-ray line is not significantly physically altered by the strong gravity or the possible presence of an intense magnetic field. Ignore any rotation of the neutron star.\n\n- (a) Compute the energy of these X-rays as they would be seen by a very distant observer,   $O_{\\infty}$ , i.e., an astronomer on Earth. (Neglect the gravitational potential of the Earth itself.)\n- (b) Suppose that another stationary observer,   $O_{r}$ , is fixed at radius r > 10 km from the center of the neutron star. Find an expression for the energy,   $E_{r}$ , of the X-ray line (coming from the surface) that would be detected by such an observer.\n- (c) If the observer at   $O_{r}$  sends X-rays of energy   $E_{r}$  to the observer back on Earth, as a report of what he/she has seen, what energy would be detected at   $O_{\\infty}$ ?", "source": "mit-ocw", "source_doc_id": "0018-problem-set-9-b803ac10-500e2c8c", "source_title": "MASSACHUSETTS INSTITUTE OF TECHNOLOGY: Physics Department", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MASSACHUSETTS INSTITUTE OF TECHNOLOGY: Physics Department", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0018-problem-set-9-b803ac10-500e2c8c:page-0004", "text": "# Problem 6\n\n“Global Positioning Satellite System (GPS)” (9×3 points)\n\nStart with equation (3) of Project A in Taylor & Wheeler, page A–3, which we will derive in lecture. Proceed to answer “Queries” 1 through 9.\n\n# Problem 7\n\n\"A Dilute Black Hole\"(3 points)\n\nTaylor & Wheeler, Problem 2–5, page 2–46.\n\n# Problem 8\n\n\"Orbital Periods Around Black Holes\" (6 points)\n\nConsider two black holes of mass   $M_{1}=1\\ M_{\\odot}$  and   $M_{2}=10^{6}\\ M_{\\odot}$  (where   $M_{\\odot}=2\\times10^{30}$  kg).\n\na. Find the Schwarzschild radius  $(R_S\\equiv 2GM / c^2)$ for each object.\n\nb. Kepler's 3rd law (for circular orbits, at least) works exactly for orbits in the Schwarzschild metric if the bookkeeper's coordinates  $r$ and  $t$ are used:  $(2\\pi / P)^2 = GMr^{-3}$, where  $P$ is the orbital period. Find the orbital period in seconds for a circular orbit at  $r$ just outside the Schwarzschild radius for an arbitrary mass (expressed in units of  $M_{\\odot}$). What are the corresponding orbital periods for the two black holes given in this problem?\n\nProblem 9 (6×3 points)\n\n“Falling into a Black Hole”\n\nYou fall radially into a black hole with   $\\tilde{E} = E/m = 1$ , i.e., starting with negligible velocity far away.\n\n1. Write down your total aging  $\\Delta \\tau$ using the GP metric from Problem 2 as an integral along an arbitrary trajectory  $r(t_{\\mathrm{ff}})$.\n\n2. Prove that the motion given by\n\n$$\n\\frac {d r}{d t _ {\\mathrm{ff}}} = - \\beta_ {r} \\tag {1}\n$$\n\nis a geodesic, i.e., maximizes your aging.\n\nHint: a simple argument suffices — no need to use the variational calculus.\n\n3. What is the relation between  $\\Delta \\tau$ and the free-fall time interval  $\\Delta t_{\\mathrm{ff}}$?\n\n4. Integrate equation (1) above to compute the time elapsed on your wristwatch between passing a radius  $r$ and when it gets destroyed at  $r \\approx 0$.\n\n5. What is the bookkeeper time interval  $\\Delta t$ between the start of your fall and your crossing the event horizon? (No calculation needed.)", "source": "mit-ocw", "source_doc_id": "0018-problem-set-9-b803ac10-500e2c8c", "source_title": "MASSACHUSETTS INSTITUTE OF TECHNOLOGY: Physics Department", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MASSACHUSETTS INSTITUTE OF TECHNOLOGY: Physics Department", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0018-problem-set-9-b803ac10-500e2c8c:page-0005", "text": "6. To get used to working with the orbital equations of motion, use them to rederive equation (1) above. Specifically, use   $d\\tau = dt_{ff}$  and these two equations as your starting point:\n\n$$\n\\left(\\frac {d r}{d \\tau}\\right) ^ {2} = \\tilde {E} ^ {2} - \\tilde {V} (\\tilde {L}, r) ^ {2},\n$$\n\n$$\n\\tilde {V} (\\tilde {L}, r) ^ {2} = \\left(1 - \\frac {2 M}{r}\\right) \\left(1 + \\frac {\\tilde {L} ^ {2}}{r ^ {2}}\\right).\n$$\n\nProblem 10(9 points)\n\n“Dropping in on a Black Hole”\n\nExercise 7, Chapter 3, page 3-30 of Taylor & Wheeler.\n\nProblem 11(6 points)\n\n\"Orbit of a Satellite With Same Clock Speed As One On Earth\"\n\nFind the orbital radius, r, of a satellite whose clock will be found to run at the same rate as that for an Earth-bound clock. Take the orbital speed to be   $v = (GM/r)^{1/2}$ , where M is the mass of the Earth. Why would you guess that the GPS satellites are placed in higher orbits?\n\nOther possibly useful pieces of information:  $R_{\\mathrm{Earth}} = 6378 \\, \\mathrm{km}$;  $M_{\\mathrm{Earth}} = 6 \\times 10^{24} \\, \\mathrm{kg}$; rotation speed of the Earth's surface equals  $\\sim 6\\%$ of the orbital speed at  $R_{\\mathrm{Earth}}$.\n\nProblem 12\n\n“Time Travel Using the Black Hole” (6+3+3 points)\n\n- 1. Do exercise 7, Chapter 4, page 4–32 of Taylor & Wheeler.\n- 2. A more realistic circular orbit to use is the “tourist” orbit with E/m = 1 that can be reached with essentially no use expenditure of rocket fuel. Derive r for this orbit using Taylor & Wheeler equations [30] and [43], then compute the time dilation factor   $d\\tau/dt$  using the same formula as you did above for the r = 6M case.\n- Hint: If you've found the correct  $r$-value in the lecture notes, you can simply verify that it satisfies equations [30] and [43] for an appropriate  $L$-value.\n- 3. Compute  $d\\tau / dt$ for the  $r = 3M$ orbit. How much energy is required to reach this orbit?", "source": "mit-ocw", "source_doc_id": "0018-problem-set-9-b803ac10-500e2c8c", "source_title": "MASSACHUSETTS INSTITUTE OF TECHNOLOGY: Physics Department", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MASSACHUSETTS INSTITUTE OF TECHNOLOGY: Physics Department", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0018-problem-set-9-b803ac10-500e2c8c:page-0006", "text": "# Optional Problem A\n\nWithout knowing it, you have almost learned how to do advanced classical mechanics with Lagrangeans, where a particle moves along a trajectory   $x(t)$  such that the “action”\n\n$$\nS \\equiv \\int_ {t _ {0}} ^ {t _ {1}} (T - V) d t\n$$\n\nis minimized. This is called the principle of least action. Here   $V = V(x(t))$  is the potential energy and   $T = \\frac{1}{2}m\\dot{x}(t)^{2}$  is the kinetic energy. Use the Euler-Lagrange equation to derive the law of motion F = ma, i.e.,\n\n$$\nm \\ddot {x} = - V ^ {\\prime} (x).\n$$\n\n# Optional Problem B\n\n“Heuristic Derivation of the Schwarzschild Metric”\n\n- (a) Read the first 4 pages of the article by Matt Visser - on the 8.033 web site.\n- (b) Follow Visser's simple derivation of the invariant interval leading to:\n\n$$\nd \\tau^ {2} = \\left[ 1 - \\frac {2 M}{r} \\right] d t ^ {2} - 2 \\sqrt {\\frac {2 M}{r}} d r d t + d r ^ {2} - r ^ {2} \\left(d \\theta^ {2} + \\sin \\theta^ {2} d \\phi^ {2}\\right)\n$$\n\nwhere we have set c = G = 1.\n\n(c) To transform this form of the metric to the usual Schwarzschild form, make the following transformation of variables:\n\n$$\nd r = d r ^ {\\prime}\n$$\n\n$$\nd t = d t ^ {\\prime} + \\alpha d r ^ {\\prime},\n$$\n\nwhere   $\\alpha$  will turn out to be a function of   $r'$ .