category stringclasses 107
values | title stringlengths 15 179 | question_link stringlengths 59 147 | question_body stringlengths 53 33.8k | answer_html stringlengths 0 28.8k | __index_level_0__ int64 0 1.58k |
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general relativity | Why is there a gravitational attraction between two objects at rest with respect to each other? | https://physics.stackexchange.com/questions/79676/why-is-there-a-gravitational-attraction-between-two-objects-at-rest-with-respect | <p>From my understanding of relativity, gravity is not a force, but a result of the curvature of spacetime. If Object1 moves past Object2, even though it's moving in a straight line, its direction may change due to the distortion caused by Object2's mass.</p>
<p>However, what about the situation where Object1 is not m... | <p>Two objects that are initially at rest with respect to each other have initially parallel <a href="https://en.wikipedia.org/wiki/World_line" rel="nofollow noreferrer">world lines</a>.</p>
<p>However, the curvature of spacetime means that world lines that are initially parallel do not remain so. This is called geod... | 500 |
general relativity | what are the direct real life applications of general relativity and quantum physics | https://physics.stackexchange.com/questions/82121/what-are-the-direct-real-life-applications-of-general-relativity-and-quantum-phy | <p>What are the direct real life applications of general relativity other than nuclear technology? </p>
<p>What I meant was, was there any technology developed based on general relativity that can benefit mankind today? </p>
<p>Secondly, are there any adverse effects quantum technology today?</p>
| 501 | |
general relativity | About divergence of a vector field and geodesic sphere | https://physics.stackexchange.com/questions/94459/about-divergence-of-a-vector-field-and-geodesic-sphere | <p>I have a question. I want to know the difference between the sphere and the geodesic sphere.
Another question:
given a vector field, $Y$, on a manifold $M$ defined by: $Y(p)=p$ for every point $p \in M$, I tried to calculate the divergence of $Y$. I think that it is equal to $n$, where $n$ is the dimension of $M$, b... | 502 | |
general relativity | What does adding a scalar field component to the Einstein field equations mean for black holes and string theory? | https://physics.stackexchange.com/questions/35863/what-does-adding-a-scalar-field-component-to-the-einstein-field-equations-mean-f | <p>If a scalar field component has to be added to the Einstein field equations (see below) to solve dark matter/energy, then how would string theory need to be modified and do black holes still exist?</p>
<p>The proposed modified equations are (ignoring physical constants)
$$ R_{ij} - \frac12 R g_{ij} = T_{ij} + \nabl... | <p>It does nothing more than saying that $T_{ij}$ can be decomposed into ''ordinary matter'' and ''scalar field.'' That equation is an assertion that the matter content of the universe contains a scalar field. </p>
| 503 |
general relativity | How to choose a solution from all possible solutions of general relativity | https://physics.stackexchange.com/questions/55435/how-to-choose-a-solution-from-all-possible-solutions-of-general-relativity | <p>So there are so many solutions for general relativity - then how does one "choose" the solution that is right one? By checking with observation? (though I also know that it is currently unknown which one is the correct solution.)</p>
| <p>Generally speaking we start with a known stress-energy tensor and boundary conditions and look for solutions for the curvature. When doing this we're not usually overloaded with possible solutions, and it's normally pretty obvious which solutions are physically relevant.</p>
<p>Where multiple physically relevant so... | 504 |
general relativity | Help me to understand this conversion (4-vectors) | https://physics.stackexchange.com/questions/60251/help-me-to-understand-this-conversion-4-vectors | <p>$u^{\mu}$ - 4-velocity</p>
<p>$b^{\mu}$ - 4-vector of magnetic field</p>
<p>$
u_{\mu}u^{\mu}=-1, \qquad u_{\mu}b^{\mu}=0
$</p>
<p>$$
u_{\beta}u^{\alpha}\nabla_{\alpha}b^{\beta}-u_{\beta}b^{\alpha}\nabla_{\alpha}u^{\beta}+\nabla_{\alpha}b^{\alpha}=0
$$
I don't understand why this equation gives this
$$
u^{\alpha}u... | <p>This is because $u_\alpha \nabla_\beta u^\alpha =0$. To show this, just act with $\nabla_\beta$ on both sides of the equality $u_\alpha u^\alpha = -1$. You get
$$
u_\alpha \nabla_\beta u^\alpha + u^\alpha \nabla_\beta u_\alpha = 0
$$
and thus $u_\alpha \nabla_\beta u^\alpha =0$ as promised.</p>
<p>Cheers!</p>
| 505 |
general relativity | What is the physical meaning of charges at light-like infinity in asymptotically flat space-times? | https://physics.stackexchange.com/questions/60748/what-is-the-physical-meaning-of-charges-at-light-like-infinity-in-asymptotically | <p>In the case of charges defined at space-like infinity, I can understand the physical meaning of them because they can be related to measurements made by a physical observer (that is an observer whose wordline is time-like). For example in four dimensions, for the Schwarschild solution, the ADM mass coincide with the... | 506 | |
general relativity | Transforming an equation to the co-vector version | https://physics.stackexchange.com/questions/61641/transforming-an-equation-to-the-co-vector-version | <p>Ok, this question is more a result of my lack of knowledge of how to manipulate equations involving index notation rather than about physics...</p>
<p>I have the geodesic equation with $U^\lambda\equiv\dot{x}^\lambda$:-</p>
<p>$$
\dot{U^\lambda} + \Gamma^\lambda_{\mu\nu} U^\mu U^\nu
$$</p>
<p>And I want to transf... | <p>You generally need to do the second thing where you sub in. For an affinely parameterized geodesic $x(\lambda) = (x^\mu(\lambda))$ we have
$$
x_\mu(\lambda)= g_{\mu\nu}(x(\lambda))x^\nu(\lambda)
$$
Denoting derivatives with respect to affine parameter by overdots, it follows that
\begin{align}
\dot x_\mu(\lambd... | 507 |
general relativity | Interval and proper time | https://physics.stackexchange.com/questions/61353/interval-and-proper-time | <p>Is the definition of $$d s^2=-d \tau^2$$ assuming that $c=1$, so that we always have $$\left({ds\over d\tau}\right)^2=-1$$?
Is there a reason for this definition?
Don't we get an imaginary ${ds\over d\tau}$?</p>
| <p>It depends on what convention you're using for the metric's signature. Some people use the metric signature (-+++), which is what you have there. The interval is then:</p>
<p>$$ds^2=-dt^2+d \mathbf{r}^2$$</p>
<p>On the other hand, some people use the (+---) convention:</p>
<p>$$ds^2=dt^2-d \mathbf{r}^2$$</p>
<p>... | 508 |
general relativity | A question about the relativity of time | https://physics.stackexchange.com/questions/8671/a-question-about-the-relativity-of-time | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://physics.stackexchange.com/questions/8659/invariant-spacetime-distance-circular-motion">Invariant spacetime - distance - Circular Motion</a> </p>
</blockquote>
<p>I understand that the closer something travels to the speed of light, that t... | 509 | |
general relativity | Calculating position in space assuming general relativity | https://physics.stackexchange.com/questions/10870/calculating-position-in-space-assuming-general-relativity | <p>Suppose two pointed masses are given in space. Suppose further that one of the masses has a given velocity at (local) time 0. Is there a way to compute its position in a future time?</p>
<p>Neglecting general relativity, I will simply compute an integral, but with general relativity, we see that the metric of the s... | <p>If the moving mass is small enough, you can do this using the geodesic equation,</p>
<p>$$\frac{\mathrm{d}^2x^\lambda}{\mathrm{d}t^2} + \Gamma^{\lambda}_{\mu\nu}\frac{\mathrm{d}x^\mu}{\mathrm{d}t}\frac{\mathrm{d}x^\nu}{\mathrm{d}t} = 0$$</p>
<p>This is essentially the general relativistic equivalent of Newton's se... | 510 |
general relativity | General relativity at 0K | https://physics.stackexchange.com/questions/14338/general-relativity-at-0k | <p>Relativistic gravity affects particle in motion, does it affect particle that are resting too? How? and if not could one say that the matter at 0K is not affected by gravity?</p>
<p>I'm not a physicist; is just a thought and probably really naive.</p>
| <p>Yes, gravity affects particles at rest, and particles at rest affect gravity.</p>
<p>In GR, the interaction between spacetime and matter and energy is described by <a href="http://en.wikipedia.org/wiki/Einstein_field_equations" rel="nofollow noreferrer">Einstein's equation</a>,
<img src="https://i.sstatic.net/gNZm... | 511 |
general relativity | How do we resolve a flat spacetime and the cosmological principle? | https://physics.stackexchange.com/questions/16090/how-do-we-resolve-a-flat-spacetime-and-the-cosmological-principle | <p>As I've said elsewhere, I've not had the opportunity to take a class in general relativity. Nonetheless, I understand that two major aspects of the standard cosmological model are the <a href="http://en.wikipedia.org/wiki/Cosmological_principle" rel="nofollow">cosmological principle</a> and the observation of a <a ... | <p>I'm not sure I understand what you don't understand. I am adding a answer since this would be too long for a comment... </p>
<p>Given the cosmological principle and the flat space time observation, the idea is that the flat spacetime is infinite, or at least very much larger than our horizon. So, yes, an observer ... | 512 |
general relativity | Intuitively, what is the source term of the Einstein field equation? | https://physics.stackexchange.com/questions/16725/intuitively-what-is-the-source-term-of-the-einstein-field-equation | <p>My copy of Feynman's "Six Not-So-Easy Pieces" has an interesting introduction by Roger Penrose. In that introduction (copyright 1997 according to the copyright page), Penrose complains that Feynman's "simplified account of the Einstein field equation of general relativity did need a qualification that he did not qui... | 513 | |
general relativity | Local Charts in General Relativity | https://physics.stackexchange.com/questions/16951/local-charts-in-general-relativity | <p>We may consider a "local" region in curved spacetime (local in respect of the spatial and the temporal coordinates). A "local inertial frame" may be constructed by some transformation that produces flat spacetime locally. This transformation produces the diagonal [1,-1,-1,-1] in an approximate manner.</p>
<p>A phys... | <p>You cannot do a coordinate transformation in which a curved space becomes flat, and this is because the nonzero curvature is intrinsically meaningful.</p>
<p>The way you define a meaningful notion of curvature in two dimensions is by drawing triangles. In flat space, the sum of the angles in a triangle is 180 degre... | 514 |
general relativity | The Light Ray Bends Round! | https://physics.stackexchange.com/questions/19429/the-light-ray-bends-round | <p>Let's consider the equation y=x in the x-y rectangular Cartesian frame in flat space time. We use the transformations in the first quadrant:
$$y=y'^2$$
$$x=x'$$
$$t=t'$$
For the first transformation we are taking the positive values of $y'$ only.
