Is the uncertainty principle just saying something about what an observer can know or is it a fundamental property of nature?

I ask this question because I have read two different quotes on the uncertainty principle that don't seem to match very well. There are similar questions around here but I would like an explanation that reconciles these two interpretations specifically:   1. Feynman talks about the uncertainty principle in one of his lectures and mentions it as the reason why electrons don't crash into the atom's nucleus: If they did they would have an exact location and momentum which is not allowed by the uncertainty principle. In saying this it is clear that the uncertainty principle is a fundamental property of nature because it has an effect on where an electron can reside.   2. Recently I read - somewhere else but I forgot where exactly - an account of the uncertainty principle where there was explained how we can measure position of a particle by firing another particle into it, the collision disturbs the velocity of the observed particle therefore we can not know its momentum anymore. Now, 2) very much seems like a limitation of what the observer can know, while 1) attributes a fundamental property of nature to it (electrons don't crash into the nucleus). What is the correct way to think about this?


A list of inconveniences between quantum mechanics and (general) relativity?

It is well known that quantum mechanics and (general) relativity do not fit well. I am wondering whether it is possible to make a list of contradictions or problems between them? E.g. relativity theory uses a space-time _continuum_ , while quantum theory uses _discrete_ states. I am not merely looking for a solution or rebuttal of such opposites, more for a survey of the field out of interest.


What are the Conflicting Predictions of General Relativity & Quantum Mechanics?

I see a lot of questions in various sites about why the 2 theories are or aren't incompatible, I'm satisfied as to why that's the case. However it has been mentioned that both theories make predictions about phenomena that contradict or are incompatible, and I've been unable to find any examples. What are the conflicting/contrary/incompatible predictions made by General Relativity versus Quantum Mechanics? Or are the claims false?


Incompatibility of GR and QM

I am told that the theories of General Relativity and Quantum Mechanics are fundamentally incompatible... Why is that? Someone explained that it had to do with the fact that quantum particles such As quarks and photons are points without volume... Leading to singularities. Do people really believe that ^ What data do we have to suggest that quarks are really zero dimensional points?


What is an example of a situation where Quantum Mechanics and Relativity do not work together?

I've learned special relativity in school last semester, and this semester we began learning about Quantum Mechanics, and my teacher told us that there was a Relativistic Schrodinger equation. I was under the impression that you couldn't use Quantum Mechanics and Relativity together, so why is it appropriate to have a Relativistic Schrodinger equation, and what is an example of a situation where QM and Relativity cannot be used together?


Quantum Mechanics and General Relativity in Macroscopic Level

Hi I read a book yesterday.The book was Brian Greene's The Elegant Universe. I learned that uncertainty principle affects space-time very microscopic levels and this affection makes conflict in General Relativity and Quantum Mechanics. I wanted to know this is the only reason that this two theoris can not be conbined or there are more reasons ?


General relativity && quantum mechanics "incompatibillity"

Now this may be utterly weird layman-physics-question, but anyways... I have read recently following: "The fundamental universe laws are everywhere the same. It's just that the manifestation (!) of (these) laws is different (!) on a specific "scale"." Which brings me to the question: Could it be that GR and QM are one and the same thing on a "different scale"? If yes (LOL), that would "explain" why it is not possible to "merge" GR and QM. Was question like this ever considered?


Sound as a use to separate molecular structures

Sound can be a destructive force. However, could it be used to separate say the Hydrogen atom from the Oxygen atoms?


Zero modes ~ zero eigenvalue modes ~ zero energy modes?