\n\n- (d) Set the coefficient of the  $dt'dr'$ term equal to zero to find  $\\alpha$.\n- (e) Finally, use the functional form of   $\\alpha$  to derive the coefficients in front of the   $dr'^{2}$  and   $dt'^{2}$  terms, and thereby show that   $ds^{2}$  takes the standard form of the Schwarzschild metric.\n\nBig hint: The metric above is simply the GP metric from problem 2 if you rename the time coordinate   $t_{ff}$ .\n\n# Optional problem C\n\n“More rigorous derivation of the Schwarzschild metric”\n\nAs indicated in the lecture notes and optional handout, the Einstein field equations for GR are\n\n$$\nG _ {\\alpha \\beta} = \\frac {8 \\pi G}{c ^ {4}} T _ {\\alpha \\beta}, \\tag {1}\n$$\n\nwhere   $G_{\\alpha\\beta}$  is the Einstein tensor and   $T_{\\alpha\\beta}$  is the “stress–energy” tensor, and G (with no indices) is Newton’s gravitational constant. For the case of trying to find the space-time", "source": "mit-ocw", "source_doc_id": "0018-problem-set-9-b803ac10-500e2c8c", "source_title": "MASSACHUSETTS INSTITUTE OF TECHNOLOGY: Physics Department", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 5, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MASSACHUSETTS INSTITUTE OF TECHNOLOGY: Physics Department", "Page 6"], "page_start": 6, "page_end": 6, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0018-problem-set-9-b803ac10-500e2c8c:page-0007", "text": "metric in the region outside of a spherically symmetric mass distribution (e.g., a neutron star or black hole), the Einstein tensor reduces to:\n\n$$\nG _ {0 0} = \\frac {1}{r ^ {2}} e ^ {2 \\Phi} \\frac {d}{d r} [ r (1 - e ^ {- 2 \\Lambda}) ];\n$$\n\n$$\nG _ {1 1} = - \\frac {1}{r ^ {2}} e ^ {2 \\Lambda} (1 - e ^ {- 2 \\Lambda}) + \\frac {2}{r} \\frac {d \\Phi}{d r}, \\tag {2}\n$$\n\nwhere the metric has been taken to be of the form:\n\n$$\nd s ^ {2} = - e ^ {2 \\Phi} d (c t) ^ {2} + e ^ {2 \\Lambda} d r ^ {2} + r ^ {2} d \\theta^ {2} + r ^ {2} \\sin \\theta^ {2} d \\phi^ {2} \\tag {3}\n$$\n\nand   $e^{2\\Phi}$  and   $e^{2\\Lambda}$  are convenient ways of writing the two unknown functions of r (only). In the region outside the mass distribution, take the stress–energy tensor to be equal to zero, and use   $G_{00}$  and   $G_{11}$  to solve for   $e^{2\\Phi}$  and   $e^{2\\Lambda}$ . Take the constant of integration from the   $G_{00}$  equation to be   $2GM/c^{2}$ , and take the constant from the   $G_{11}$  equation to be 0.\n\n# Optional problem D\n\n“Non-Relativistic Keplerian Orbits”\n\nIn the very weak-field limit (the Kepler problem), the equation for the conserved energy becomes:\n\n$$\nE = \\frac {1}{2} m \\left(\\frac {d r}{d t}\\right) ^ {2} - \\frac {G M m}{r} + \\frac {L ^ {2}}{2 m r ^ {2}}, \\tag {4}\n$$\n\nwhere L is the angular momentum constant associated with the orbit, E is a negative quantity for a bound orbit, and m is the mass of the orbiting particle. The distances of closest and farthest approach of the orbiting body occur when dr/dt = 0. These are the points where a line of constant E intersects the effective potential curve. Solve the resulting quadratic equation to find:\n\n$$\nr _ {\\max, \\min} = - \\frac {G M m}{2 E} \\left[ 1 \\pm \\left(1 + \\frac {2 L ^ {2} E}{G ^ {2} M ^ {2} m ^ {3}}\\right) ^ {1 / 2} \\right]. \\tag {5}\n$$\n\nThe leading term is defined as the “semimajor axis”, a, of the binary orbit, while the square root term is the “orbital eccentricity”, i.e.,   $r_{\\max} = a(1 + e)$  and   $r_{\\min} = a(1 - e)$ . Show that:\n\n$$\nL ^ {2} = a (1 - e ^ {2}) G M m ^ {2} \\tag {6}\n$$\n\nand\n\n$$\nE = - \\frac {G M m}{2 a}. \\tag {7}\n$$\n\nNote that the energy of a Keplerian orbit depends only on the semimajor axis, and not on the orbital eccentricity. The physical parameters L and E thereby uniquely determine the orbital shape – which turns out to be an ellipse.", "source": "mit-ocw", "source_doc_id": "0018-problem-set-9-b803ac10-500e2c8c", "source_title": "MASSACHUSETTS INSTITUTE OF TECHNOLOGY: Physics Department", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 6, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MASSACHUSETTS INSTITUTE OF TECHNOLOGY: Physics Department", "Page 7"], "page_start": 7, "page_end": 7, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0018-problem-set-9-b803ac10-500e2c8c:page-0008", "text": "“Simple Gravitational Lens System”\n\nConsider an astronomical object, S, sufficiently distant   $(D_{\\mathrm{S}})$  that it appears pointlike to an observer at O. Now introduce a pointlike gravitational lens at a distance   $D_{L}$  from the observer. The unperturbed angular separation between the source and the lens is   $\\beta$  as indicated in the sketch. In the presence of the gravitational lens, light from the source can travel the heavy-line path shown in the sketch, pass a distance of closest approach b to the lens, and then be deflected by an angle   $\\delta$  so that it subsequently passes through O. The apparent angular distance between the lens and the image of the source I, is   $\\theta$  (also indicated on the sketch).\n\nThere are two convenient approximations that one can make in doing this problem. (1) All angles are taken to very small such that   $\\tan x \\simeq \\sin x \\simeq x$ . (2) The light path from the source to the observer may be considered as two straight-line segments with a deflection of angle   $\\delta$  taking place at the point of closest approach to the lens.\n\nUtilize the following steps to derive the relation between the angles   $\\theta$  and   $\\beta$ :\n\n$$\n\\delta = \\frac {4 G M}{c ^ {2} b}; \\alpha = \\frac {h}{D _ {\\mathrm{S}}}; \\theta = \\frac {b}{D _ {\\mathrm{L}}}; \\delta = \\frac {h}{D _ {\\mathrm{LS}}}.\n$$\n\n(a) In particular, show that:\n\n$$\n\\beta = \\theta - \\frac {\\theta_ {\\mathrm{E}} ^ {2}}{\\theta},\n$$\n\nwhere\n\n$$\n\\theta_ {E} ^ {2} = \\frac {4 G M}{c ^ {2}} \\frac {D _ {\\mathrm{LS}}}{D _ {\\mathrm{L}} D _ {\\mathrm{S}}},\n$$\n\nwhere   $\\theta_{E}$  is defined as the angle of the “Einstein Ring”, i.e., the value of   $\\theta$  when   $\\beta \\rightarrow 0$  (the lens and the source are along a line).\n\n(b) Show that there are two solutions for   $\\theta$ , the apparent position of the source (i.e., the image) for each value of   $\\beta$ , and find these two solutions. Make a sketch, analogous to the one above, to indicate what the geometry of these two solutions looks like.\n\n# Optional problem F\n\n“Gravitational Acceleration on the Spherical Shell”\n\nExercise 9, Chapter 3, page 3-31 of Taylor & Wheeler.\n\n# Optional problem G\n\n“Energy Measured by a Shell Observer”", "source": "mit-ocw", "source_doc_id": "0018-problem-set-9-b803ac10-500e2c8c", "source_title": "MASSACHUSETTS INSTITUTE OF TECHNOLOGY: Physics Department", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 7, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MASSACHUSETTS INSTITUTE OF TECHNOLOGY: Physics Department", "Page 8"], "page_start": 8, "page_end": 8, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0018-problem-set-9-b803ac10-500e2c8c:page-0009", "text": "Start with the general expression for   $d\\tau/dt$  in the Schwarzschild metric:\n\n$$\n\\frac {d \\tau}{d t} = \\left[ \\left(1 - \\frac {2 M}{r}\\right) - \\left(1 - \\frac {2 M}{r}\\right) ^ {- 1} \\left(\\frac {d r}{d t}\\right) ^ {2} - r ^ {2} \\left(\\frac {d \\phi}{d t}\\right) ^ {2} \\right] ^ {1 / 2},\n$$\n\nand show that\n\n$$\n\\frac {d \\tau}{d t} = \\left\\{\\left(1 - \\frac {2 M}{r}\\right) \\left[ 1 - \\left(\\frac {d r _ {s}}{d t _ {s}}\\right) ^ {2} - r ^ {2} \\left(\\frac {d \\phi}{d t _ {s}}\\right) ^ {2} \\right] \\right\\} ^ {1 / 2}.