The equation of y=x in the transformed condition:</p>... | <p>Your coordinate transformation is double-valued, and the line y=0 is singular, the transformation turns around there. This is the reason that you see turning around. When we say coordinate transformations are allowed, they are restricted to be 1-1 and nonsingular Jacobian, so that they are differentiable and differe... | 515 |
general relativity | Pseudo-Superluminal Motion and the Synchronization of Clocks | https://physics.stackexchange.com/questions/27908/pseudo-superluminal-motion-and-the-synchronization-of-clocks | <p>Let's consider two points A an B separated by a finite distance in curved space time. A light ray flashes across an infinitesimally small spatial interval at B. </p>
<p>Metric:
$$ds^2=g_{00}dt^2-g_{11}dx^2-g_{22}dy^2-g_{33}dz^2$$ ----------- (1)</p>
<p>We may write,</p>
<p>$$ds^2=dT^2-dL^2$$
Where,</p>
<p>$dT= \... | <p>This is more or less correct, the intention is clear, but you are saying it in a little bit of a clumsy way.</p>
<p>Your first statement is that if you have light crossing a certain distance, the distance is the same from the point of view of point A and point B, but the time taken to traverse the distance is diffe... | 516 |
general relativity | How exactly does a real object (e.g. a triangle) behave considering the effects of non-euclidian geometry? | https://physics.stackexchange.com/questions/419477/how-exactly-does-a-real-object-e-g-a-triangle-behave-considering-the-effects | <p>The question is somewhat related to
<a href="https://physics.stackexchange.com/questions/220055/what-is-the-sum-of-the-angles-of-a-triangle-on-earth-orbit">What is the sum of the angles of a triangle on Earth orbit?</a> but still not quite what I think of.</p>
<p>Consider a real world triangle made of straight ste... | <p>I’m not sure if I understand you question good enough, so let me try.</p>
<p>There are several ways to look at the curvature in the vicinty of a central mass. As you talk about a triangle consider the bending of light in relation to the center of the mass. So heuristically it’s clear that the sum of the internal an... | 517 |
general relativity | Angle of a hanging ball in system trying to approach speed of light | https://physics.stackexchange.com/questions/432698/angle-of-a-hanging-ball-in-system-trying-to-approach-speed-of-light | <p>A common example of acceleration is a ball hanging from the top of the car. The angle this hanging ball makes from zero is dependent on the acceleration of the car.</p>
<p>What happens as we <em>allow</em> the car to <em>attempt</em> to approach the speed of light at constant acceleration?</p>
<p>My expectation is... | <p>You have come to correct conclusion, but the reasoning is not correct.</p>
<p>Indeed, from the point of view of observer who is staying still the acceleration of the car is decreasing to 0 as the speed of car approaches <span class="math-container">$c$</span>.</p>
<p>But you can't just use non-relativistic approac... | 518 |
general relativity | Mathematical question on Mathisson-Papapetrou-Dixon equations | https://physics.stackexchange.com/questions/443795/mathematical-question-on-mathisson-papapetrou-dixon-equations | <p>I am studying about Mathisson-Papapetrou-Dixon equations which govern the motion of a test particle around a central massive object in the pole-dipole approximation.</p>
<p>Given that <span class="math-container">$S_a=-\frac{1}{2}\epsilon_{abcd}V^bS^{cd}$</span> I want to prove that <span class="math-container">$S^... | <p>In Minkowski signature (-,+,+,+) we have
<span class="math-container">$$
\epsilon_{0123}= - \epsilon^{0123}
$$</span>
so I think your identity should read
<span class="math-container">$$
\epsilon_{abcd}\epsilon^{aijk}= -\delta^i_b\delta^j_c \delta^k_d\pm {\rm perms.}
$$</span>
which is minus what you have have. The... | 519 |
general relativity | Future pointing light cones in Black Hole in Schwarzschild Coordinates | https://physics.stackexchange.com/questions/448884/future-pointing-light-cones-in-black-hole-in-schwarzschild-coordinates | <p>In examining black holes in Schwarzchild Coords (ie without resorting to other coords) the r coord becomes timeline within the event horizon and the t coord spacelike.</p>
<p>Therefore the light cone is tilted by 90 degrees. However, how do we say which direction in r is future and which is past? (textbooks jump to... | <p>You can't expect an explanation if you stick to singular coordinates,
where there is no smooth way to go from without to within.</p>
<p>Shift to well-behaved coordinates (e.g. Kruskal-Szekeres). Then
light-cones behaviour is quite simple and you may easily see which
their relationship must be wrt to Schwarzschild c... | 520 |
general relativity | Are the effects of general relativity accounted for in the calculation of the trajectory of the PARKER probe? | https://physics.stackexchange.com/questions/485500/are-the-effects-of-general-relativity-accounted-for-in-the-calculation-of-the-tr | <p>I think about the precession of the perihelion of the trajectory as the probe comes closer to the sun than Mercury, which was the first successful test for the general relativity.</p>
| 521 | |
general relativity | Flat space limit of metric blows up in different coordinates | https://physics.stackexchange.com/questions/498407/flat-space-limit-of-metric-blows-up-in-different-coordinates | <p>Consider the metric </p>
<p><span class="math-container">$$ds^2=dv^2-dt^2+v^2 \mathcal{R}^2 d\Omega^2_3,$$</span></p>
<p>where <span class="math-container">$d\Omega^2_{3}$</span> is the metric of the <span class="math-container">$3$</span>-sphere, and <span class="math-container">$\mathcal{R}$</span> is the radius... | 522 | |
general relativity | Consistent initial data for evolution in time with the Newman-Penrose formalism | https://physics.stackexchange.com/questions/531467/consistent-initial-data-for-evolution-in-time-with-the-newman-penrose-formalism | <p>In the <a href="http://www.scholarpedia.org/article/Spin-coefficient_formalism" rel="nofollow noreferrer">Newman-Penrose formalism</a>, one rewrites the Einstein equations in terms of a system of linear transport equations for the Newman-Penrose scalars. I am considering the initial value problem for this system of ... | 523 | |
general relativity | Projection of 4D geodesic in spacetime onto 3D space is an ellipse | https://physics.stackexchange.com/questions/552566/projection-of-4d-geodesic-in-spacetime-onto-3d-space-is-an-ellipse | <p>In the Wikipedia article on "Geodesics in general relativity" we can find the following statement:</p>
<blockquote>
<p>In general relativity, gravity can be regarded as not a force but a
consequence of a curved spacetime geometry where the source of
curvature is the stress–energy tensor (representing matter, ... | 524 | |
general relativity | Regarding nothingness and motion in the ocean of Matter Energy | https://physics.stackexchange.com/questions/658803/regarding-nothingness-and-motion-in-the-ocean-of-matter-energy | <p>I am but a humble college dropout with a keen interest in physics, but one way or another I feel I have discovered some key information that will lead to a solid theory of everything that should satisfy all parties it concerns, that is, if I can correctly format and word it. My hope here is simply some respectfully ... | <p>If you don't know all the reasons why people think they are right, you'll never be able to convince them that they are wrong.</p>
<p>So I suggest first learning more about how most physicists see the Universe, if you want any hope of convincing them otherwise.</p>
<p>Theories of everything do attempt to connect empt... | 525 |
general relativity | Does a volume stay constant when freely falling? | https://physics.stackexchange.com/questions/709750/does-a-volume-stay-constant-when-freely-falling | <p>In general relativity, if a volume of particles moves unrestricted through spacetime, is their volume always conserved?</p>
<p>Say we let a collection of particles at rest wrt each other, fall freely in a gravitational field. Will tidal forces keep the volume they occupy constant? My intuition says yes, but how do w... | <p>In general no, the volume is not conserved. To take an extreme example, consider a shell of particles surrounding the Earth. As they freely fall, their volume will definitely decrease!</p>
<p>EDIT: my example is extreme, of course. Baez at <a href="https://math.ucr.edu/home/baez/einstein/node5.html" rel="noreferrer"... | 526 |
general relativity | Correct my understanding of the logical flow of the formulation of General Relativity | https://physics.stackexchange.com/questions/731841/correct-my-understanding-of-the-logical-flow-of-the-formulation-of-general-relat | <p>Correct and direct me when I am wrong.