There have been several Phys.SE questions on the topic of zero modes. Such as, e.g.,   * **zero-modes** (What are zero modes?, Can massive fermions have zero modes?),    * **majorana-zero-modes** (Majorana zero mode in quantum field theory),   * **path-integral-with-zero-energy-modes** (Path integral with zero energy modes), etc. Here I would like to understand further **whether "Zero Modes" may have physically different interpretations** and **what their consequences are** , or **how these issues really are the same, related or different.** There at least 3 relevant issues I can come up with: ## (1) **Zero eigenvalue modes** By definition, **Zero Modes** means zero eigenvalue modes, which are modes $\Psi_j$ with zero eigenvalue for some operator $O$. Say, $$O \Psi_j = \lambda_j \Psi_j,$$ with some $\lambda_a=0$ for some $a$. This can be Dirac operator of some fermion fields, such as $$(i\gamma^\mu D^\mu(A,\phi)-m)\Psi_j = \lambda_j \Psi_j$$ here there may be nontrivial gauge profile $A$ and soliton profile $\phi$ in spacetime. If zero mode exists then with $\lambda_a=0$ for some $a$. **In this case, however, as far as I understand, the energy of the zero modes may not be zero.** This zero mode contributes nontrivially to the path integral as $$\int [D\Psi][D\bar{\Psi}] e^{iS[\Psi]}=\int [D\Psi][D\bar{\Psi}] e^{i\bar{\Psi}(i\gamma^\mu D^\mu(A,\phi)-m)\Psi } =\det(i\gamma^\mu D^\mu(A,\phi)-m)=\prod_j \lambda_j$$ In this case, if there exists $\lambda_a=0$, then we need to be very careful about the possible long range correlation of $\Psi_a$, seen from the path integral partition function ( **any comments at this point?** ). ## (2) **Zero energy modes** If said the operator $O$ is precisely the hamiltonian $H$, i.e. the $\lambda_j$ become energy eigenvalues, then the zero modes becomes zero energy modes: $$ H \Psi_j= \lambda_j \Psi_j $$ if there exists some $\lambda_a=0$. ## (3) **Zero modes $\phi_0$ and conjugate momentum winding modes $P_{\phi}$** In the chiral boson theory or heterotic string theory, the bosonic field $\Phi(x)$ $$ \Phi(x) ={\phi_{0}}+ P_{\phi} \frac{2\pi}{L}x+i \sum_{n\neq 0} \frac{1}{n} \alpha_{n} e^{-in x \frac{2\pi}{L}} $$ contains zero mode $\phi_0$. * * * Thus: **Are the issues (1),(2) and (3) the same, related or different physical issues?** If they are the same, why there are the same? If they're different, how they are different? I also like to know when people consider various context, which issues they are really dealing with: such as the **Jackiw-Rebbi** model, the **Jackiw- Rossi** model and **Goldstone-Wilczek** current computing induced quantum number under soliton profile, **Majorana zero energy modes**, such as the Fu- Kane model (arXiv:0707.1692), Ivanov half-quantum vortices in p-wave superconductors (arXiv:cond-mat/0005069), or the issue with **fermion zero modes under QCD instanton** as discussed in Sidney Coleman's book ``Aspects of symmetry''. ps. since this question may be a bit too broad, it is totally welcomed that anyone attempts to firstly answer the question partly and add more thoughts later.


Negative probabilities in quantum physics

Negative probabilities are naturally found in the Wigner function (both the original one and its discrete variants), the Klein paradox (where it is an artifact of using a one-particle theory) and the Klein-Gordon equation. The question is if there is a general treatment of quasi-probability distributions, besides naively using 'legit' probabilistic formulas? For example, is there a theory saying which measurements are allowed, so to screen negative probabilities? _Additionally_ , is there an intuition behind negative probabilities? (Providing other examples than ones mentioned in the question can illuminate the issue.)


Smaller free remote control telescopes?

There are several online services that let you control a large telescope (eg, lightbuckets.com and slooh.com), even some that are free (eg, telescope.org). Unfortunately, the pay services are expensive, and you get very little reserved time on the scopes. The free services are painfully slow: I had to wait several months for a picture of Jupiter I wanted. Has anyone set up automated (ideally free) remote access to a smaller telescope? I'd much rather play with a 10" telescope in real-time than a larger one in limited/delayed time. In fact, even a CCD or camera would be nice for wide-angle shots.


Is it possible to recover Classical Mechanics from Schrödinger's equation?

Let me explain in details. Let $\Psi=\Psi(x,t)$ be the wave function of a particle moving in a unidimensional space. Is there a way of writing $\Psi(x,t)$ so that $|\Psi(x,t)|^2$ represents the probability density of finding a particle in classical mechanics (using a Dirac delta function, perhaps)?


Classical limit of quantum mechanics

I have heard that one can recover classical mechanics from quantum mechanics in the limit the $\hbar$ goes to zero. How can this be done? (Ideally, I would love to see something like: as $\hbar$ goes to zero, the position wavefunction reduces to a delta function and that the Schrodinger equation/Feynman path integral reduces to the Newtonian/Lagrangian/Hamiltonian equations of motion.)


How to prove Newton's second law with quantum mechanics?

Newton's second law claims that $F=ma$. In terms of quantum mechanics, the equality can be written as $ \frac{d\langle p \rangle}{dt} = -\langle \nabla V(x) \rangle$. How can I prove this with non-relativistic Schrodinger Equation? I've calculated the expectation of $p=mv$, and tried to take the derivative under the integral. And there I'm stuck.


Classical Limit of the Feynman Path Integral

I understand that in the limit that h_bar goes to zero, the Feynman path integral is dominated by the classical path, and then using the stationary phase approximation we can derive an approximation for the propagator which is a function of the classical trajectory (see http://www.blau.itp.unibe.ch/lecturesPI.pdf pg 46). I am under the impression that this further implies that the particle follows the classical trajectory but I don't understand how the above mentioned fact implies this. The propagator describes the time-evolution of the wavefunction, so I would think that this classical limit form of the propagator should give a time- evolution in which the wavefunction follows the classical trajectory, but I have not been able to find such work. Moreover, even this statement itself is problematic since the wavefunction describes a probability distribution and not a single trajectory. $\textbf{New Edit:}$ In section 7 of Feynman's paper introducing the path integral (see http://imotiro.org/repositorio/howto/artigoshistoricosordemcronologica/1948c%20-FEYNMAN%201948C%20Invention%20of%20the%20path%20integral%20formalism%20for%20quantum%20mechanics.pdf) he discusses the classical limit. It appears that the key to understanding why the fact that the classical path dominates the path integral further implies that the particle follows the classical trajectory may be found in Feynman's remark on pg 21: "Now we ask, as $\hbar → 0$ what values of the intermediate coordinates $x_i$ contribute most strongly to the integral? These will be the values most likely to be found by experiment and therefore will determine, in the limit, the classical path." However, I don't understand why "These will be the values most likely to be found by experiment" ?