\n$$\n\nCarefully examine all the terms and argue that the quantity in square brackets ('[]') is, in fact,  $\\gamma_{\\mathrm{shell}}^{-2}$, and therefore:\n\n$$\n\\frac {d \\tau}{d t} = \\left(1 - \\frac {2 M}{r}\\right) ^ {1 / 2} \\frac {1}{\\gamma_ {\\mathrm{shell}}}.\n$$\n\nFinally, combine this relation with our expression for the conserved quantity,  $E$:\n\n$$\n\\frac {E}{m _ {0}} = \\left(1 - \\frac {2 M}{r}\\right) \\frac {d t}{d \\tau},\n$$\n\nto relate  $E$ and  $E_{\\mathrm{shell}}$ as follows:\n\n$$\nE = \\left(1 - \\frac {2 M}{r}\\right) ^ {1 / 2} E _ {\\mathrm{shell}},\n$$\n\nwhere  $E_{\\mathrm{shell}} \\equiv m_0 \\gamma_s$, the relativistic energy measured by the shell observer.\n\n# Optional problem H\n\n\"Speed of Light in a Schwarzschild Metric\"\n\nFollow the derivation of equations (14) and (15) in the Boxed Exercise “Motion of Light in Schwarzschild Geometry”, page 5–8 of the Taylor & Wheeler book.\n\n- (a) From these two equations derive equation 5-16 on page 5-7.\n\n(b) For purely radial motion of the photon, show that the speed of light as “reckoned” by the bookkeeper is:\n\n$$\n\\frac {d r}{d t} = \\pm \\left(1 - \\frac {2 G M}{c ^ {2} r}\\right) c.\n$$\n\n(c) In the weak field limit, find the bookeeper's time for a photon to move radially (with respect to a central   $1 M_{\\odot}$  star) from a distance of   $r = 10^{8}$  m to   $r = 10^{10}$  m. How much longer does this take than for a photon making the same trip in the absence of the star?", "source": "mit-ocw", "source_doc_id": "0018-problem-set-9-b803ac10-500e2c8c", "source_title": "MASSACHUSETTS INSTITUTE OF TECHNOLOGY: Physics Department", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 8, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["MASSACHUSETTS INSTITUTE OF TECHNOLOGY: Physics Department", "Page 9"], "page_start": 9, "page_end": 9, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0019-quiz1-2006-sol-8d307061-32d1976d:page-0001", "text": "# Massachusetts Institute of Technology\n\n# Department of Physics\n\nPhysics 8.033\n\n17 October 2006\n\n# Quiz 1\n\nName: (Last, First) ____ (please print).\n\nRecitation number (circle one): 1 2 3\n\n- Record all answers and show all work in this exam booklet. If you need extra space, use the back of the opposing page.\n- All scratch paper must be handed in with the exam, but will not be graded.\n- No materials besides pencils and erasers are allowed (no calculators, notes, books, pets, etc.)\n- Whenever possible, try to solve problems using general analytic expressions. Plug in numbers only as a last step.\n- Please make sure to answer all sub-questions.\n- Good luck!\n\n| Problem | Max | Grade | Grader |\n| --- | --- | --- | --- |\n| 1 | 25 |  |  |\n| 2 | 25 |  |  |\n| 3 | 25 |  |  |\n| 4 | 25 |  |  |\n| Total | 100 |  |  |", "source": "mit-ocw", "source_doc_id": "0019-quiz1-2006-sol-8d307061-32d1976d", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0019-quiz1-2006-sol-8d307061-32d1976d:page-0002", "text": "(a). (4 pts) Professor F. Ishy claims to have discovered a new force of magnitude  $F = A|\\mathbf{r}_1 + \\mathbf{r}_2|$ between protons, where  $\\mathbf{r}_1$ and  $\\mathbf{r}_2$ are the position vectors of the protons and  $A$ is a constant that he's trying to get named in his honor.\n\n- (A) Is this force translationally invariant? YES / NO (circle one) No\n- (B) Is this force invariant under rotations around  $\\mathbf{r} = \\mathbf{0}$? YES / NO (circle one) Yes\n\n(b). (4 pts) Professor C. Rank claims that a charge at   $(\\mathbf{r}_{1}, t_{1})$  will contribute to the air pressure at   $(\\mathbf{r}_{2}, t_{2})$  by an amount   $B \\sin[C(|\\mathbf{r}_{2} - \\mathbf{r}_{1}|^{2} - c^{2}|t_{2} - t_{1}|^{2})]$ , where B and C are constants.\n\n- (A) Is this effect Galilean invariant? YES / NO (circle one) No\n- (B) Is this effect Lorentz invariant? YES / NO (circle one) Yes\n\n(c). (1 pt) To 1 significant figure, the speed of light in meters/second is ____ c =\n\n$$\n3 \\times 1 0 ^ {8} \\mathrm{m/s}\n$$\n\n(d). (2 pts) From which two postulates did Einstein derive special relativity?\n\n(a) Laws of physics must be valid in all intertial frames (b) The speed of light is a constant, independent of observer\n\n(e). (11 pts) Indicate whether each of the following statements are true of false.\n\n- (A) The proper length of a ruler is Lorentz invariant. TRUE / FALSE (circle one) True\n- (B) The wave equation is Galilean invariant. TRUE / FALSE (circle one) False\n- (C) The kinetic energy of a particle is Lorentz invariant. TRUE / FALSE (circle one) False\n- (D) The acceleration of a particle is Galilean invariant. TRUE / FALSE (circle one) True\n- (E) A ping-pong ball moving near the speed of light still looks spherical. TRUE / FALSE (circle one) True\n- (F) X-rays travel faster than microwaves. TRUE / FALSE (circle one) False\n- (G) If you could send a signal faster than light, then there's a frame where you could send a signal backward in time. TRUE / FALSE (circle one) True\n- (H) If two twins are reunited, who is oldest may be frame-dependent. TRUE / FALSE (circle one) False\n- (I) No experiment inside an isolated sealed lab in space can determine its orientation. TRUE / FALSE (circle one) True\n- (J) No experiment inside an isolated sealed lab in space can determine its velocity. TRUE / FALSE (circle one) True\n- (K) No experiment inside an isolated sealed lab in space can determine its acceleration. TRUE / FALSE (circle one) False\n\n(f). (3 pts) List three pieces of observational evidence supporting special relativity. Cosmic ray muons, atomic bombs, GPS measuring time slowdown", "source": "mit-ocw", "source_doc_id": "0019-quiz1-2006-sol-8d307061-32d1976d", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0019-quiz1-2006-sol-8d307061-32d1976d:page-0003", "text": "Your goal in this problem is to completely fill out the table below.\n\n|  | S (Chris) | S' (Zoe) | S'' (Train) |\n| --- | --- | --- | --- |\n| xB | L/γ1 | L | Lγ2/γ1 |\n| ctB | 0 | -β1L | -γ2β2L/γ1 |\n| xC | γ1L | L | γ1γ2L(1 - β1β2) |\n| ctC | γ1β1L | 0 | γ1γ2L(β1 - β2) |\n\nTo save time, note that all three frames have the same spacetime origin (at event A) and that there is no need to draw spacetime diagrams.\n\n(a). (2 pts) Based on the text below, fill in the entries corresponding to   $X_{B}$  and   $X_{C}^{\\prime}$  in the table above (no calculation needed!). Chris is standing next to a barn taking measurements as Zoe runs through the barn in the x-direction holding a pole horizontally in the direction of motion. In Zoe's frame   $S^{\\prime}$, the pole has length L, the event when the rear of the pole is aligned with the entrance to the barn is   $\\mathbf{X}_{A}^{\\prime} = (x_{A}^{\\prime}, ct_{A}^{\\prime}) = (0, 0)$, and the location of the front of the pole at that same time is   $\\mathbf{X}_{C}^{\\prime} = (L, 0)$. In Chris' frame S, Zoe is running at speed   $\\beta_{1}$, the rear of the pole is aligned with the entrance to the barn (event A) at   $\\mathbf{X}_{A} = (x_{A}, ct_{A}) = (0, 0)$, and the front of the pole is aligned with the barn exit (event B) at   $\\mathbf{X}_{B} = (x_{B}, ct_{B}) = (L/\\gamma_{1}, 0)$, where   $\\gamma_{1} = 1/\\sqrt{1 - \\beta_{1}^{2}}$.