That is how I understand the logical flow of the formulation of the general theory of relativity:
First we have the equivalence principle, that one tells us that if want to see the effect of gravity just figure out the Physics in some frame accelerating with g in a semi flat sp... | <blockquote>
<p>First we have the equivalence principle, that one tells us that if want to see the effect of gravity just figure out the Physics in some frame accelerating with g in a semi flat space</p>
</blockquote>
<p>This is ok, except it is important to emphasize that the equivalence principle is a local principle... | 527 |
general relativity | Relativity: A modification on Sea Tower experiment | https://physics.stackexchange.com/questions/223645/relativity-a-modification-on-sea-tower-experiment | <p>I first read about it on A Brief History of Time(Stephen Hawking). In 1962, a relativity experiment was executed: identical (classical) watches put on a water tower, one is on very high, other one is at the bottom. And of course, after a certain time, the one which is closer to ground was slower than the other.</p>
... | <p>The upper clock tries to force the lower clock to speed up, by exerting a force on the shaft, the force is produced by a motor.</p>
<p>The lower clock tries to force the upper clock to slow down, by exerting a force on the shaft, the force is produced by the mechanism that limits the clocks rate in normal use.</p>
... | 528 |
general relativity | Which Photon would win the race? | https://physics.stackexchange.com/questions/289025/which-photon-would-win-the-race | <p><a href="https://i.sstatic.net/VDHUg.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/VDHUg.jpg" alt="enter image description here"></a>Imagine that the Sun is not rotating. It also has a tunnel throughout its body exactly through the core. Please disregard any other effect then gravity.
From a far awa... | <p>It seems to me that your central question is: <em>Does the particular lensed path light travels on affect its arrival time?</em> If so, the answer is a resounding <strong>yes</strong>, with evidence from astronomy! For example, the Twin Quasar (<a href="https://en.wikipedia.org/wiki/Twin_Quasar" rel="nofollow norefe... | 529 |
general relativity | Motion of a planet around a spherically distributed mass | https://physics.stackexchange.com/questions/596430/motion-of-a-planet-around-a-spherically-distributed-mass | <p>I was studying a book which claims that, after a lot of math, the metric of the isotropic spacetime around a spherically symmetric is, approximately <span class="math-container">$$ds^2 = -A(r)dt^2 + B(r)dt^2 + r^2 d\Omega^2$$</span></p>
<p>So, the motion of a planet around this mass is given by the Lagrangian L</p>... | <p>The Lagrangian of a single particle in General Relativity (see <a href="https://en.wikipedia.org/wiki/Relativistic_Lagrangian_mechanics#Lagrangian_formulation_in_general_relativity" rel="nofollow noreferrer">Wikipedia</a>, for example) is given by:</p>
<p><span class="math-container">$$\mathcal{L} = -mc^2\sqrt{g_{\m... | 530 |
general relativity | No mass in the Big Bang? | https://physics.stackexchange.com/questions/605193/no-mass-in-the-big-bang | <p>Roger Penrose says in the Big Bang there was no mass!!
<a href="https://www.youtube.com/watch?v=OFqjA5ekmoY" rel="nofollow noreferrer">https://www.youtube.com/watch?v=OFqjA5ekmoY</a>
because E=M.c2. ( minute ca. 6). So where came the mass from.</p>
| <p>The big bang was before the spontaneous symmetry breaking, so there was no massive particles.</p>
| 531 |
general relativity | Where do these vectors come from? | https://physics.stackexchange.com/questions/607861/where-do-these-vectors-come-from | <p>I am reading <a href="http://www.fis.puc.cl/%7Erbenguri/3.pdf" rel="nofollow noreferrer">this paper</a> (link to pdf) by Benguria et al. <em>Aspects of the Hamiltonian dynamics of interacting grvitational gauge andd Higgs fields with application to spherical symmetry</em></p>
<p>At page 13 of the paper the authors s... | 532 | |
general relativity | If gravity impacts length measurements, and length measurements impact gravity, how do we resolve net gravity? (paradox I can't resolve) | https://physics.stackexchange.com/questions/608614/if-gravity-impacts-length-measurements-and-length-measurements-impact-gravity | <p>Here's the concept. We see a very dense 1.5 km radius asteroid, and my friend Charlie and I fly up in our spaceship to check it out. I fly close on my bathroom scale, equipped with rocket thruster, and hover above the non-rotating asteroid. I have 100 kg mass.</p>
<p>My scale reads 2.05864e15 Newtons, so gravitat... | <p>There's an unfortunate tendency in descriptions of relativity (special especially) to say that observers "disagree" about measured quantities, as though they'd actually get into an argument about it, along the lines of "<span class="math-container">$Δx=2\text{m}$</span>!" "No, <span class="m... | 533 |
general relativity | Using relativity to suppress classical inertial acceleration | https://physics.stackexchange.com/questions/611328/using-relativity-to-suppress-classical-inertial-acceleration | <p>This is a question from a mathematics student trying to visualize the fact that general relativity is based on a concept of, “identifying gravity with the curvature of spacetime” (sincere apologies for probable physics inaccuracies and useless details).</p>
<p>Suppose an absolute spacetime, a time-dependent mass dis... | <p>This is the simplest way I know to see the basis of General Relativity. It sacrifices math for simplicity.</p>
<p>See <a href="https://physics.stackexchange.com/q/178417/37364">Why can't I do this to get infinite energy?</a> for the basis of General Relativity. This tells you the physical reason why time runs slower... | 534 |
general relativity | How does the gravitational field of a massive body affect the orbit of that body around yet another massive body? | https://physics.stackexchange.com/questions/574129/how-does-the-gravitational-field-of-a-massive-body-affect-the-orbit-of-that-body | <p>If I understand the relativistic explanation of gravitation - that it is curvature of spacetime - then a particle left alone will travel along that curved spacetime in a path that depends on the local curvature at each instant in time (right?). If so, then the orbit of a small body without significant effects on th... | <p>Yes, the gravitational field of the smaller body does have an impact on its own orbit. This effect is known as the “gravitational self force”. The gravitational field of the smaller body can be separated in two parts. The first is the “direct” field of the body. This is akin to the field it would produce in complete... | 535 |
general relativity | Misconception in the implications of the gravitational redshift experiments | https://physics.stackexchange.com/questions/575694/misconception-in-the-implications-of-the-gravitational-redshift-experiments | <p>In both Sean Carroll's book and Misner,Wheeler,Thorne, the authors take the gravitational redshift experiments to show that spacetime has a curved geometry.