Can Newton's laws be explained by Quantum Physics?

I have only basic knowledge of physics. Could you please explain to me if a "Quantum" laws can theoretically (perhaps in the future?) be used to explain everything in macro levels? I'm having problems to understand how we can have "two physics" at the same time. are Newton's laws only a simplification?


Classical limit of the path integral formulation of quantum mechanics

It is well-known that if $S \gg \hbar$, then the classical path dominates the Feynman path integral. But is there some to show that if $S\gg\hbar$, then the particle's trajectory will approach the classical path?


From Quantum Mechanics to Classical Mechanics

Is it possible, and has it been attempted, to use quantum mechanics to deduce Newtonian, macroscopic level mechanics laws as was the case of statistical mechanics deriving thermodynamic relations?


How are the Lagrange equation and Feynmann path integral related?

My question is, where could I get some more info on how the Euler-Lagrange equations are related $$ \delta S [y(x)] =0 $$ with the Feynmann path integral formulation $ \int D[y(X)]e^{iS[y(x)]/\hbar} $ I believe that when $ \hbar \to 0$ (semiclassical), only the points that satisfy (1) contribute to the path integral. If someone can give me more info about it :D thanks


How to go from Quantum World to Classical World?

> **Possible Duplicate:**   >  Is it possible to recover Classical Mechanics from Schrodinger’s equation?   >  Classical Limit of the Feynman Path Integral In the quantum world we don't have specific trajectories, the particle so to speak goes through all possible paths. In the classical and macroscopic world we have definite paths, and usually one specific trajectory is assigned to a body's motion. How would you go from a trajectoryless world to trajectoried world? Are there any theories about this bridge between the two worlds? I guess there should be such a theory, cause one world is the building block of the other.


How do inflationary models predict the generation of gravitational waves during the inflationary period?

Recent results from the BICEP2 experiment have produced a lot of talk about the primordial gravitational waves produced during the inflationary period. I would like to have some explanation about how inflationary models predict the generation of these gravitational waves. Have these gravitational waves been described as metric perturbations around a de Sitter spacetime? Are they predicted using a semiclassical gravity formalism so as to take into account the quantization of a scalar field (the inflaton) in such "perturbed" spacetime?


What is the speed of electrical current in salt water?

I am wondering about a specific question regarding the speed at which an electrical current traverses through salt-water / saline. By this I do _not_ mean the electron drift speed - I mean, at what speed would a current travelling from an anode to a cathode immersed in salt water be? A ball park figure will do. Thank you.


Is Electromagnetic Mass Possible?