\n\n(b). (7 pts) Compute   $X_{B}^{\\prime}$  and   $X_{C}$  and fill in the corresponding entries in the table above.\n\n$$\nx _ {B} ^ {\\prime} = \\gamma_ {1} (x _ {B} - \\beta_ {1} c t _ {B}) = \\gamma_ {1} L / \\gamma_ {1} = L\n$$\n\n$$\nc t _ {B} ^ {\\prime} = \\gamma_ {1} (c t _ {B} - \\beta_ {1} x _ {B}) = - \\gamma_ {1} \\beta_ {1} L / \\gamma_ {1} = - \\beta_ {1} L\n$$\n\n$$\nx _ {C} = \\gamma_ {1} (x _ {C} ^ {\\prime} + \\beta_ {1} c t _ {C} ^ {\\prime}) = \\gamma_ {1} L \\quad (N o t e i n v e r s e t r a n s f o r m f r o m S ^ {\\prime} t o S)\n$$\n\n$$\nc t _ {C} = \\gamma_ {1} (c t _ {C} ^ {\\prime} + \\beta_ {1} x _ {C} ^ {\\prime}) = \\gamma_ {1} \\beta_ {1} L \\quad (N o t e i n v e r s e t r a n s f o r m f r o m S ^ {\\prime} t o S)\n$$\n\n(c). (8 pts) A train with the rest of the 8.033 students passes by, and Chris measures its speed to be  $\\beta_{2}$ in the  $x$-direction. The train's frame  $S''$ is aligned such that  $\\mathbf{X}_A'' = (0,0)$. Compute  $\\mathbf{X}_B''$ and  $\\mathbf{X}_C''$ in terms of  $L$,  $\\gamma_1$,  $\\beta_1$,  $\\gamma_2$, and  $\\beta_2$, and fill in the corresponding entries in the table above.\n\nThe trick here is to transform all events from the S frame to the   $S''$  frame\n\n$$\nx _ {B} ^ {\\prime \\prime} = \\gamma_ {2} (x _ {B} - \\beta_ {2} c t _ {B}) = \\gamma_ {2} L / \\gamma_ {1}\n$$\n\n$$\nc t _ {B} ^ {\\prime \\prime} = \\gamma_ {2} (c t _ {B} - \\beta_ {2} x _ {B}) = - \\gamma_ {2} \\beta_ {2} L / \\gamma_ {1}\n$$\n\n$$\nx _ {C} ^ {\\prime \\prime} = \\gamma_ {2} (x _ {C} - \\beta_ {2} c t _ {C}) = \\gamma_ {2} (\\gamma_ {1} L - \\beta_ {2} \\gamma_ {1} \\beta_ {1} L) = \\gamma_ {1} \\gamma_ {2} L (1 - \\beta_ {1} \\beta_ {2})\n$$\n\n$$\nc t _ {C} ^ {\\prime \\prime} = \\gamma_ {2} (c t _ {C} - \\beta_ {2} x _ {C}) = \\gamma_ {2} (- \\gamma_ {1} \\beta_ {1} L - \\beta_ {2} \\gamma_ {1} L) = \\gamma_ {1} \\gamma_ {2} L (\\beta_ {1} - \\beta_ {2})\n$$\n\n(d). (4 pts) Show that for  $\\beta_{2} = 0$, the students on the train agree with Chris.\n\n$$\n\\beta_ {2} = 0, \\gamma_ {2} = 1\n$$\n\n$$\nx _ {B} ^ {\\prime \\prime} = L / \\gamma_ {1}\n$$\n\n$$\nc t _ {B} ^ {\\prime \\prime} = 0\n$$\n\n$$\nx _ {C} ^ {\\prime \\prime} = \\gamma_ {1} L\n$$\n\n$$\nc t _ {C} ^ {\\prime \\prime} = \\gamma_ {1} \\beta_ {1} L\n$$\n\n(e). (4 pts) Show that for   $\\beta_{2} = \\beta_{1}$  the students on the train agree with Zoe.\n\n$$\n\\beta_ {2} = \\beta_ {1}, \\gamma_ {2} = \\gamma_ {1}\n$$\n\n$$\nx _ {B} ^ {\\prime \\prime} = \\gamma_ {1} L / \\gamma_ {1} = L\n$$\n\n$$\nc t _ {B} ^ {\\prime \\prime} = - \\gamma_ {1} \\beta_ {1} L / \\gamma_ {1} = - \\beta_ {1} L\n$$\n\n$$\nx _ {C} ^ {\\prime \\prime} = \\gamma_ {1} ^ {2} L (1 - \\beta_ {1} ^ {2}) = L\n$$\n\n$$\nc t _ {C} ^ {\\prime \\prime} = \\gamma_ {1} ^ {2} L (\\beta_ {1} - \\beta_ {1}) = 0\n$$", "source": "mit-ocw", "source_doc_id": "0019-quiz1-2006-sol-8d307061-32d1976d", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0019-quiz1-2006-sol-8d307061-32d1976d:page-0004", "text": "For a particle of rest mass   $m_{0}$  in a potential   $V(x)$ , show that out of all trajectories   $x(t)$  between two events A and B, the one maximizing the quantity\n\n$$\n\\int_ {t _ {A}} ^ {t _ {B}} \\left[ \\frac {1}{\\gamma} + \\frac {V (x)}{m _ {0} c ^ {2}} \\right] d t\n$$\n\nsatisfies the relation\n\n$$\n\\frac {d}{d t} (m _ {0} \\gamma \\dot {x}) = - V ^ {\\prime} (x)\n$$\n\n(the relativistic version of Newton's second law). Here  $\\gamma \\equiv 1 / \\sqrt{1 - \\dot{x}^2 / c^2}$,  $\\dot{x} \\equiv dx / dt$ and  $V' = dV / dx$. Solution:  $L(x) = \\left[\\frac{1}{\\gamma} + \\frac{V(x)}{m_0c^2}\\right]$. The Euler-Lagrange equation implies\n\n$$\n\\frac {d}{d t} \\left[ \\frac {\\partial L}{\\partial \\dot {x}} \\right] - \\frac {\\partial L}{\\partial x} = 0 \\tag {3.1}\n$$\n\n$$\n\\Rightarrow \\frac {d}{d t} \\left[ \\frac {m _ {0} c ^ {2}}{\\sqrt {1 - \\dot {x} ^ {2} / c ^ {2}}} \\frac {\\dot {x}}{c ^ {2}} \\right] + \\frac {\\partial V}{\\partial x} = 0 \\tag {3.2}\n$$\n\n$$\n\\Rightarrow \\frac {d}{d t} \\left[ \\frac {m _ {0} v}{\\sqrt {1 - \\dot {x} ^ {2} / c ^ {2}}} \\right] = - \\frac {d V}{d x} \\tag {3.3}\n$$", "source": "mit-ocw", "source_doc_id": "0019-quiz1-2006-sol-8d307061-32d1976d", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0019-quiz1-2006-sol-8d307061-32d1976d:page-0005", "text": "In a parallel universe, the Boston team made the playoffs.\n\n- (a). (6 pts) Manny Relativirez hits the ball and starts running towards first base at speed   $\\beta$ . How fast is he running, given that he sees third base   $45^{\\circ}$  to his left (as opposed to straight to his left before he started running)? Assume that he is still very close to home plate.\n- Using the aberration formula with  $\\cos \\theta' = -1 / \\sqrt{2}$,  $\\beta = 1 / \\sqrt{2}$\n\n(b). (7 pts) A player standing on third base is wearing red socks emitting light of wavelength   $\\lambda_{red}$ . What wavelength does Manny see? What color are the socks according to Manny, in the approximation that   $\\lambda_{green} = \\lambda_{red}/2^{1/4}$ ,   $\\lambda_{blue} = \\lambda_{red}/\\sqrt{2}$ ,   $\\lambda_{violet} = \\lambda_{red}/\\sqrt{3}$ ?\n\nUsing the doppler shift formula,  $\\lambda' = \\lambda/\\sqrt{2}$\n\n(c). (5 pts) In Manny's frame, the ball is moving with speed  $c / \\sqrt{2}$ towards first base. How fast is it going in the rest frame of the stadium?\n\nUsing the relativistic velocity addition formula,   $\\beta = \\sqrt{8/9}$\n\n(d). (5 pts) Later in the game, a squabble erupts. An outfielder catches a fly ball at   $(x_{A}, y_{A}, z_{A}, ct_{A}) = (50\\text{m}, 40\\text{m}, 2\\text{m}, cT)$  (event A) and Manny starts to run from second base at   $(x_{B}, y_{B}, z_{B}, ct_{B}) = (30\\text{m}, 30\\text{m}, 0, 0)$  (event B), where cT = 30m, which corresponds to   $T \\approx 90$  nanoseconds. The outfielder claims that Manny is out by virtue of running too soon, whereas Manny claims that B preceded A is his frame. You are the umpire and must settle this. Compute the spacetime interval between A and B and indicate whether it is (circle one) TIMELIKE / SPACELIKE / NULL\n\nThe interval is timelike since  $\\Delta t^2 >\\Delta r^2$\n\n(e). (2 pts) Is Manny correct? YES / NO / DEPENDS ON HIS VELOCITY (circle one)\n\nIn one sentence, why?\n\nNo, because there's no relativity of simulatneity for time like seperations.", "source": "mit-ocw", "source_doc_id": "0019-quiz1-2006-sol-8d307061-32d1976d", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0020-quiz2-2006-sol-9ae7db86-59935af5:page-0001", "text": "# Massachusetts Institute of Technology\n\n# Department of Physics\n\nPhysics 8.033\n\n21 November 2006\n\n# Quiz 2\n\nName: (Last, First) ____ (please print).