There is a paper by Harvey R Brown(published in American Journal of Physics) that challenges this viewpoint by pointing out how this is a misconception. A singl... | <p>MTW certainly didn't miss this issue. Carroll may have. The paper you linked quotes Carroll's textbook as saying of the Pound-Rebka experiment:</p>
<blockquote>
<p>simple geometry seems to imply that the [emission and reception intervals] must be the same. But of course they are not; the gravitational redshift impli... | 536 |
general relativity | Would gravity still act if all objects in a closed system somehow became stationary? | https://physics.stackexchange.com/questions/581982/would-gravity-still-act-if-all-objects-in-a-closed-system-somehow-became-station | <p>I have been trying to understand the implications of general relativity. I unfortunately don't have a good knowledge of advanced topics and I may have made some silly assumptions.</p>
<p>As far as I understand, spacetime dictates the trajectory of an object, and the object curves spacetime. Objects follow the shorte... | <ol>
<li></li>
</ol>
<p>By 'stationary', you would mean 'stationary with respect to the spatial axes of a certain inertial frame of reference'. However, a stationary object would still 'move' along the time axis -- in fact, it will 'move' as fast as it can (=at the speed of light) along the time axis, if it is stationa... | 537 |
general relativity | Coordinate Invariant Divergence | https://physics.stackexchange.com/questions/584772/coordinate-invariant-divergence | <p>I'm reading the book "Einstein Gravity in a nutshell" by Anothy Zee and I'm a bit stuck on one of the steps in the derivation for divergence in an arbitrary coordinate system. The proof goes as follows,</p>
<p>since we know
<span class="math-container">$$W^\mu\partial_\mu\phi$$</span>
where <span class="m... | <p>The integrated out term is a surface integral
<span class="math-container">$$
\int W^\mu \phi \sqrt g \,dS_\mu
$$</span>
at infinity (and not what you have written). <span class="math-container">$\phi$</span> is arbitrary, and as always in these types of arguments, can be taken to be zero at infinity. So the integra... | 538 |
general relativity | Why are these vectors perpendicular? | https://physics.stackexchange.com/questions/591847/why-are-these-vectors-perpendicular | <p>I am reading <a href="https://doi.org/10.1016/S0370-2693(00)01125-4" rel="nofollow noreferrer">this paper</a> <em>The Bardeen model as a nonlinear magnetic monopole</em> by Eloy Ayón-Beato Alberto Garcı́a. A the end where the authors prove the that the weak energy condition is satisfied, they say that the vector
<sp... | <p>Assuming that <span class="math-container">$\mathbf F$</span> is the Faraday tensor, then it is antisymmetric by definition. Therefore,</p>
<p><span class="math-container">$$E_\lambda X^\lambda = F_{\lambda \mu}X^\mu X^\lambda$$</span></p>
<p>is the contraction of an antisymmetric object <span class="math-container... | 539 |
general relativity | 1) When can a time coordinate be separated in the interval (General relativity) ? 2) Unclear proper time expression | https://physics.stackexchange.com/questions/594680/1-when-can-a-time-coordinate-be-separated-in-the-interval-general-relativity | <ol>
<li><p>One has that <span class="math-container">$ds^{2} = g_{ij}(x)dx^{i}dx^{j}$</span>. I often see that the interval is re-expressed with a time "seperation" of the form:
<span class="math-container">$$
ds^{2} = g_{00}(x)dt^{2} + \tilde{g}_{ab}dx^{a}dx^{b} \;\; a,b = 1,2,3
$$</span>
When can this be ... | <p><strong>1)</strong></p>
<p>I believe this can be done always, but I am not sure. You need to get rid of the cross terms <span class="math-container">$g_{0a}$</span> and you have 4 coordinate transformations at your disposal to get rid of the 3 metric functions, while keeping <span class="math-container">$g_{00}$</sp... | 540 |
general relativity | Gravity in compact space, like three-torus or a ball with ends identified | https://physics.stackexchange.com/questions/595287/gravity-in-compact-space-like-three-torus-or-a-ball-with-ends-identified | <p>What does gravity look like in a compact space, such as a universe with spatial periodic boundary conditions equivalent to a 3-torus, or a ball with opposite points on the surface of the ball identified? In particular, what is the equivalent to the Schwarzschild vacuum solution to the Einstein field equations? I hav... | 541 | |
general relativity | Make Newman-Penrose scalars definite spin weight | https://physics.stackexchange.com/questions/526164/make-newman-penrose-scalars-definite-spin-weight | <p>Reading through the original <a href="https://aip.scitation.org/doi/10.1063/1.1666410" rel="nofollow noreferrer">GHP paper</a>, I notice that some of the <a href="http://www.scholarpedia.org/article/Spin-coefficient_formalism" rel="nofollow noreferrer">Newman-Penrose scalars</a> do not have definite spin weight. For... | 542 | |
general relativity | Does curvature of spacetime changes the wavelength of light? | https://physics.stackexchange.com/questions/525469/does-curvature-of-spacetime-changes-the-wavelength-of-light | <p>Suppose we do an experiment on earth and light a monochromatic light near a highly dense object. Does it cause any change in the wavelength of light? </p>
| 543 | |
general relativity | Rotating dust in Gödel universe | https://physics.stackexchange.com/questions/525772/rotating-dust-in-g%c3%b6del-universe | <p>I have a question regarding the Gödel metric. Supposedly the Gödel universe is filled with rotating pressure-less dust.
However, checking different sources, it seems like Einstein's field equations are satisfied in this case for a perfect fluid without pressure and 4-velocity <span class="math-container">$u^{\mu}=\... | <p>Coordinate velocities in GR are generally meaningless. They have no special physical interpretation. To find out whether a particular spacetime has rotation, you need to consider some coordinate-independent criterion. <a href="https://en.wikipedia.org/wiki/G%C3%B6del_metric#Rigid_rotation" rel="nofollow noreferrer">... | 544 |
general relativity | Gravitational motion of 2 point masses in free space | https://physics.stackexchange.com/questions/526989/gravitational-motion-of-2-point-masses-in-free-space | <p>I came across this question:</p>
<p><strong>"If there are two point masses in free space(i.e., there is no other mass/force/field acting in their vicinity), will those two point masses get closer to each other, or will they remain stationary as they are?"</strong></p>
<p>I approached this question through Newtonia... | <p>if the two masses do not circle with the right speed around their center of mass, the will move to get closer to each other, in no case they will stay at there initial distance. So the statement you heard is false.
trula</p>
| 545 |
general relativity | Does the Unruh Effect violate observer independence in General Relativity? | https://physics.stackexchange.com/questions/528820/does-the-unruh-effect-violate-observer-independence-in-general-relativity | <p>Observer independence means that the physic involved is independent of the reference frame of the observers but observers can't agree on the vacuum temperature due to the the Unruh Effect, does this not violate GR's principle? If we can't agree on the energy we can't agree on curvature, correct?</p>
| <p>If you are at rest on the surface of the earth you receive the same Unruh temperature <span class="math-container">$T = \hbar g/(2\pi c k_b)$</span> as if you were in a rocket accelerating with <span class="math-container">$a=1g$</span> (see <a href="https://www.youtube.com/watch?v=qPKj0YnKANw&t=9m26s" rel="nofo... | 546 |
general relativity | If a photon does not experience time, how is it affected by gravitational pull of masses that might not yet be in its path? | https://physics.stackexchange.com/questions/532814/if-a-photon-does-not-experience-time-how-is-it-affected-by-gravitational-pull-o | <p>Ok, I apologize in advance if this has been asked before. I tried googling for it, but didnt find anything related. Im a comp sci guy, not a physics guy, so its possible Im not googling the right terms. </p>
<p>So my understanding is that: </p>
<p>A) Photons of light do not experience time in their reference frame... | <p>In the frame of an observer, a photon follows <a href="https://en.wikipedia.org/wiki/Geodesics_in_general_relativity" rel="nofollow noreferrer">a geodesic.</a> . In the observer's framemork the moving masses create moving geodesics for the incoming photon which is traveling with the speed of light in vacuum. In the ... | 547 |
general relativity | Term for radius and gradient of spacetime distortion? | https://physics.stackexchange.com/questions/541651/term-for-radius-and-gradient-of-spacetime-distortion | <p>A black hole would distort spacetime time to a greater degree than planet Earth. That is, both the radius and the gradient of the distortion are greater. </p>
<p>Is there a term that combines "greater radius" and "greater gradient" into one?</p>
<p>In layman terms, a black hole distorts spacetime more "aggressivel... | <p>Spacetime distortion is measured by the <a href="https://en.wikipedia.org/wiki/Riemann_curvature_tensor" rel="nofollow noreferrer">Riemann curvature tensor</a> <span class="math-container">$R_{\mu\nu\lambda\kappa}$</span>. This tensor has 256 components, but various symmetries reduce the number of independent compon... | 548 |
general relativity | Do gravitational fields exist in vacuum region? | https://physics.stackexchange.com/questions/137153/do-gravitational-fields-exist-in-vacuum-region | <p>I was reading about "vacuum solution" in wiki, <a href="http://en.wikipedia.org/wiki/Vacuum_solution_(general_relativity)" rel="noreferrer">http://en.wikipedia.org/wiki/Vacuum_solution_(general_relativity)</a>. There are some words I'm confused.</p>
<blockquote>
<p>1.In general relativity, a vacuum solutio... | <p>That article's choice of words could certainly be improved. Basically yes, $T$ captures all the non-gravitational "stuff."</p>
<hr>
<p>The idea of a "gravitational field" doesn't really fit in to GR. There is stress-energy $T$ everywhere, and there is a metric $g$ everywhere, and that's really all you need to defi... | 549 |
general relativity | Gravitational time dilation, does time of the observer at a lower gravitational potential looked slowed down in the frame of the higher one | https://physics.stackexchange.com/questions/154061/gravitational-time-dilation-does-time-of-the-observer-at-a-lower-gravitational | <p>This question is mainly inspired after watching the movie known as Interstellar</p>
<p>We knew that for time dilation caused by relativistic motion between A and B. A will measure <strong>B's clocks slowing down</strong>, and B will measure <strong>A's clock slowing down</strong> by the same rate, while they both me... | <p>Yes, the observers on Earth will measure the astronaut's clocks to be be running slow while the astronauts will measure the clocks on Earth to be running fast. So the situation is asymmetric.</p>
<p>The situation is asymmetric because the two sets of clocks are in different environments. Specifically there is a (gr... | 550 |
general relativity | How would Einstein calculate acceleration of a ball? | https://physics.stackexchange.com/questions/561508/how-would-einstein-calculate-acceleration-of-a-ball | <p>Let's say that we have two homogeneous spherical balls - one with mass <span class="math-container">$m_1=1000m$</span> and radius <span class="math-container">$r_1=1000r$</span> and second with mass <span class="math-container">$m$</span> and radius <span class="math-container">$r$</span>. Distance between centers o... | 551 | |
general relativity | An easy way to determine the space-like and time-like paths on a spacetime manifold based on linear algebra? | https://physics.stackexchange.com/questions/565750/an-easy-way-to-determine-the-space-like-and-time-like-paths-on-a-spacetime-manif | <p>I think I have found a way to easily understand time-like and space-like paths with the contect of a little linear algebra. My question is: is my understanding, below, correct?</p>
<p>When I learned General Relativity, one source of some confusion was how to find the actual time or distance from a metric. In particu... | <p>You say: "It is very easy to show that there is a linear transformation, P, of coordinates that will put M in "canonical" form, with all -1s or +1s on the diagonal." This is true at any one point. However, it's not true in a neighborhood. For any small 4-dimensional region of spacetime, <span cla... | 552 |
general relativity | What will be the "Einstein field equations" for two or three bodies? | https://physics.stackexchange.com/questions/566281/what-will-be-the-einstein-field-equations-for-two-or-three-bodies | <p>In general theory of relativity the Einstein field equations e.g. relate the geometry of space-time with the distribution of one body within it.