If the sinusoidal electric component of a light wave were off-set to one side of the magnetic component and then the smaller "lobe" were to cancel out with much of the larger side, then where would the energy go? Would it not form a closed loop much like a mass-bearing string? Could the electrical energy not be converted into a gravitational field to bend space-time over the length of the wave to form a 1-D "string" along the junction of the perpendicular E and B fields? This would be much like folding a sheet of paper in half so that one edge protrudes past the other. The protruding edge being the electric component and the rest being electrical energy converted into mass/gravity (comparable to a mass-bearing string). The magnetic component could then arise from Lorentz symmetry as described by Lubos: > If you only start with the $E_z$ electric field, the component $F_{03}$ is > nonzero. However, when you boost the system in the $x$-direction, you mix > the time coordinate $0$ with the spatial $x$-coordinate $1$. Consequently, a > part of the $F_{03}$ field is transformed into the component $F_{13}$ which > is interpreted as the magnetic field $B_y$, up to a sign. **Is the described "string wave" model possible?** I realize that I haven't provided a mechanism to explain the proposed model. However "I do not reject food because I do not understand digestion." -Oliver Heavenside From here on the rest of this post is merely a compilation of the "evidence" that circumstantially supports the proposed idea. Please do not feel required to address these topics. They are here because I think electromagnetic mass is real. I realize this borders on "promoting unaccepted theories," but it is merely my way of assessing the possibility. I promise not to bring this up again on SE if it is refuted. **The concept seems compatible with the standard model:** a.) an origin of charge has been proposed. b.) eliminating one lobe has reduced the "spin" or magnetic field by 1/2 relative to a light wave. c.) Lepton number 1 applies due to the "strong interaction" of the remaining electric component d.) a mechanism of mass has been proposed. **Maxwells equations seem to fit:** The Laws of Electromagnetism have geometric relations incorporated into them that naturally arise from the proposed electron-as-an-EM-wave model: the dot product and the cross product specifically. Ampere's Law Ampere’s Law describes the magnetic field produced by the flow of electrons along a wire. The negative components of electrons flowing along the wire should repel each other, which would mean that the negative components should protrude from the wire like the dorsal fins of sharks swimming parallel and breaking the surface. Since the electrons are travelling in the same direction along the wire the magnetic components (in the direction of the side fins of the shark analogy) should be tangent to the surface of the wire, which results in a circular magnetic field around the wire just as Ampere’s Law and the Biot-Savart Law predict. Faraday's Law Faraday’s Law describes the electric voltage produced in a coil of wire as a magnetic field through it changes. The voltage is proportional to the number of loops of wire, which is counter-intuitive/non-conservative. Why should the voltage depend on the number of loops? An explanation naturally arises from the proposed model. Electrons inherently have velocity in the form of a "Poynting-like" vector. When the loop of wire encounters a changing magnetic field the "poyinting vectors" align and under the right orientations they align with the wire loops and thereby form electric current. ![enter image description here](http://i.stack.imgur.com/eyCkq.jpg) In the bottom orientation the magnet produces no EMF along the wire. In the top orientation the EMF is along the wire as seen in generators. Schrodinger's statistical model won because Maxwells equations had already divorced physics from first causes. Saying that the surface covered area of a loop was the cause of EMF instead of the poynting vectors of electrons adding over the distance of the circumference. Gauss's Laws Gauss’ laws of electricity and magnetism are easily integrated with the proposed model as the net magnetic flux is zero and the net charge is unchanged. However Coulombs Law does not hold true at the subatomic level. The force between two charged electrons is modified by the presence of other electrons if electrons are not point charges, but electric components of an EM-wave. Coulomb's law Coulomb's law for point charges: $$F=k\frac{q_1 q_2}{r^2}=\frac{1}{4\pi\epsilon_0}\frac{qq_0}{r^2}\hat{u}$$ Does not hold at the subatomic level for all directions if a blip of negative charge sticks off one side of a mass-bearing electromagnetic wave. This naturally provides a classical explanation for "quantum tunnelling:" certain orientations of subatomic particles behave unlike point charges. **Atomic Orbitals** Take one electron and one proton and place them near each-other. The proposed model suggests an intrinsic and intuative reason why electrons don’t fall into the nucleus, which current theories lack. The velocity of an imbalanced EM wave is perpendicular to both the electric and magnetic components, which means that the radial acceleration of attraction towards the nucleus experienced by the electric component of an electron is always perpendicular to its velocity or the “Poynting vector” of an electron. All that is left for the electron to do is to set up the lowest energy standing wave possible. This also suggests that the proton rotates with the electron's orbit so that the "fins" constantly point toward eachother. Spherical harmonics should arise naturally from this arrangement and approximate to the schrodinger equation. **Relativistic Explanation of the Lorentz Force** Using relativistic tensors: "If you only start with the $E_z$ electric field, the component $F_{03}$ is nonzero. However, when you boost the system in the $x$-direction, you mix the time coordinate $0$ with the spatial $x$-coordinate $1$. Consequently, a part of the $F_{03}$ field is transformed into the component $F_{13}$ which is interpreted as the magnetic field $B_y$, up to a sign\cite{Lubos}." If the electron is an E-M wave as proposed and the magnetic components align with the extern magnetic field, then performing the reverse of the above transformation should convert the external magnetic field into an electric component in the $E_z$ direction so that the electron feels the equivalent to a charge perpendicular to its direction of motion, which explains the Lorentz force. **Numerous Other Explanations** There are many other phenomena that seem to fit with the proposed model. I'm just running out of steam! **A Little History** Electrons were viewed as "matter waves" in DeBrogli's model. "Matter waves" is a polite way of saying: something is waving but we don't know what. Heisenberg came along and said that despite not knowing what is waving we can assume that the wave doesn't have any undiscovered properties that would allow for knowing both position and momentum (an arrogant assertion!). Schrodinger then came along and noticed that tweaking spherical harmonics provided a reasonable model of atomic orbitals (statistically). That all led to the Copenhagen interpretation, which Einstein called the "Born-Heisenberg tranquilizing philosophy, or religion." Electrons have continued to be treated as point charges or matter waves as is convenient for interpreting experimental results ever since. QED and QCD have introduced "virtual photons" in explaining the interactions of "point charges" and light, etc. Surely "virtual photons" would be unnecessary if the point charges were instead modelled as EM waves themselves. Particle physicists have invented a "higgs field" which they propose vibrates to create mass. The higgs field seems like adding epicycles: sorta unnecessary when there is a simpler and better alternative (that I have not really understood yet!). It has been obvious for decades that physics has stalled while trying to model hadrons as point charges. String theory has been a lonely success story waiting to happen. Electromagnetic string-waves are the future! (unless someone refutes me :-).