\n\nRecitation number (circle one): 1 2 3\n\n- Record all answers and show all work in this exam booklet. If you need extra space, use the back of the page.\n- All scratch paper must be handed in with the exam, but will not be graded.\n- This exam is closed book. You may use your handwritten notes if they are clearly labeled with your name and you hand them in with your exam.\n- Whenever possible, try to solve problems using general analytic expressions. Plug in numbers only as a last step.\n- Please make sure to answer all sub-questions.\n- Good Luck!\n\n| Problem | Max | Grade | Grader |\n| --- | --- | --- | --- |\n| 1 | 25 |  |  |\n| 2 | 25 |  |  |\n| 3 | 25 |  |  |\n| 4 | 25 |  |  |\n| Total | 100 |  |  |", "source": "mit-ocw", "source_doc_id": "0020-quiz2-2006-sol-9ae7db86-59935af5", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0020-quiz2-2006-sol-9ae7db86-59935af5:page-0002", "text": "(a). (5 pts) Consider the following reactions:\n\n- (A) ____ A 100 MeV photon decays into an electron-positron pair. answer:E\n- (B) ____ A neutron decays into an electron-positron pair and a photon. answer:B\n- (C) ____ A neutron decays into a proton, an electron and a neutrino. answer: L\n- (D) ____ A proton decays into a neutron, a positron and a neutrino. answer:E\n- (E) ____ A neutron decays into a proton and a photon. answer: Q\n\nFor each one, write one of the letters from the option list below.\n\n- L violates lepton number conservation\n- B violates baryon number conservation\n- P violates parity conservation\n- E violates energy-momentum conservation\n- Q violates charge conservation\n- N violates none of the above conservation laws\n\n(b). (9 pts) Give each of the following quantities to the nearest power of 10 (don't show calculations, being off by one power of 10 is OK):\n\n- (A) ____ Age of our universe when most He nuclei were formed answer:1 min\n- (B) ____ Age of our universe when hydrogen atoms formed. answer:400000 yrs\n- (C) ____ Age of our universe today. answer:10 Gyr\n- (D) ____ Number of stars in our Galaxy. answer:1e11\n- (E) ____ Light travel time to closest star (Sun!:) in minutes. answer:8\n- (F) ____ Hydrogen binding energy in eV/c $^{2}$.answer:13.6\n- (G) ____ Electron mass in eV/c² .answer:500000\n- (H) ____ Neutron mass in eV/c  $^{2}$ . answer:10  $^{9}$\n- (I) ____ Light travel time to 2nd closest star in years. answer:3\n\n(c). (9 pts) Indicate whether each of the following statements are true of false.\n\n- (A) TRUE / FALSE If our Universe is only X billion years old, then we can only see objects that are now less than X billion light years away answer:F\n- (B) TRUE / FALSE Space must be infinite, because it cannot end with a boundary without more space on the other side. answer:F\n- (C) TRUE / FALSE Leptons do not feel the strong interaction. answer:T\n- (D) TRUE / FALSE No experiment inside an isolated sealed lab in space can distinguish between whether it is uniformly accelerating or in a uniform gravitational field. answer:T\n- (E) TRUE / FALSE A clock by the ceiling runs faster than one by the floor. answer:T\n- (F) TRUE / FALSE Hubble's law implies that the Big Bang was an explosion localized near the comoving position of our Galaxy.\n- (G) TRUE / FALSE The expansion of our galaxy is governed by the Friedmann equation. answer:F\n- (H) TRUE / FALSE Two galaxies can recede from each other faster than the speed of light.\nanswer:T\n- (I) TRUE / FALSE We know that our entire observable universe was once at infinite density answer:F\n\n(d). (2 pts) A tritium   $\\left(\\mathrm{H}^{3}\\right)$  nucleus contains ____ up quarks and ____ down quarks. answer:   $p+2n = 4 + 5$", "source": "mit-ocw", "source_doc_id": "0020-quiz2-2006-sol-9ae7db86-59935af5", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0020-quiz2-2006-sol-9ae7db86-59935af5:page-0003", "text": "In the Sun, one of the processes in the He fusion chain is   $p + p + e^{-} \\rightarrow d + \\nu$ , where d is a deuteron. Make the approximations that the deuteron rest mass is   $2m_{p}$ , and that   $m_{e} \\approx 0$  and   $m_{\\nu} \\approx 0$ , since both the electron and the neutrino have negligible rest mass compared with the proton rest mass   $m_{p}$ .\n\n(a). For the arrangement shown in the figure, where (in the lab frame) the two protons have the same energy  $\\gamma m_{p}$ and impact angle  $\\theta$, and the electron is at rest, calculate the energy  $E_{\\nu}$ of the neutrino in the rest frame of the deuteron in terms of  $\\theta$,  $m_{p}$ and  $\\gamma$.\n\nanswer: Use the fact that the quantity  $E^2 - p^2 c^2$ is invariant. In the deutron's rest frame, after the collison:\n\n$$\nE ^ {2} - p ^ {2} c ^ {2} = (2 m _ {p} c ^ {2} + E _ {\\nu}) ^ {2} - E _ {\\nu} ^ {2} \\tag {2.1}\n$$\n\n$$\n= 4 m _ {p} ^ {2} c ^ {4} + 4 m _ {p} c ^ {2} E _ {\\nu} = 4 m _ {p} c ^ {2} (m _ {p} c ^ {2} + E _ {\\nu}) \\tag {2.2}\n$$\n\nIn the lab frame, before collison:\n\n$$\nE ^ {2} - p ^ {2} c ^ {2} = (2 E _ {p}) ^ {2} - (2 p _ {p} \\cos \\theta c) ^ {2} \\tag {2.3}\n$$\n\n$$\n= (2 \\gamma m _ {p} c ^ {2}) ^ {2} - (2 \\gamma \\beta m _ {p} \\cos \\theta c ^ {2}) ^ {2} \\tag {2.4}\n$$\n\nUse   $\\gamma^{2}\\beta^{2}=(\\gamma^{2}-1)$  in the second term and simplify the algebra to find\n\n$$\nE ^ {2} - p ^ {2} c ^ {2} = 4 m _ {p} ^ {2} c ^ {4} (\\gamma^ {2} - (\\gamma^ {2} - 1) \\cos^ {2} \\theta) (2. 5)\n$$\n\nEquating the invariants in the two frames, we have\n\n$$\n4 m _ {p} c ^ {2} (m _ {p} c ^ {2} + E _ {\\nu}) = 4 m _ {p} ^ {2} c ^ {4} (\\gamma^ {2} - (\\gamma^ {2} - 1) \\cos^ {2} \\theta) \\tag {2.6}\n$$\n\n$$\n\\Rightarrow E _ {\\nu} = m _ {p} c ^ {2} \\left(\\gamma^ {2} - 1\\right) \\sin^ {2} \\theta \\tag {2.7}\n$$\n\n(b). For the special case where the deuteron remains at rest in the lab frame and  $\\theta = 30^{\\circ}$, solve for  $\\gamma$ and calculate the energy of all particles (the deuteron, the neutrino, one of the protons) in terms of the proton rest mass  $m_p$.\n\nanswer:The deutron's rest frame is the lab frame. Also,  $\\theta = 30^{\\circ}$. Use conservation of energy, along with the result from the previous part to find:\n\n$$\n2 \\nu m _ {p} c ^ {2} = 2 m _ {p} c ^ {2} + E _ {\\nu} \\tag {2.8}\n$$\n\n$$\n= 2 m _ {p} c ^ {2} + m _ {p} c ^ {2} \\left(\\gamma^ {2} - 1\\right) / 4 \\tag {2.9}\n$$\n\n$$\n\\Rightarrow 2 \\gamma = 2 + \\gamma^ {2} / 4 - 1 / 4 \\tag {2.10}\n$$\n\n$$\n\\Rightarrow \\gamma = 7, 1 \\tag {2.11}\n$$\n\n$\\gamma = 1$ is obviously not the solution. Thus,  $\\gamma = 7$ and the energies are:  $E_{p} = 7m_{p}, E_{\\nu} = 12m_{p}, E_{d} = 2m_{p}$", "source": "mit-ocw", "source_doc_id": "0020-quiz2-2006-sol-9ae7db86-59935af5", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0020-quiz2-2006-sol-9ae7db86-59935af5:page-0004", "text": "In an inertial fram  $S$, the position  $\\mathbf{r}_q$ of a point charge  $q$ moves according to  $\\mathbf{r}_q(t) = v\\hat{\\mathbf{z}} t$, i.e. with velocity  $v$ in the  $\\hat{\\mathbf{z}}$-direction, passing the origin at  $t = 0$. In the moving frame  $S'$ where the charge is at rest at the origin, Coulomb's law states that the electric field is\n\n$$\nE ^ {\\prime} = A \\frac {\\mathbf {r} ^ {\\prime}}{r ^ {\\prime 3}},\n$$\n\nwhere  $A = q / 4\\pi \\varepsilon_0$. Show that in the frame  $S$, the electric field at  $t = 0$ is s\n\n$$\n\\mathbf {E} = A \\frac {(1 - \\beta^ {2})}{(1 - \\beta^ {2} \\sin^ {2} \\theta) ^ {3 / 2}} \\frac {\\mathbf {r}}{r ^ {3}},\n$$\n\nwhere   $\\theta$  is the usual polar angle (  $z = r \\cos \\theta$ ,   $x^{2} + y^{2} = r^{2} \\sin^{2} \\theta$ ).