<span class="math-container">$$R_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}R+g_{\mu\nu}\Lambda=\dfrac{8\pi G}{c^4}T_{\mu\nu}.$$</span></p>
<p>What will be the "Einstein field equ... | <p>They are already included, that is a <strong>field</strong> equation. So the energy-momentum tensor on the right must include the distribution of energy (matter) and momentum in your model, all of it. Normally it is used the other way around, one takes a particular form of the energy momentum tensor to model a singl... | 553 |
general relativity | When is a spacetime a black hole? | https://physics.stackexchange.com/questions/233034/when-is-a-spacetime-a-black-hole | <p>While reading $\textrm{Present status of the Penrose Inequality}$ by Marc Mars, 2009, I was confused with the following statement:</p>
<blockquote>
<p>... in order to determine whether a space-time is a black hole, detailed knowledge of its global future behaviour is required.</p>
</blockquote>
<p>Why is that ?<... | <p>The most common definition of a black home is the portion of the spacetime manifold $\mathcal{M}$ that is $\mathcal{M} - J^-(ℐ^+)$, the manifold minus the causal past of null future infinity. That is, it's the region of spacetime where no signal can escape to infinity at some point in the future. This requires you t... | 554 |
general relativity | Elementary question about non-Euclidean geometry in general relativity: "cannot move about without changing shape" | https://physics.stackexchange.com/questions/244842/elementary-question-about-non-euclidean-geometry-in-general-relativity-cannot | <p>One basic result of general geometry (from math) in curved spaces or on curved surfaces is that if you are in a surface of variable curvature, things like the Euclidean congruence postulates and theorems for triangles and other simple figures fail, and the reason this is is because those are actually strong statemen... | <p>It isn't necessarily objects that change. This is a 4 dimensional space with 3 "space" dimensions and 1 time dimension. </p>
<p>The definition of curvature can be stated as when you go around what should be a rectangle, you don't come back to the same place. Or equivalently, if you take the two different paths to t... | 555 |
general relativity | Can we distinguish between two mass distributions in spacetime having the same effect over a test partlicle | https://physics.stackexchange.com/questions/244884/can-we-distinguish-between-two-mass-distributions-in-spacetime-having-the-same-e | <p>Einstein's equation is</p>
<p>$$8πT_{ab}=G_{ab}$$</p>
<p>where the left side contains the stress-energy tensor and the right side contains the Einstein tensor. </p>
<p>Is there exactly one unique stress-energy tensor corresponding to a given spacetime curvature? Or is it possible for one curvature (i.e. one space... | 556 | |
general relativity | I can't understand where the minus sign in the second equation is coming from | https://physics.stackexchange.com/questions/696194/i-cant-understand-where-the-minus-sign-in-the-second-equation-is-coming-from | <p><span class="math-container">$R_{\mu v}=\frac{\partial}{\partial x^{\lambda}} \Gamma_{\mu v}^{\lambda}+\Gamma_{\mu \lambda}^{\eta} \Gamma_{v \eta}^{\lambda}$</span></p>
<p>equation (2) after the multiplication of the meteric tensor</p>
<p><span class="math-container">$g^{\nu\sigma}R_{\mu v}=\frac{\partial}{\partial ... | <p>Metric compatibility
<span class="math-container">$$
\nabla_\lambda g_{\mu\nu} =\partial_\lambda g_{\mu\nu}+ g_{\alpha\nu}{\Gamma^\alpha}_{\mu\lambda}+ g_{\mu\alpha}{\Gamma^\alpha}_{\nu\lambda}=0. $$</span></p>
| 557 |
general relativity | Penrose diagram of a star in AdS? | https://physics.stackexchange.com/questions/527460/penrose-diagram-of-a-star-in-ads | <p>Often the Penrose diagram is drawn for pure AdS or AdS-Schwarzschild black hole. What is the Penrose diagram for a star that is not collapsing in AdS?</p>
| 558 | |
general relativity | Does GR reflect Aristotelian time? | https://physics.stackexchange.com/questions/167044/does-gr-reflect-aristotelian-time | <p>This is really a philosophical question, which I've already asked at Phil.SE but I'm asking here with more physics detail.</p>
<p>In Newtonian Mechanics, space and time are independent of each other, and of motion; in that the geometry of space time is independent of the distribution of matter and energy and it's mo... | <p>There are a couple notions of time in GR. Typically when you write down a space you do so in some set of coordinates $(t,x,y,z)$. The $t$ there is the <em>coordinate time</em>. It marches forward forever, and doesn't care at all what the matter and energy content of the system may be. This seems very similar to ... | 559 |
general relativity | Derivation of Teukolsky equation doesn't match | https://physics.stackexchange.com/questions/713899/derivation-of-teukolsky-equation-doesnt-match | <p>I am trying to derive Teukolsky equation on Kerr spacetime using SageMath and</p>
<p>(1) [(Δ+3γ−γ∗+4μ+μ∗)(D+4ϵ−ρ)−(δ∗−τ∗+β∗+3α+4π)(δ−τ+4β)−3ψ2]ψ4=0.</p>
<p>Like he did in his <a href="https://articles.adsabs.harvard.edu/pdf/1973ApJ...185..635T" rel="nofollow noreferrer">paper</a>. But I did not have success, the exp... | 560 | |
general relativity | Does curvature of spacetime depend upon the "mass" or "density" of a object? | https://physics.stackexchange.com/questions/275562/does-curvature-of-spacetime-depend-upon-the-mass-or-density-of-a-object | <p>Suppose we have a object with mass "M" with small density and a object with same mass "M" but different density (like a large density). Does the curvature of spacetime same for the two object with same mass but different densities?</p>
| <p>To start with, as pointed out in the comments, your question is not very precise. I will assume you are referring to spherically symmetric bodies. Then Birkhoff's theorem implies that the exterior is described by the Schwarzschild solution, so as long as we are in the exterior of both objects they are indeed equiva... | 561 |
general relativity | Is the curvature of space-time a smooth function everywhere ? (except at black holes) | https://physics.stackexchange.com/questions/1628/is-the-curvature-of-space-time-a-smooth-function-everywhere-except-at-black-h | <p>Is the curvature of space-time a smooth function everywhere (except at black holes) in view of General relativity. By 'smooth' it is meant that it possesses derivatives of all order at a given point. </p>
| <p>No, not at the boundary of a solid object like a planet. There's a step function in the stres-energy tensor, and so you'll have a step function in the Riemann tensor.</p>
| 562 |
general relativity | The Matter-Vacuum Boundary in General Relativity | https://physics.stackexchange.com/questions/3751/the-matter-vacuum-boundary-in-general-relativity | <p>A previous <a href="https://physics.stackexchange.com/questions/1628/is-the-curvature-of-space-time-a-smooth-function-everywhere-except-at-black-ho">Stack question</a> (before I joined) asking about continuity in GR received replies which suggested that Curvature would be discontinuous at say a planetary boundary (a... | <p>Roy, your wishful thinking is manifestly impossible. If the tensor $T_{\mu\nu}$ is discontinuous, and it surely is on the surface of a solid, then Einstein's equations guarantee that the Einstein tensor $G_{ab}$ is discontinuous as well - up to a normalization, it's the same tensor. It follows that the Ricci tensor ... | 563 |
general relativity | Is there an energy density limit in GR? | https://physics.stackexchange.com/questions/7771/is-there-an-energy-density-limit-in-gr | <p>I am speaking about GR with classical fields and energy. One question, spread over three increasingly strict situations:</p>
<p>Is there an energy density limit in GR? (literally, can the energy density have an arbitrarily large value at some point in space at some point in time)</p>
<p>Is there an energy density ... | <p>The answer is NO. There is no energy density limit (for all three questions).</p>
<p>The easiest way to see this is that the energy density is just the $T^{00}$ component of the stress energy tensor. The solution in GR depends on the <em>full</em> stress energy tensor, so it is not enough to just talk about the e... | 564 |
general relativity | Decomposing geodetic/de Sitter effect into Thomas precession and spatial curvature | https://physics.stackexchange.com/questions/8043/decomposing-geodetic-de-sitter-effect-into-thomas-precession-and-spatial-curvatu | <p>According to Rindler the geodetic effect can be considered as consisting of Thomas precession combined with the effect of moving through curved space.</p>
<p>Wolfgang Rindler (2006) Relativity: special, general, and cosmological (2nd Ed.) p234</p>
<p>However according to Misner, Thorne, and Wheeler, Gravitation, p... | <p>The difference between Rindler's wording and the MTW wording is just a difference in the choice of coordinates.</p>