How exactly does time slow down near a black hole? I have heard this as a possible way of time traveling, and I do understand that it is due in some way to the massive gravity around a black hole, but how exactly does that massive gravity slow down time?


Does a photon exert a gravitational pull?

I know a photon has zero rest mass, but it does have plenty of energy. Since energy and mass are equivalent does this mean that a photon (or more practically, a light beam) exerts a gravitational pull on other objects? If so, does it depend on the frequency of the photon?


How does gravity effects both time and light if they have no mass

I've been reading about how black holes can effect both time and light with gravity. So I was wondering, doesn't something have to have mass to be effected by gravity? And if so, does this mean both light and time have mass? And if not, how can gravity effect something that has no mass?


Is light affected by gravity? Why?

I would like to know if light is affected by gravity, also, I would like to know what is the correct definition of gravity: "A force that attracts bodies with mass" or "a force that attracts bodies with energy, such as light"? Is light massless after all?


What causes light to travel?

What is the force that causes it to move and why does it maintain the speed for so long? If it has no mass, why is it effected by mass?


Can electromagnetic radiation (i.e. photons) produce gravity?

I don't want to play with physical laws in a frivolous way. Assuming that the nature of matter and energy is the same, can a high density of highly energetic photons produce a gravity force? We do know that radiation is affected by space-time distortions, or in another way "feels gravity". Why do photons can (or cannot) produce a gravity field?


Why can light (photons) bends in a curve through space without mass?

I've heard that light can form a curve if they travel near high-mass stars or even a black hole with strong gravity. Which is according to this Newtonian formula $$\large F_{g}=\dfrac{Gm_1m_2}{r^2}.$$ But I've also heard that photons do not have (rest) mass! So it doesn't fit that equation anymore! But why can photons be pulled by gravities without (rest) mass? Could someone explain that?


Explain how (or if) a box full of photons would weigh more due to massless photons

I understand that mass-energy equivalence is often misinterpreted as saying that mass can be converted into energy and vice versa. The reality is that energy is always manifested as mass in some form, but I struggle with some cases: **Understood Nuclear Decay Example** In the case of a simple nuclear reaction, for instance, the total system mass remains the same since the mass deficit (in rest masses) is accounted for in the greater relativistic masses of the products per $E=\Delta m c^2$. When a neutron decays and you are left with a fast proton and a relativistic electron. If you could weigh those two without slowing them down, you would find it weighed as much as the original neutron. **Light in a Box** This becomes more difficult for me when moving to massless particles like photons. Photons can transmit energy from one heavy particle to another. When a photon is absorbed the relativistic mass (not the rest mass) of the (previously stationary) particle that absorbs it increases. But if my understanding is correct, the energy must still be manifested as mass _somehow_ while the photon is in-flight, in spite of the fact that the photon does not have mass. So let's consider a box with the interior entirely lined with perfect mirrors. I have the tare weight of the box with no photons in it. When photons are present the box has an additional quantifiable amount of energy (quantified below) due to the in-flight photons. Say there are $N$ photons... obviously assume $N$ is large. $$\Delta m = \frac{ E }{ c^2 } = \frac{ N h }{ \lambda c}$$ Interactions are limited to reflections with the wall, which manifest as a constant pressure on the walls. If I hold this box in a constant gravitational field (like the surface of Earth) then there will be a gradient in the pressure that pushes down slightly. Is this correct? Wouldn't there still technically be mass as the photons are in-flight, which would cause its own gravitational field just as all matter does? How is this all consistent with the assertion that photons are massless? Is it really correct to say that photons don't have mass? It seems to be a big stretch. Please offer a more complete and physically accurate account of this mirror- box.


If photons are deflected by a strong gravitational field, then how come photons do not have mass?

> **Possible Duplicate:**   >  Explain how (or if) a box full of photons would weigh more due to massless > photons It has been proved and showed through experiments that light can be bent by the Sun or any other body with considerable mass. Also light is nothing but photons. So can these photons be attracted by massive bodies if they have no mass?


What is the cause the light is affected by gravity?

I know that photons have no mass and that a photons exist only moving at the speed of light. So what is the cause that a massive astronomical object can bend a ray of light? I have two thoughts, but I am confused which of these, if any is correct:   1. A gravitational field of a massive astronomical object curves spacetime and affects the light traveling near this object **indirectly**. That is, for external viewers the light bends, but for the light itself it is still traveling in a straight line. So I mean, that the coordinates in that place are curved, but only for the external viewer.   2. Energy of a photon is equivalent to mass, so the gravity field interacts with photons **directly** the same way, as if they would have mass (equivalent to the energy).


Does non-matter energy curve spacetime?

I know that matter (mass) curves spacetime, but do other forms of energy do the same? I.e. is matter the only form of energy that curves spacetime?