\n\nanswer: Let us convert all quantities to the cartesian coordinates. In the frame  $S'$, the components of the electric field are:\n\n$$\n\\vec {E} ^ {\\prime} = E _ {x} ^ {\\prime} \\hat {x} ^ {\\prime} + E _ {y} ^ {\\prime} \\hat {y} ^ {\\prime} + E _ {z} ^ {\\prime} \\hat {z} ^ {\\prime} \\tag {3.1}\n$$\n\n$$\nE _ {x} ^ {\\prime} = \\frac {A x ^ {\\prime}}{\\left(x ^ {\\prime 2} + y ^ {\\prime 2} + z ^ {\\prime 2}\\right) ^ {3 / 2}} \\tag {3.2}\n$$\n\n$$\nE _ {y} ^ {\\prime} = \\frac {A y ^ {\\prime}}{\\left(x ^ {\\prime 2} + y ^ {\\prime 2} + z ^ {\\prime 2}\\right) ^ {3 / 2}} \\tag {3.3}\n$$\n\n$$\nE _ {z} ^ {\\prime} = \\frac {A z ^ {\\prime}}{(x ^ {\\prime 2} + y ^ {\\prime 2} + z ^ {\\prime 2}) ^ {3 / 2}} \\tag {3.4}\n$$\n\nWe can now Lorentz transform the fields and coordinates from   $S'$  to S. First the coordinates,\n\n$$\nx = x ^ {\\prime} \\tag {3.5}\n$$\n\n$$\ny = y ^ {\\prime} \\tag {3.6}\n$$\n\n$$\n\\gamma z = z ^ {\\prime} \\tag {3.7}\n$$\n\nand then the fields,\n\n$$\nE _ {x} = \\gamma E _ {x} ^ {\\prime} = \\frac {A \\gamma x}{\\left(x ^ {2} + y ^ {2} + \\gamma^ {2} z ^ {2}\\right) ^ {3 / 2}} \\tag {3.8}\n$$\n\n$$\nE _ {y} = \\gamma E _ {y} ^ {\\prime} = \\frac {A \\gamma y}{\\left(x ^ {2} + y ^ {2} + \\gamma^ {2} z ^ {2}\\right) ^ {3 / 2}} \\tag {3.9}\n$$\n\n$$\nE _ {z} = E _ {z} ^ {\\prime} = \\frac {A \\gamma z}{\\left(x ^ {2} + y ^ {2} + \\gamma^ {2} z ^ {2}\\right) ^ {3 / 2}} \\tag {3.10}\n$$\n\nNote that the primed coordinates have been converted to the unprimed ones using the coordinate transformation. The total magnitude for the electric field in the S frame can be obtained from\n\n$$\nE ^ {2} = E _ {x} ^ {2} + E _ {y} ^ {2} + E _ {z} ^ {2} \\tag {3.11}\n$$\n\n$$\n= \\frac {A ^ {2} \\gamma^ {2} r ^ {2}}{\\left(x ^ {2} + y ^ {2} + \\gamma^ {2} z ^ {2}\\right) ^ {3}} \\tag {3.12}\n$$\n\n$$\n= \\frac {A ^ {2} r ^ {2}}{(1 - \\beta^ {2}) (x ^ {2} + y ^ {2} + z ^ {2} / (1 - \\beta^ {2})) ^ {3}} \\tag {3.13}\n$$\n\n$$\n\\Rightarrow E = \\frac {A (1 - \\beta^ {2})}{r ^ {2} (1 - \\beta^ {2} \\sin^ {2} \\theta) ^ {3 / 2}} \\tag {3.14}\n$$\n\nSince the electric field always has to be radial,   $\\vec{E}=|E|\\hat{r}$ .", "source": "mit-ocw", "source_doc_id": "0020-quiz2-2006-sol-9ae7db86-59935af5", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0020-quiz2-2006-sol-9ae7db86-59935af5:page-0005", "text": "(a). (10 pts) Consider a particle coasting in the r-direction (i.e., with constant   $\\theta$  and   $\\phi$ ) in a flat FRW metric, with no non-gravitational forces acting on it. Use variational calculus to prove that   $p \\propto 1/a$  (here   $p = m_{0}\\gamma u$  is its momentum and   $u \\equiv a\\dot{r}$  is its velocity relative to nearby comoving observers). answer: The particle only has a radial motion   $\\Rightarrow d\\theta = d\\phi = 0$ . Also, the universe is flat   $\\Rightarrow k = 0$ . Thus, the FRW metric becomes:\n\n$$\nd \\tau^ {2} = d t ^ {2} - a ^ {2} d r ^ {2} \\tag {4.1}\n$$\n\n$$\n\\Rightarrow \\Delta \\tau = \\int d \\tau = \\int \\sqrt {d t ^ {2} - a ^ {2} d r ^ {2}} (4. 2)\n$$\n\n$$\n= \\int d t \\sqrt {1 - a ^ {2} \\dot {r} ^ {2}} = \\int d t f (t, r, \\dot {r}) \\tag {4.3}\n$$\n\nThe interval   $\\Delta\\tau$  has to be extremized. The Euler lagrange equations give:\n\n$$\n\\frac {\\partial f}{\\partial r} - \\frac {d}{d t} \\left[ \\frac {\\partial f}{\\partial \\dot {r}} \\right] = 0 \\tag {4.4}\n$$\n\n$$\n\\Rightarrow \\frac {d}{d t} \\left[ \\frac {a ^ {2} \\dot {r}}{\\sqrt {1 - a ^ {2} \\dot {r} ^ {2}}} \\right] = 0 \\tag {4.5}\n$$\n\n$$\n\\Rightarrow \\frac {a ^ {2} \\dot {r}}{\\sqrt {1 - a ^ {2} \\dot {r} ^ {2}}} = \\text { constant } \\tag {4.6}\n$$\n\nIdentifying   $a\\dot{r}=u$  leave us with   $\\gamma u\\propto1/a\\Rightarrow p=m_{0}\\gamma u\\propto1/a$ .\n\n- (b). (2 pts) Given a), the value of  $u$ in the limit  $a \\to 0$ is ____. answer:1(c)\n- (c). (2 pts) Given a), the value of u in the limit   $a \\rightarrow \\infty$  is ____. answer:0\n- (d). (2 pts) Thus relative to comoving observers, your results show that an object without external forces in an expanding universe\n\n(circle one) REMAINS IN UNIFORM MOTION / SLOWS DOWN / ACCELERATES.\nanswer:slows down\n\n(e). (3 pts) Starting with the answer from a), derive how the wavelength   $\\lambda$  of a photon depends on a. Your answer should be of the form   $\\lambda \\propto (\\text{function of } a)$ .\n\nanswer: For a photon,   $p = hk/2\\pi \\propto 1/\\lambda$ . So   $\\lambda \\propto 1/p \\propto a$ .\n\n(f). (6 pts) Solve the Friedmann equation\n\n$$\nH ^ {2} = \\frac {8 \\pi G}{3} \\rho - \\frac {k c ^ {2}}{a ^ {2}}\n$$\n\nto obtain a solution of the form   $a(t) \\propto (\\text{function of } t)$  for the case where space is flat and the density is dominated by photons, and compute the age of the universe at the time when   $H^{-1} = 30$  seconds.\n\nanswer: For a flat universe domainated by photons,\n\n$$\n\\left(\\frac {\\dot {a}}{a}\\right) ^ {2} = H ^ {2} \\propto a ^ {- 4} \\tag {4.7}\n$$\n\n$$\n\\Rightarrow \\dot {a} \\propto a ^ {- 1} \\tag {4.8}\n$$\n\n$$\n\\Rightarrow a = A t ^ {1 / 2} \\tag {4.9}\n$$\n\n$$\n\\Rightarrow H = \\dot {a} / a = 1 / 2 t \\tag {4.10}\n$$\n\nThus,   $t = H^{-1}/2 = 15$ seconds.", "source": "mit-ocw", "source_doc_id": "0020-quiz2-2006-sol-9ae7db86-59935af5", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0021-quiz-1-02f708e8-d90b0cf8:page-0001", "text": "# Massachusetts Institute of Technology\n\n# Department of Physics\n\nPhysics 8.033\n\n17 October 2006\n\n# Quiz 1\n\nName: (Last, First) ____ (please print).\n\nRecitation number (circle one): 1 2 3\n\n- Record all answers and show all work in this exam booklet. If you need extra space, use the back of the opposing page.\n- All scratch paper must be handed in with the exam, but will not be graded.\n- No materials besides pencils and erasers are allowed (no calculators, notes, books, pets, etc.)\n- Whenever possible, try to solve problems using general analytic expressions. Plug in numbers only as a last step.\n- Please make sure to answer all sub-questions.\n- Good luck!\n\n| Problem | Max | Grade | Grader |\n| --- | --- | --- | --- |\n| 1 | 25 |  |  |\n| 2 | 25 |  |  |\n| 3 | 25 |  |  |\n| 4 | 25 |  |  |\n| Total | 100 |  |  |", "source": "mit-ocw", "source_doc_id": "0021-quiz-1-02f708e8-d90b0cf8", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0021-quiz-1-02f708e8-d90b0cf8:page-0002", "text": "(a). (4 pts) Professor F. Ishy claims to have discovered a new force of magnitude  $F = A|\\mathbf{r}_1 + \\mathbf{r}_2|$ between protons, where  $\\mathbf{r}_1$ and  $\\mathbf{r}_2$ are the position vectors of the protons and  $A$ is a constant that he's trying to get named in his honor.