<p><strong>Thomas precession in STR</strong></p>
<p>First, what is the Thomas precession? It is a special relativistic effect so the original derivation of the Thomas precession only applies in flat ... | 565 |
general relativity | Perturbation of a Schwarzschild Black Hole | https://physics.stackexchange.com/questions/8307/perturbation-of-a-schwarzschild-black-hole | <p>If we have a perfect Schwarzschild black hole (uncharged and stationary), and we "perturb" the black hole by dropping in a some small object. For simplicity "dropping" means sending the object on straight inward trajectory near the speed of light.</p>
<p>Clearly the falling object will cause some small (time depen... | <p>Your intuitive picture is basically correct. If you perturb a black hole it will respond by "ringing". However, due to the emission of gravitational waves and because you have to impose ingoing boundary conditions at the black hole horizon, the black hole will not ring with normal-modes, but with quasi-normal modes ... | 566 |
general relativity | A question on an assumption of space-time | https://physics.stackexchange.com/questions/10329/a-question-on-an-assumption-of-space-time | <blockquote>
<p>"A four-dimensional differentiable (Hausdorff and paracompact) manifold $M$ will be called a space time if it possesses a pseudo-Riemannian metric of hyperbolic normal signature $(+,-,-,-)$ and a time orientation. There will be no real loss of generality in physical applications if we assume that $M$ ... | <p>Dear Rajesh, in reality, physics sometimes works with continuous functions that are not infinitely differentiable - for example look at the energy of the beam at atlas.ch (click at the "Status" button in the middle) when they ramp it up - there are all kinds of discontinuities.</p>
<p>But an arbitrary function that... | 567 |
general relativity | Is Einstein's 1916 General Relativity paper a recommended way to start learning about the subject? | https://physics.stackexchange.com/questions/14241/is-einsteins-1916-general-relativity-paper-a-recommended-way-to-start-learning | <p>If a person has a good grounding in classical mechanics, electrodynamics and special relativity, is Einstein's 1916 paper a recommended way of learning about the subject?</p>
<p>After looking through it briefly, I like what I see because he explains all about tensors from first principles. On the other hand, I'm no... | <p>No. It is not a good starting point. If nothing else, modern notation is very different from Einstein's original notation. Old notation left a lot to be desired about separating tensors from tensor components, if nothing else. </p>
<p>There has also been a lot of new insight into topology, surface charges, the ... | 568 |
general relativity | Schwarzschild metric | https://physics.stackexchange.com/questions/15187/schwarzschild-metric | <p>Why, if the Schwarzschild metric is a vacuum solution ($T_{\mu\nu}=0$) , do textbooks state that $T=\rho c^{2}$ when approximating Poisson's Equation from the Einstein Field Equations? </p>
<p>Thank you.</p>
| <p>There is a coordinate slicing known as the Kerr-Schild coordinate system where one can look at the Hamiltonian constraint $16\pi \rho = {}^{3}R - K^{ab}K_{ab} + K^{2}$, and find that the left hand side has the same singularity that you would find in $\nabla \cdot E = \rho$ when you put in the $E$ for a point charge.... | 569 |
general relativity | Is 4-velocity normalized to -1 even for non-geodesic timelike curves? | https://physics.stackexchange.com/questions/16852/is-4-velocity-normalized-to-1-even-for-non-geodesic-timelike-curves | <p>In Hartle's General Relativity book ("Gravity"), one of the problems (chapter 8 problem 6) is to prove that $g_{\mu\nu}u^\mu u^\nu$ is conserved along geodesics (really not hard to show), where $u^\mu$ is the 4-velocity. My question is: Isn't it true that $g_{\mu\nu}u^\mu u^\nu$ is equal to $-1$ for <em>any</em> tim... | <p>I emailed my TA and here was his answer, which I think makes sense:</p>
<p>While it <em>is</em> true that a curve which is everywhere timelike can be parametrized so that its tangent vector has unit norm, it is also possible to draw a curve which starts out timelike and then becomes null or spacelike, so its norm w... | 570 |
general relativity | Proper distance and embedding diagrams? | https://physics.stackexchange.com/questions/18553/proper-distance-and-embedding-diagrams | <p>I'm trying to understand proper distance equation in Schwarzschild spacetime.</p>
<p>$d\sigma=\frac{dr}{\left(1-\frac{R_{S}}{r}\right)^{1/2}}$.</p>
<p>I'm sure I'm missing something really obvious here, but how do I use this to find the coordinate distance $r$
for a particular proper distance $\sigma$
. For exa... | <p>You have the Schwarzschild metric</p>
<p>$ds^2=(1-R_s/r)c^2dt^2-(1-R_s/r)^{-1}dr^2-r^2(d\theta^2+sin^2\theta d\phi^2)$</p>
<p>For an equatorial orbit, put $\theta=\pi/2; d\theta=0$.
The proper distance between two events is defined as the integral of $ds$ along a spacelike path between them. I'm guessing the two ... | 571 |
general relativity | Projective Transformations in GR | https://physics.stackexchange.com/questions/18616/projective-transformations-in-gr | <p>A Thought Experiment:</p>
<p>We are in flat spaceime provided with a reference frame—a rectangular Cartesian frame. The coordinate labels[the spatial labels] are visible to us. Each spatial point is provided with a clock—and the different clocks are synchronized wrt to each other. Gravity is now turned on and made ... | <p>What you are talking about are not what are normally called projective transformations, but nonphysical coordinate changes, these are gauge changes in GR. Einstein was famously confused about this for years, beginning in 1913 when he wrote about the "Hole Argument" in General Relativity. He concluded that it is impo... | 572 |
general relativity | Modification of de Donder gauge | https://physics.stackexchange.com/questions/19684/modification-of-de-donder-gauge | <p>The de Donder gauge is often used to simplify the linearised equations of motion of general relativity. If the metric is linearised as $g_{ab} = \bar g_{ab} + \gamma_{ab}$, then the de Donder gauge reads<br>
$\nabla^a(\gamma_{ab} - \frac{1}{2}\bar g_{ab}\gamma) = 0$.</p>
<p>The partial differential equation for the... | <p>It may be far too late for this to be of any value to you, but in this paper by Mora et al: <a href="http://arxiv.org/abs/1205.4468" rel="nofollow">http://arxiv.org/abs/1205.4468</a>, it is shown that such a gauge choice does exist (for any $n$ in your notation).</p>
| 573 |
general relativity | What should be the equation satisfied by The Momentum commutators in a curved background? | https://physics.stackexchange.com/questions/21628/what-should-be-the-equation-satisfied-by-the-momentum-commutators-in-a-curved-ba | <p>This may be obvious but I have limited experience in physics , The generators of Spatial translation symmetry commutes with each other i.e [P(i),P(j)] = 0 but if Spacetime is a curved manifolds then the value of the commutator should not be zero but some invariant property related to curvature i.e a Function of the ... | <p>The commutator you are interested in is non-trivial if you generalize the
translations to curvilinear coordinates. For a vector function $A^{\alpha
}\left( x\right) $, a «translation» along $dx^{\mu}$ is the following
transformation:
$$
A^{\alpha}\rightarrow A^{\alpha}-dx^{\mu}D_{\mu}A^{\alpha}
$$
(where $D_{\mu}$... | 574 |
general relativity | Tension on a cable in a gravitational field | https://physics.stackexchange.com/questions/22292/tension-on-a-cable-in-a-gravitational-field | <p>Consider a mass 'm' suspended in the gravitational field of a massive star. Assuming the Schwarzschild metric it is easy to calculate the gravitational acceleration at the location of the mass and thus the tension in the cable. The question is: how does this tension propagate up the cable?</p>
<p>I've tried to ap... | <p>Your naïve interpretation of the work equation doesn't quite make sense in this context. Consider that the standard formula for work gives that $W=\int {\vec F}\cdot d{\vec x}$. At minimum, the presence of the dot product in the above equation should not be ignored, and we should interpret, for a radial force, the... | 575 |
general relativity | Derivation of the Gauss-Codazzi equation | https://physics.stackexchange.com/questions/23749/derivation-of-the-gauss-codazzi-equation | <p>I'm interested in the derivation of the Gauss equation (Gauss-Codazzi). Usually we consider the definition of the Riemann tensor on the hypersurface.</p>
<p>$$^{(n-1)}R_{abc}^{~~~~~~~d}~w_d=[D_a,D_b]w_c$$</p>
<p>where $D$ is the connexion associated to the induced metric ($h$) on the hypersurface.</p>
<p>We have<... | 576 | |