What are the factors affecting the spacetime curvature?

Large masses in space as stars and planets cause a curvature in the spacetime fabric. What are the factors that affect this curvature? Is it only mass? And can we conclude these factors using Tensors?


Light and Gravity - bending of light around a massive body

Well, as I have read, a massive body can cause light to bend around itself due to its gravitational attraction. What I don't understand is how, as the Newtonian formula for the force of gravitational attraction is $$F = \frac{Gm_1m_2}{r^2}.$$ As photons do not have (rest) mass, shouldn't there be no attraction between light and the celestial bodies, and therefore no bending? Surely I am missing a key point. Please point out my logical flaw?


How to find the Green's Functions for time-dependent inhomogeneous Klein-Gordon equation?

I'm trying to find the Green's functions for time-dependent inhomogeneous Klein-Gordon equation which is : \begin{align*} \left[ - \nabla ^2 + \frac{1}{c^2} \dfrac{\partial ^2}{\partial t^2} + \kappa ^2 \right] \psi({\mathbf{r},t )} = \rho(\mathbf{r},t) \end{align*} It has been mentioned in the question that I can find the Green's functions : \begin{align*} &G_R(\mathbf{r} , t , \mathbf{r'} , t') = \dfrac{c}{8 \pi ^2 \mathbf{R} i }\dfrac{d}{d\mathbf{R}} \int_{- \infty}^{+\infty} \dfrac{e^{ i \frac{R}{c} \sqrt{q^2 - k^2 c^2}}}{\sqrt{q^2 - k^2 c^2}} e^{ - iq (t - t')} dq \\\ &G_A(\mathbf{r} , t , \mathbf{r'} , t') = -\dfrac{c}{8 \pi ^2 \mathbf{R} i }\dfrac{d}{d\mathbf{R}} \int_{- \infty}^{+\infty} \dfrac{e^{- i \frac{R}{c} \sqrt{q^2 - k^2 c^2}}}{\sqrt{q^2 - k^2 c^2}} e^{ - iq (t - t')} dq \end{align*} using the fourier transform, but when I use the fourier transform I don't gain the proper answer. The fourier transform which I use is the one which is generally given as : \begin{align*} f(r) = \dfrac{1}{\sqrt{2 \pi}} \int_{- \infty}^{\infty} e^{ik.r}\hat{f}(k) dk  \end{align*} but from this transform I cannot find $G_A$ and $G_R$. Is there another transform which I should use to find the Green's functions? **Edit** The Green's function which I wind up with is : \begin{align*} G_A(\mathbf{r} , t , \mathbf{r'} , t') = \dfrac{1}{(2\pi)^4} \int d^3\mathbf{k} dk' \frac{1}{k^2} e^{i\mathbf{k}.(\mathbf{r} - \mathbf{r'})}e^{ik'(t-t')} \end{align*} which is not even similar to the answer given here!


Speed of light, breaking the barrier

when the sound barrier is broken, a series of concentric waves of sound is produced.Does it mean when the speed of light barrier is broken, a ripple of photons are created in the space-time fabric?


Electric Force is to Magnetic Force as Gravitational Force is to ...?

One can no nothing about the magnetic force and yet arrive at it by taking the relativistic effects of a current and a moving charge system into account. I ask whether there exists such an inherent force in case of gravity.


Physics of tsunami: the relationship between wavelength, sea depth and the height of the water

If I understand correctly, when an earthquake occurs, energy will be transferred to the water, resulting in water waves. As the waves reach seashore, because the sea depth is getting shallower and wavelength is getting shorter, the height of the wave gets push up, resulting in tsunami. In other words in deep sea, water won't get pushed up as high as the water in shallow seashore. Is my understanding correct? Is there a quantitative way to express the physics behind all this?


Planck's Law in terms of wavelength

I am drawing a blank when it comes to equation transformation. Wikipedia gives two equations for the spectral radiance of black body:   * First as a function of frequency $\nu$: $$I(\nu, T) = \frac{2 h \nu^3}{c^2}\cdot\frac{1}{e^\frac{h \nu}{k T} - 1}$$   * Then as a function of wavelength $\lambda$ : $$I'(\lambda, T) = \frac{2hc^2}{\lambda^5}\cdot\frac{1}{e^\frac{h c}{\lambda k T}-1}$$ And I don't see how they get $\lambda^5$ term. I'm assuming that the transformation is just $\nu \rightarrow c/\lambda$, but that gives $$ \frac{2 h \nu^3}{c^2} \Rightarrow \frac{2 hc}{\lambda^3} \neq \frac{2hc^2}{\lambda^5} $$ Similar transformation happens at other parts in the article also. I'm obviously missing something, likely completely trivial.