\n\n- (A) Is this force translationally invariant? YES / NO (circle one)\n- (B) Is this force invariant under rotations around  $\\mathbf{r} = \\mathbf{0}$? YES / NO (circle one)\n\n(b). (4 pts) Professor C. Rank claims that a charge at   $(\\mathbf{r}_{1}, t_{1})$  will contribute to the air pressure at   $(\\mathbf{r}_{2}, t_{2})$  by an amount   $B \\sin[C(|\\mathbf{r}_{2} - \\mathbf{r}_{1}|^{2} - c^{2}|t_{2} - t_{1}|^{2})]$ , where B and C are constants.\n\n- (A) Is this effect Galilean invariant? YES / NO (circle one)\n- (B) Is this effect Lorentz invariant? YES / NO (circle one)\n\n(c). (1 pt) To 1 significant figure, the speed of light in meters/second is ____\n\n(d). (2 pts) From which two postulates did Einstein derive special relativity?\n\n(e). (11 pts) Indicate whether each of the following statements are true of false.\n\n- (A) The proper length of a ruler is Lorentz invariant. TRUE / FALSE (circle one)\n- (B) The wave equation is Galilean invariant. TRUE / FALSE (circle one)\n- (C) The kinetic energy of a particle is Lorentz invariant. TRUE / FALSE (circle one)\n- (D) The acceleration of a particle is Galilean invariant. TRUE / FALSE (circle one)\n- (E) A ping-pong ball moving near the speed of light still looks spherical. TRUE / FALSE (circle one)\n- (F) X-rays travel faster than microwaves. TRUE / FALSE (circle one)\n- (G) If you could send a signal faster than light, then there's a frame where you could send a signal backward in time. TRUE / FALSE (circle one)\n- (H) If two twins are reunited, who is oldest may be frame-dependent. TRUE / FALSE (circle one)\n- (I) No experiment inside an isolated sealed lab in space can determine its orientation. TRUE / FALSE (circle one)\n- (J) No experiment inside an isolated sealed lab in space can determine its velocity. TRUE / FALSE (circle one)\n- (K) No experiment inside an isolated sealed lab in space can determine its acceleration. TRUE / FALSE (circle one)\n\n(f). (3 pts) List three pieces of observational evidence supporting special relativity.", "source": "mit-ocw", "source_doc_id": "0021-quiz-1-02f708e8-d90b0cf8", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0021-quiz-1-02f708e8-d90b0cf8:page-0003", "text": "Your goal in this problem is to completely fill out the table below.\n\n|  | S (Chris) | S' (Zoe) | S'' (Train) |\n| --- | --- | --- | --- |\n| xB |  |  |  |\n| ctB |  |  |  |\n| xC |  |  |  |\n| ctC |  |  |  |\n\nTo save time, note that all three frames have the same spacetime origin (at event A) and that there is no need to draw spacetime diagrams.\n\n(a). (2 pts) Based on the text below, fill in the entries corresponding to   $X_{B}$  and   $X_{C}^{\\prime}$  in the table above (no calculation needed!). Chris is standing next to a barn taking measurements as Zoe runs through the barn in the x-direction holding a pole horizontally in the direction of motion. In Zoe's frame   $S^{\\prime}$, the pole has length L, the event when the rear of the pole is aligned with the entrance to the barn is   $\\mathbf{X}_{A}^{\\prime} = (x_{A}^{\\prime}, ct_{A}^{\\prime}) = (0, 0)$, and the location of the front of the pole at that same time is   $\\mathbf{X}_{C}^{\\prime} = (L, 0)$. In Chris' frame S, Zoe is running at speed   $\\beta_{1}$, the rear of the pole is aligned with the entrance to the barn (event A) at   $\\mathbf{X}_{A} = (x_{A}, ct_{A}) = (0, 0)$, and the front of the pole is aligned with the barn exit (event B) at   $\\mathbf{X}_{B} = (x_{B}, ct_{B}) = (L/\\gamma_{1}, 0)$, where   $\\gamma_{1} = 1/\\sqrt{1 - \\beta_{1}^{2}}$.\n\n(b). (7 pts) Compute   $X_{B}^{\\prime}$  and   $X_{C}$  and fill in the corresponding entries in the table above.\n\n(c). (8 pts) A train with the rest of the 8.033 students passes by, and Chris measures its speed to be  $\\beta_{2}$ in the  $x$-direction. The train's frame  $S''$ is aligned such that  $\\mathbf{X}_A'' = (0,0)$. Compute  $\\mathbf{X}_B''$ and  $\\mathbf{X}_C''$ in terms of  $L$,  $\\gamma_{1}$,  $\\beta_{1}$,  $\\gamma_{2}$, and  $\\beta_{2}$, and fill in the corresponding entries in the table above.\n\n(d). (4 pts) Show that for  $\\beta_{2} = 0$, the students on the train agree with Chris.\n\n(e). (4 pts) Show that for   $\\beta_{2} = \\beta_{1}$  the students on the train agree with Zoe.", "source": "mit-ocw", "source_doc_id": "0021-quiz-1-02f708e8-d90b0cf8", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0021-quiz-1-02f708e8-d90b0cf8:page-0004", "text": "For a particle of rest mass   $m_{0}$  in a potential   $V(x)$ , show that out of all trajectories   $x(t)$  between two events A and B, the one maximizing the quantity\n\n$$\n\\int_ {t _ {A}} ^ {t _ {B}} \\left[ \\frac {1}{\\gamma} + \\frac {V (x)}{m _ {0} c ^ {2}} \\right] d t\n$$\n\nsatisfies the relation\n\n$$\n\\frac {d}{d t} (m _ {0} \\gamma \\dot {x}) = - V ^ {\\prime} (x)\n$$\n\n(the relativistic version of Newton's second law). Here  $\\gamma \\equiv 1 / \\sqrt{1 - \\dot{x}^2 / c^2}$,  $\\dot{x} \\equiv dx / dt$ and  $V' = dV / dx$.", "source": "mit-ocw", "source_doc_id": "0021-quiz-1-02f708e8-d90b0cf8", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0021-quiz-1-02f708e8-d90b0cf8:page-0005", "text": "# Question 4: Relativistic Red Sox\n\n[25 Points]\n\nIn a parallel universe, the Boston team made the playoffs.\n\n(a). (6 pts) Manny Relativirez hits the ball and starts running towards first base at speed   $\\beta$ . How fast is he running, given that he sees third base   $45^{\\circ}$  to his left (as opposed to straight to his left before he started running)? Assume that he is still very close to home plate.\n\n(b). (7 pts) A player standing on third base is wearing red socks emitting light of wavelength   $\\lambda_{red}$ . What wavelength does Manny see? What color are the socks according to Manny, in the approximation that   $\\lambda_{green} = \\lambda_{red}/2^{1/4}$ ,   $\\lambda_{blue} = \\lambda_{red}/\\sqrt{2}$ ,   $\\lambda_{violet} = \\lambda_{red}/\\sqrt{3}$ ?\n\n(c). (5 pts) In Manny's frame, the ball is moving with speed  $c / \\sqrt{2}$ towards first base. How fast is it going in the rest frame of the stadium?\n\n(d). (5 pts) Later in the game, a squabble erupts. An outfielder catches a fly ball at   $(x_{A}, y_{A}, z_{A}, ct_{A}) = (50\\text{m}, 40\\text{m}, 2\\text{m}, cT)$  (event A) and Manny starts to run from second base at   $(x_{B}, y_{B}, z_{B}, ct_{B}) = (30\\text{m}, 30\\text{m}, 0, 0)$  (event B), where cT = 30m, which corresponds to   $T \\approx 90$  nanoseconds. The outfielder claims that Manny is out by virtue of running too soon, whereas Manny claims that B preceded A is his frame. You are the umpire and must settle this.\n\nCompute the spacetime interval between A and B and indicate whether it is (circle one) TIMELIKE / SPACELIKE / NULL\n\n(e). (2 pts) Is Manny correct? YES / NO / DEPENDS ON HIS VELOCITY (circle one)\nIn one sentence, why?", "source": "mit-ocw", "source_doc_id": "0021-quiz-1-02f708e8-d90b0cf8", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0022-quiz-2-612421c8-b8808961:page-0001", "text": "# Massachusetts Institute of Technology\n\n# Department of Physics\n\nPhysics 8.033\n\n21 November 2006\n\n# Quiz 2\n\nName: (Last, First) ____ (please print).\n\nRecitation number (circle one): 1 2 3\n\n- Record all answers and show all work in this exam booklet. If you need extra space, use the back of the page.