general relativity | The Light Cone in GR-----A Flickering One? | https://physics.stackexchange.com/questions/24514/the-light-cone-in-gr-a-flickering-one | <p>Events in relativity[SR or GR]are marked by coordinate values and not by physical values.We write a metric for motion along the x-axis:
$$ds^2=g_{00}dt^2-g_{11}dx^2$$ ----------- (1)</p>
<p>For physical values we may write:</p>
<p>$$ds^2=dT^2-dL^2$$ ------------ (2)</p>
<p>Where $dT^2=g_{00}dt^2$ and $dL^2=g_{11}... | <p>The equation for determining the light-cone in the case where there are two varying metric coefficients is the one you wrote down---</p>
<p>$$ {dx\over dt} = \sqrt{g_{00}\over g_{11}} $$</p>
<p>And this does make a curved line in the coordinate-labelled spacetime. This is just stating that light will bend in respo... | 577 |
general relativity | what is relation between time and space in general relativity? | https://physics.stackexchange.com/questions/30877/what-is-relation-between-time-and-space-in-general-relativity | <p>there is a relation between time and space in special theory of relativity:
$$t^2c^2-L^2=\tau^2.c^2$$
what is relation between time and space in general relativity?</p>
| <p>The remarkable property of spacetime in GR is that it is <em>locally</em> that of SR. Or, more technically, tangent to every event in the curved spacetime of GR is an SR spacetime. What this means is that, to first order, the line element at any event can be put into the (differential) form of SR in some coordinat... | 578 |
general relativity | Is a volumetric rate frame-invariant in general relativity? | https://physics.stackexchange.com/questions/38986/is-a-volumetric-rate-frame-invariant-in-general-relativity | <p>Imagine that I have a radioactive material with a long half life. The atoms in this material decay at a certain rate $R$. The rate is the decay constant times the number density $R = \lambda N $. It has dimensionality:</p>
<p>$$ \left( \frac{ \text{decays} }{m^3 s} \right) $$</p>
<p>Imagine that the material i... | <p>I don't know whether it applies to all physically possible metrics, but the volumetric decay rate you define does stay constant in a Schwarzschild metric. Well, it does if the box is small compared to the curvature i.e. the time dilation etc is constant thoughout the box. I would need to think more about what happen... | 579 |
general relativity | Do Christoffel symbols commute? | https://physics.stackexchange.com/questions/41437/do-christoffel-symbols-commute | <p>Do <a href="http://mathworld.wolfram.com/ChristoffelSymbol.html" rel="nofollow">Christoffel symbols</a> commute? For example, does $\Gamma^{e}_{db}\Gamma^{c}_{ea} = \Gamma^{c}_{ea}\Gamma^{e}_{db}$?</p>
| <p>In classical theory, all observables commute. The components $\Gamma^a_{bc}$ are just real numbers so of course that they commute.</p>
<p>In quantum theory, they don't commute. It's probably a bit laborious to calculate the commutator.</p>
| 580 |
general relativity | How do the Einstein's differential equation of the curvature of spacetime come out of Einstein's field equation? | https://physics.stackexchange.com/questions/45307/how-do-the-einsteins-differential-equation-of-the-curvature-of-spacetime-come-o | <p>The classical theory of spacetime geometry that we call gravity consists of the Einstein equation, which relates the curvature of spacetime to the distribution of matter and energy in spacetime.
$ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$
<strong>Mathematically</strong>, how do the Einstein's differential equation of the curv... | <p>$ds^{2} = g_{ab}dx^{a}dx^{b}$ isn't the Einstein equation. It's just the equation for what arc length is. It's the definition of the metric tensor, pretty much. You're implicitly using it if you've ever done calculus in three dimensions.</p>
<p>The Einstein equation and the Einstein field equations are the same ... | 581 |
general relativity | How does one write the Einstein field equations in terms of Ricci tensor? | https://physics.stackexchange.com/questions/50141/how-does-one-write-the-einstein-field-equations-in-terms-of-ricci-tensor | <p>How can I go from the 'standard' Einstein equations
$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = \frac{8\pi G}{c^4}T_{\mu\nu}$ to these equations:
$R_{\mu\nu} = \frac{8\pi G}{c^4}(T_{\mu\nu} - \frac{1}{2}g_{\mu\nu}T)$?</p>
| <p>Take the trace of the equation by contracting it with $g^{\mu\nu}$:</p>
<p>$$
g^{\mu\nu}R_{\mu\nu}-\dfrac{1}{2}g^{\mu\nu}g_{\mu\nu}R=\dfrac{8\pi G}{c^4}g^{\mu\nu}T_{\mu\nu}
$$</p>
<p>As $g^{\mu\nu}R_{\mu\nu} = R$, $g^{\mu\nu}T_{\mu\nu} \equiv T $ and $g^{\mu\nu}g_{\mu\nu} = 4$, the previous equation gives you $R =... | 582 |
general relativity | General Relativity Equivalence | https://physics.stackexchange.com/questions/55496/general-relativity-equivalence | <p>Is Einsteins Equivalence theorem in General Relativity correct? It seems to me that it neglects the fact that gravitational acceleration depends upon separation distance squared, thus neglecting the effect of tidal forces. </p>
<p>For example, as I sit on earth, I experience the affect of earth's gravity; Although ... | <p>The equivalence principle, as stated correctly by Einstein, says that these two situations are equivalent:</p>
<ul>
<li>An uniformly accelerating observer in the absence of a gravitational field</li>
<li>A free falling observer in an uniform gravitational field</li>
</ul>
<p>So, as you noted, this does not apply t... | 583 |
general relativity | General Relativity Paradox - Different local times of two frames a constant distance apart | https://physics.stackexchange.com/questions/56998/general-relativity-paradox-different-local-times-of-two-frames-a-constant-dist | <ul>
<li>Suppose there is a habitable star with a significantly large mass, and thus a huge gravitation field. It has a clock on it that ticks each local second. And it also has a mirror. This is Star A.</li>
<li>Suppose there is another habitable star with a much smaller mass, also with a clock, called Star B. </li... | <p>Time is not the only measurement that is affected by a gravitational field. What makes you think that A and B measure the same distance between them? It helps to think about how you would actually measure the distance to a faraway object. If you are patient, you could do this measurement by bouncing a light beam off... | 584 |
general relativity | Understanding Einstein's field equation | https://physics.stackexchange.com/questions/61179/understanding-einsteins-field-equation | <p><a href="http://en.wikipedia.org/wiki/Einstein_field_equations">Einstein's field equation</a>:</p>
<p>$$G_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu} - g_{\mu\nu}\Lambda$$</p>
<p>I'm trying to understand each of the terms in this equation intuitively, but I'm struggling.</p>
<p>Basically, I want to understand how the... | <p>$G_{\mu \nu}$ is the Einstein tensor, and is calculated by the following:</p>
<p>$$G_{\mu \nu} = R_{\mu \nu} -\frac{1}{2}g_{\mu \nu}R$$</p>
<p>where $R_{\mu \nu}$ is the Ricci curvature tensor, $g_{\mu \nu}$ the metric tensor and $R$ is the trace of the curvature tensor with respect to the metric, i.e. $g^{\mu \nu... | 585 |
general relativity | Timelike/null generic condition in general relativity | https://physics.stackexchange.com/questions/70193/timelike-null-generic-condition-in-general-relativity | <p>My question concerns the following definition</p>
<blockquote>
<p><strong>Definition:</strong> The <em>timelike</em> (resp. null) <em>generic condition</em> in GR is fulfilled if
$$u_{[\alpha} R_{\rho]\mu \nu [\sigma}u_{\beta]}u^\mu u^\nu \ne 0$$
at some point of each timelike (resp. null) geodesic with tan... | <p>Carroll has this to say about this condition</p>
<blockquote>
<blockquote>
<p>These fancy conditions simply serve to exclude very special metrics for which the curvature consistently vanishes in some directions - Carroll P. 242-243</p>
</blockquote>
</blockquote>
<p>While that answers your question, it doe... | 586 |
general relativity | Conservation of Energy and Birkhoff's theorem | https://physics.stackexchange.com/questions/71952/conservation-of-energy-and-birkhoffs-theorem | <p>I am reading the original paper by Bondi, van der Berg and Metzner (<a href="http://rspa.royalsocietypublishing.org/content/269/1336/21" rel="nofollow">link</a>) regarding gravitational waves in asymptotically flat axisymmetric spacetimes. In the introduction, he makes the following comment - </p>
<blockquote>
<p... | <p>All you need to do is set up a multipole expansion of the gravitational waveform. You'll find that the monopole moment is proportional to the time derivative of the mass of the stress-energy tensor, and the dipole moment is proportional to the second time derivative of the momentum from the stress-energy tensor, bo... | 587 |