Wien's displacement law in frequency domain

When I tried to derive the Wien's displacement law I used Planck's law for blackbody radiation: $I_\nu = \frac{8 \pi \nu^2}{c^3} \frac{h \nu}{e^{h\nu/k_bT}-1}$ Asking for maximum: $\frac{dI_\nu}{d \nu}=0:~0= \frac{\partial}{\partial\nu}(\frac{\nu^3}{e^{h \nu/k_bT}-1}) = \frac{3\nu^2(e^{h \nu/k_bT}-1) - \nu ^3h/k_bT \cdot e^{h\nu/k_bT}}{(e^{h \nu/k_bT}-1)^2}$ It follows that numerator has to be $0$ and looking for $\nu>0$: $3(e^{h \nu/k_bT}-1) - h \nu/k_bT \cdot e^{h\nu/k_bT}=0$ Solving for $\gamma=h\nu/k_bT$: $3 (e^\gamma-1) - \gamma e^\gamma=0 \rightarrow \gamma=2.824$ Now I look at the wavelength domain: $\lambda = c/\nu:~ \lambda =\frac{h c}{\gamma k_b} \frac{1}{T}$ but from Wien's law $\lambda T = b$ I expect that $hc/\gamma k_b$ is equal to $b$ which is not: $\frac{h c}{\gamma k_b}= 0.005099$, where $b = 0.002897$ Why the derivation from frequency domain does not correspond the maximum in wavelength domain? I tried to justify it with chain rule: $\frac{dI}{d\lambda} = \frac{dI}{d\nu} \frac{d \nu}{d \lambda} = \frac{c}{\nu^2} \frac{dI}{d \nu}$ where I see that $c/\nu^2$ does not influence where $dI_\lambda/d \lambda$ is zero.


Different versions of Planck's law

For a presentation in physics I am going to talk about black body radiation since our book just mentions Planck's law, Wien's displacement law and Stefan- Boltzmann's law. I want to derive Wien's law and Stefan-Boltzmann's law from Planck's law. I have managed to derive Stefan-Boltzmann's law from: $$u(\nu)=\frac{2\pi h \nu^3}{c^2}\frac{1}{e^\frac{h\nu}{kT}-1}$$ by integrating over all $\nu$ from $0$ to $\infty$. However, my textbook says: $$u(\lambda,T)=\frac{2\pi hc^2 }{\lambda^5}\frac{1}{e^\frac{hc}{\lambda kT}-1}$$ If I can get from the second formula to the first, I will be happy alltough, a simple and informal derivation of Plack's law would be very satisfying. I also managed to derive Wien's law by derivation and setting equal to zero by using: $$u_\lambda=\frac{8\pi hc }{\lambda^5}\frac{1}{e^\frac{hc}{\lambda kT}-1}$$ What are the differences? And please, since I am Norwegian and do not know very advanced physics, do not use any advanced expression without explaining them. Thanks!


What exactly means "is a singlet under $SU(N)$"

I don't get a grip of what that exactly means. What IS an abstract singlet, doublet,... under $SU(N)$ or other groups?


What does this notation mean?

Terminology question. Reading this, in the middle of the second page, when it says > Left-handed quarks form 3 (3; 2; + 1 6 ) multiplets Qn (n = 1; 2; 3); What does this (3;2;+1/6) mean?


Notation for Standard Model Charges?

Does anybody know what these following numbers describing an electron $(1, 1, -1)$ represent in $SU(3) \times SU(2) \times U(1)$? Or, these numbers that describe an up quark: $(3, 1, 2/3)$? I'm really confused!


What happens to the Lagrangian of the Dirac theory under charge conjugation?