\n- All scratch paper must be handed in with the exam, but will not be graded.\n- This exam is closed book. You may use your handwritten notes if they are clearly labeled with your name and you hand them in with your exam.\n- Whenever possible, try to solve problems using general analytic expressions. Plug in numbers only as a last step.\n- Please make sure to answer all sub-questions.\n- Good Luck!\n\n| Problem | Max | Grade | Grader |\n| --- | --- | --- | --- |\n| 1 | 25 |  |  |\n| 2 | 25 |  |  |\n| 3 | 25 |  |  |\n| 4 | 25 |  |  |\n| Total | 100 |  |  |", "source": "mit-ocw", "source_doc_id": "0022-quiz-2-612421c8-b8808961", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0022-quiz-2-612421c8-b8808961:page-0002", "text": "(a). (5 pts) Consider the following reactions:\n\n- (A) ____ A 100 MeV photon decays into an electron-positron pair.\n- (B) ____ A neutron decays into an electron-positron pair and a photon\n- (C) ____ A neutron decays into a proton, an electron and a neutrino.\n- (D) ____ A proton decays into a neutron, a positron and a neutrino.\n- (E) ____ A neutron decays into a proton and a photon.\n\nFor each one, write one of the letters from the option list below.\n\n- L violates lepton number conservation\n- B violates baryon number conservation\n- P violates parity conservation\n- E violates energy-momentum conservation\n- Q violates charge conservation\n- N violates none of the above conservation laws\n\n(b). (9 pts) Give each of the following quantities to the nearest power of 10 (don't show calculations, being off by one power of 10 is OK):\n\n- (A) ____ Age of our universe when most He nuclei were formed\n- (B) ____ Age of our universe when hydrogen atoms formed\n- (C) ____ Age of our universe today\n- (D) ____ Number of stars in our Galaxy\n- (E) ____ Light travel time to closest star (Sun!:) in minutes\n- (F) ____ Hydrogen binding energy in eV/ $c^{2}$\n- (G) ____ Electron mass in eV/c²\n- (H) ____ Neutron mass in eV/c²\n- (I) ____ Light travel time to 2nd closest star in years\n\n(c). (9 pts) Indicate whether each of the following statements are true of false.\n\n- (A) TRUE / FALSE If our Universe is only X billion years old, then we can only see objects that are now less than X billion light years away\n- (B) TRUE / FALSE Space must be infinite, because it cannot end with a boundary without more space on the other side.\n- (C) TRUE / FALSE Leptons do not feel the strong interaction.\n- (D) TRUE / FALSE No experiment inside an isolated sealed lab in space can distinguish between whether it is uniformly accelerating or in a uniform gravitational field.\n- (E) TRUE / FALSE A clock by the ceiling runs faster than one by the floor.\n- (F) TRUE / FALSE Hubble's law implies that the Big Bang was an explosion localized near the comoving position of our Galaxy.\n- (G) TRUE / FALSE The expansion of our galaxy is governed by the Friedmann equation.\n- (H) TRUE / FALSE Two galaxies can recede from each other faster than the speed of light.\n- (I) TRUE / FALSE We know that our entire observable universe was once at infinite density\n\n(d). (2 pts) A tritium   $\\left(\\mathrm{H}^{3}\\right)$  nucleus contains ____ up quarks and ____ down quarks.", "source": "mit-ocw", "source_doc_id": "0022-quiz-2-612421c8-b8808961", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0022-quiz-2-612421c8-b8808961:page-0003", "text": "In the Sun, one of the processes in the He fusion chain is   $p + p + e^{-} \\rightarrow d + \\nu$ , where d is a deuteron. Make the approximations that the deuteron rest mass is   $2m_{p}$ , and that   $m_{e} \\approx 0$  and   $m_{\\nu} \\approx 0$ , since both the electron and the neutrino have negligible rest mass compared with the proton rest mass   $m_{p}$ .\n\n(a). For the arrangement shown in the figure, where (in the lab frame) the two protons have the same energy   $\\gamma m_{p}$  and impact angle   $\\theta$ , and the electron is at rest, calculate the energy   $E_{\\nu}$  of the neutrino in the rest frame of the deuteron in terms of   $\\theta$ ,   $m_{p}$  and   $\\gamma$ .\n\n(b). For the special case where the deuteron remains at rest in the lab frame and   $\\theta = 30^{\\circ}$ , solve for   $\\gamma$  and calculate the energy of all particles (the deuteron, the neutrino, one of the protons) in terms of the proton rest mass   $m_{p}$ .", "source": "mit-ocw", "source_doc_id": "0022-quiz-2-612421c8-b8808961", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0022-quiz-2-612421c8-b8808961:page-0004", "text": "# Question 3: Coulomb's Law generalized\n\n[25 Points]\n\nIn an inertial fram  $S$, the position  $\\mathbf{r}_q$ of a point charge  $q$ moves according to  $\\mathbf{r}_q(t) = v\\hat{\\mathbf{z}}t$, i.e. with velocity  $v$ in the  $\\hat{\\mathbf{z}}$-direction, passing the origin at  $t = 0$. In the moving frame  $S'$ where the charge is at rest at the origin, Coulomb's law states that the electric field is\n\n$$\nE ^ {\\prime} = A \\frac {\\mathbf {r} ^ {\\prime}}{r ^ {\\prime 3}},\n$$\n\nwhere  $A = q / 4\\pi \\varepsilon_0$. Show that in the frame  $S$, the electric field at  $t = 0$ is s\n\n$$\n\\mathbf {E} = A \\frac {(1 - \\beta^ {2})}{(1 - \\beta^ {2} \\sin^ {2} \\theta) ^ {3 / 2}} \\frac {\\mathbf {r}}{r ^ {3}},\n$$\n\nwhere   $\\theta$  is the usual polar angle (  $z = r \\cos \\theta$ ,   $x^{2} + y^{2} = r^{2} \\sin^{2} \\theta$ ).", "source": "mit-ocw", "source_doc_id": "0022-quiz-2-612421c8-b8808961", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0022-quiz-2-612421c8-b8808961:page-0005", "text": "(a). (10 pts) Consider a particle coasting in the  $r$-direction (i.e., with constant  $\\theta$ and  $\\phi$) in a flat FRW metric, with no non-gravitational forces acting on it. Use variational calculus to prove that  $p \\propto 1/a$ (here  $p = m_0 \\gamma u$ is its momentum and  $u \\equiv a \\dot{\\mathbf{r}}$ is its velocity relative to nearby comoving observers).\n\n(b). (2 pts) Given a), the value of u in the limit   $a \\rightarrow 0$  is ____.\n\n(c). (2 pts) Given a), the value of u in the limit   $a \\rightarrow \\infty$  is ____.\n\n(d). (2 pts) Thus relative to comoving observers, your results show that an object without external forces in an expanding universe\n(circle one) REMAINS IN UNIFORM MOTION / SLOWS DOWN / ACCELERATES.\n\n(e). (3 pts) Starting with the answer from a), derive how the wavelength   $\\lambda$  of a photon depends on a. Your answer should be of the form   $\\lambda \\propto (\\text{function of } a)$ .\n\n(f). (6 pts) Solve the Friedmann equation\n\n$$\nH ^ {2} = \\frac {8 \\pi G}{3} \\rho - \\frac {k c ^ {2}}{a ^ {2}}\n$$\n\nto obtain a solution of the form   $a(t) \\propto (\\text{function of } t)$  for the case where space is flat and the density is dominated by photons, and compute the age of the universe at the time when   $H^{-1} = 30$  seconds.", "source": "mit-ocw", "source_doc_id": "0022-quiz-2-612421c8-b8808961", "source_title": "Massachusetts Institute of Technology: Department of Physics", "domain": "physics", "subdomain": "relativity", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Massachusetts Institute of Technology: Department of Physics", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}