general relativity | About an Einstein equation | https://physics.stackexchange.com/questions/72054/about-an-einstein-equation | <p>This is a question about an historical theory of gravitation, studied by Einstein quite a bit <em>before</em> he settled on General Relativity. At that time, Einstein did not know that gravity was a consequence of curved space-time. He identified the variations of gravity with the variations of light speed in a grav... | <p>There is a derivation of the equations above given by Giulini (may be more pedagogical?), You can look at it at :</p>
<p><a href="http://ae100prg.mff.cuni.cz/presentations/Giulini_Domenico.pdf" rel="nofollow">http://ae100prg.mff.cuni.cz/presentations/Giulini_Domenico.pdf</a></p>
<p>As you will see he arrives at th... | 588 |
general relativity | Axial symmetry constraints on the metric | https://physics.stackexchange.com/questions/72074/axial-symmetry-constraints-on-the-metric | <p>I am reading the paper on Gravitational Waves in General Relativity. VII. Waves from Axi-Symmetric Isolated Systems by H. Bondi, M. G. J. van der Burg, A. W. K. Metzner. (<a href="http://rspa.royalsocietypublishing.org/content/269/1336/21" rel="nofollow noreferrer">link</a>) Here is a quote(s) from that paper</p>
<b... | <p>I finally figured it out. It's simple really. He requires that we have radial null geodesics right? This implies that the geodesic equation
$$
\frac{d^2 x^\lambda}{d \tau^2} + \Gamma^\lambda_{\mu\nu} \frac{d x^\mu}{d \tau} \frac{d x^\nu}{d \tau} = 0
$$
should be solved by $(u,\theta,\phi)$ constant. Plugging this in... | 589 |
general relativity | Are orbits reversible in general relativity? | https://physics.stackexchange.com/questions/72359/are-orbits-reversible-in-general-relativity | <p>It seems if I reverse velocities then things begin orbiting backwards, at least in classical mechanics.</p>
<p>From <a href="https://en.wikipedia.org/wiki/Orbital_mechanics#Laws_of_astrodynamics" rel="nofollow">here</a>:</p>
<blockquote>
<p>Every orbit and trajectory outside atmospheres is in principle reversibl... | <p>Yes, the Schwarzchild space-time is reversible. Closed orbits and the like will stay closed in the time-reversed system. </p>
<p>There is, of course one obvious flaw: what about the horizon? Things go in, but they do not go out. Well, the answer to that is that the full spacetime doesn't JUST include a black ho... | 590 |
general relativity | Is it possible that a matter field has a dependent on non-radial space-like coordinate in a spacetime with spherical symmetry? | https://physics.stackexchange.com/questions/78608/is-it-possible-that-a-matter-field-has-a-dependent-on-non-radial-space-like-coor | <p>After the work from Breitenlohner and Freedman, we know matter fields in asymptotically AdS spacetime can be stable out of the black hole under some special conditions.</p>
<p><strong>My question:</strong> In such a spherically symmetric spacetime, could the matter field has the non-radial variable of the coordinat... | 591 | |
general relativity | How strong is the spacetime curvature at distance $d$ for a nonmoving point mass? | https://physics.stackexchange.com/questions/92769/how-strong-is-the-spacetime-curvature-at-distance-d-for-a-nonmoving-point-mass | <p>Consider a point mass $A$ with mass $m$ in empty space. The point mass $A$ does not have a velocity and does not rotate.
Since gravity is symmetric for nonmoving objects, the spacetime curvature around $A$ is also symmetric.</p>
<p>So at a distance $d$ from the point mass $A$ how strong is the curvature $C$ ?</p>
... | <p>It sounds as if you just want the acceleration given by the non-relativistic equation from Newton's law:</p>
<p>$$ a = \frac{GM}{r^2} $$</p>
<p>where $M$ is the mass of the object generating the gravitational field (strictly speaking this equation only applies when the mass of the accelerating object is much less ... | 592 |
general relativity | Is it possible to express "free"-ness of a time-like world line without referring to "tangent space" (but only directly to causal relations )? | https://physics.stackexchange.com/questions/93037/is-it-possible-to-express-free-ness-of-a-time-like-world-line-without-referrin | <p>I don't know much about <a href="http://ncatlab.org/nlab/show/tangent+bundle" rel="nofollow noreferrer"><i>tangent spaces</i>, or <i>tangent vectors</i>, "as such"</a>; nor about <a href="http://en.wikipedia.org/wiki/Schild%27s_ladder#affine_parametrization" rel="nofollow noreferrer"><i>affine parametrization</i></a... | <p>Since I'm in a car right now I can't double check my answer, but I'll try my best not to lie to you. </p>
<p>To begin with I am a little unclear as to what you are asking. In most circumstances (i.e if the region under consideration is geodesically complete, which in physics is the case if there is no singularity p... | 593 |
general relativity | Sobolev norm for Schwarzschild metric | https://physics.stackexchange.com/questions/100996/sobolev-norm-for-schwarzschild-metric | <p>Considering a static spacetime of the metric form
\begin{equation}
\mathrm{d}s^{2}=-V^{2}\mathrm{d}t^{2}+h_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j}
\end{equation}
with a timelike killing field $\xi^{\mu}=(\partial_{t})^{\mu}$ we
can choose a function space on each constant time hypersurface $\Sigma$
as$\mathcal{H}=\left\... | <p>Notice that if $r<2M$, the coordinate $t$ turns out to be <strong>spacelike</strong>, so surfaces at $t$-constant are not appropriate for imposing, for instance, Cauchy data.
(I guess that your Sobolev norms are used to deal with the Cauchy problem). </p>
<p>If you are dealing with the whole Kruskal manifold yo... | 594 |
general relativity | The border of the System | https://physics.stackexchange.com/questions/111766/the-border-of-the-system | <p>In General Relativity, if the system accelerates, the inside of the system and the outside of the system will have different speed of time.</p>
<p>Where is the boundary of the system? </p>
<p>If a human accelerates closer to the speed of light, does that mean the human exist inside of their specific system? </p>
| 595 | |
general relativity | Stationary and Static | https://physics.stackexchange.com/questions/112670/stationary-and-static | <p>I have some confusion about the concept of stationary and static.</p>
<p>A metric $g$ is called <strong>stationary</strong> if there is a time like Killing vector $K$.</p>
<p>$g$ is called <strong>static</strong> is further HSO (hyper surface orthogonal), i.e. there exists a foliation of hyper surface orthogonal t... | 596 | |
general relativity | What is the effective physical difference between a massive region of a polarized vacuum and a region of curved space-time? | https://physics.stackexchange.com/questions/114091/what-is-the-effective-physical-difference-between-a-massive-region-of-a-polarize | <p>What is the <em>effective</em> physical difference between a large region of curved space-time and an equally large region of a polarized vacuum? Consider the fact that vacuum polarization amounts to an effective deviation in speed of light in a local region due to the deviation in electric permittivity of free spa... | 597 | |
general relativity | Hyper surface orthogonal vector in Boyer-Lindquist coordinate | https://physics.stackexchange.com/questions/117325/hyper-surface-orthogonal-vector-in-boyer-lindquist-coordinate | <p>The Boyer-Lindquist coordinate coordinate of the Kerr Solution is
$$
ds^2=\left(1-\frac{2Mr}{\Sigma}\right)dt^2+\frac{4Mar\sin^2\theta}{\Sigma}dtd\phi - \frac{\Sigma}{\Delta}dr^2-\Sigma d\theta^2-\left(r^2+a^2+\frac{2Ma^2r\sin^2\theta}{\Sigma}\right)\sin^2\theta d\phi^2
$$</p>
<p>Let
$$
K=\frac{\partial}{\partial t... | <p>Your equivalence is not an equivalence. From the way you write, I think that your mistake comes from a problem with index notation. For the Levi-Civita connection, we have $$\nabla_a g_{\mu\nu} = 0. \tag{1}$$ But this does not mean that the $a$-derivative of the $\mu\nu$ component of $g$ is $0$. It means that the $a... | 598 |
general relativity | Why do the space time get curved around a massive object?What problems do we face if we do not consider the space time to be curved? | https://physics.stackexchange.com/questions/132481/why-do-the-space-time-get-curved-around-a-massive-objectwhat-problems-do-we-fac | <p>As far as I have the knowledge of GTR that a mass bends the space time around it.But why does this bend occur?The example from real life that when a mass is placed on a net then the net bends but it us very difficult for me to visualise the situation of bending of spacetime due to a mass.What is actually happening?W... | <p>As photons have energy, gravity affects light rays, turning their path from straight to curved, and changing their energy (frequency/color).</p>
<p>In classical relativity light always travels in a straight zero-lenght line, with phase speed = $c$.
If you include gravity in your relativistic model you this is not t... | 599 |
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