Consider a charge conjugation operator which acts on the Dirac field($\psi$) as   $$\psi_{C} \equiv \mathcal{C}\psi\mathcal{C}^{-1} = C\gamma_{0}^{T}\psi^{*}$$ Just as we can operate the parity operator on the Lagrangian, and we say that a theory has a symmetry if $$\mathcal{P}\mathcal{L}(t,x^{i})\mathcal{P}^{-1} = \mathcal{L}(t,-x^{i})$$ Suppose we operate $\mathcal{C}$ on the Dirac Lagrangian what should we get? $$\mathcal{C}\mathcal{L}_{Dirac}(x^{\mu})\mathcal{C}^{-1} = \mathcal{L}^{*}_{Dirac}(x^{\mu}) ?$$ in analogy to the transformation of the scalar field $\phi$ under charge conjugation. On a same note one can ask what equation should $\psi_{C}$ satisfy? Should it satisfy the conjugated Dirac equation as $$(i\gamma^{\mu}\partial_{\mu} + m)\psi_{C} = 0 ?$$ If so can someone give me the physical interpretation for it. I am asking this question as I want to explicitly use $\psi_{C}$ and check whether it keeps the Dirac Lagrangian invariant. I have done a calculation by substituting $\psi_{C}$ in the Dirac equation and have found it is not satisfying as shown below. $$(i\gamma_{\mu}\partial_{\mu} - m)\psi_{C} = i(\gamma_{\mu}C\gamma_{0}^{T})\partial^{\mu}\psi^{*} - (C\gamma_{0}^{T})m\psi^{*}$$ We will use $C^{-1}\gamma_{\mu}C = - \gamma_{\mu}^{T}$ and $\\{\gamma_{\mu}^{T},\gamma_{\nu}^{T}\\} = 2g_{\mu\nu}$.   Consider \begin{align} \gamma_{\mu}C\gamma_{0}^{T} &= CC^{-1}\gamma_{\mu}C\gamma_{0}^{T} \\\ &= -C\gamma_{\mu}^{T}\gamma_{0}^{T} \\\ &= C\gamma_{0}^{T}\gamma_{\mu}^{T} \end{align} Hence substituting back we will get   \begin{align} (i\gamma_{\mu}\partial_{\mu} - m)\psi_{C} &= C\gamma_{0}^{T}(i\gamma_{\mu}^{T}\partial^{\mu}\psi^* - m\psi^*) \\\ &= C\gamma_{0}^{T}[(i\gamma_{\mu}^{T}\partial^{\mu}\psi^* - m\psi^*)^{T}]^{T} \\\ &= C\gamma_{0}^{T}(i\psi^{\dagger}\gamma_{\mu}\partial^{\mu} - m\psi^{\dagger})^{T} \\\ &= C\gamma_{0}^{T}[(i\bar{\psi}\gamma_{0}\gamma_{\mu}\partial^{\mu} - m\bar{\psi}\gamma_{0})]^{T} \\\ \end{align} Now \begin{align} \gamma_{0}\gamma_{\mu}\partial^{\mu} &= (\gamma_{0}\partial^{t} + \gamma_{i}\partial^{i})\gamma_{0} \end{align} If we substitute back we will get \begin{align} (i\gamma_{\mu}\partial_{\mu} - m)\psi_{C} &= C\gamma_{0}^{T}[\\{i\bar{\psi}(\gamma_{0}\partial^{t} + \gamma_{i}\partial^{i}) - m\bar{\psi}\\}\gamma_{0}]^T \\\ &\neq 0 \end{align}


String tension in vertical circular motion

Suppose that I have a point mass attached to a massless string and I am rotating it vertically. That means The mass is in uniform circular motion and the path of its motion is vertical circle. How does the tension change with respect to the position of the mass. More specifically is the tension in the string is only due to circular motion ($mv^2/R$) or gravity plays a part in it ($mv^2/R$ + something due to weight)?


What is our location relative to the Big Bang?

Given what we know about space, time and the movement of galaxies, have we or can we determine what our position is in relation to the projected location of the Big Bang? I've read some introductory papers on the superstructure and galaxy cluster movements, but none of them specifically mentioned space in terms of relative or absolute positions relating to the original position of the Big Bang. So my question is, does our current understanding of the structure and workings of the Universe give us a good enough estimate to determine our location relative to the Big Bang or can we never guess at it, since every viewpoint in our universe looks the same in every direction?


The ideal trampoline

Suppose we have a mass attached to the top of an ideal (linear and massless) spring oriented vertically in a uniform gravitational field, and on top of that mass there is another mass resting on it. The two masses are not attached at all, so they will lose contact with each other as the normal force is about to become negative. Also suppose that once the two masses separate and collide again, they undergo perfectly elastic collisions. First of all, is there a name for systems like this? It seems like an "ideal trampoline" to me but searching for that doesn't yield much. Has anyone ever discussed it in a book? Second of all, is this system chaotic? For sufficiently small oscillations, of course, the masses remain in contact the whole time and you get simple harmonic oscillation, but above some threshold the free mass will keep bouncing off the spring-attached mass and it's quite nontrivial to figure out what eventually happens. Do you get interesting things like period doubling?


Learning physics online?

I'm thinking of following some kind of education in physics online. I have a master degree in Computer Science and have reasonable good knowledge in physics. I would like a program of 1-2 years and I'm more interested in particle physics. Is there any good online program that offer something similar?


How would a newtons cradle act in zero gravity?

I imagine that the first ball would strike the rest as normal, but what would the last ball do, without gravity to swing it back?


Can an Incandescent Bulb be considered as a heater?

Due to an experiment, I need a small heater (around 70 to 100 watts). I intend to use an incandescent bulb so it can act as a heater. What I wonder here is will a 70 watts Incandescent Bulb be equal to a 70 watts heater?


Intensity of unpolarized light

If the electromagnetic field of an unpolarized plane wave is written as $$\bar{E}(t,\bar{x})=(\bar{E}_{0x}+\bar{E}_{0y}e^{i\delta(t)})e^{i(\bar{k}\bar{x}-\omega t)}$$ $$\bar{B}(t,\bar{x})=\frac{1}{\omega}\bar{k}\times\bar{E}(t,\bar{x})$$ where $\delta(t)$ is a random phase shift, then the intensity of this light is given by the time-average of the norm of the Poynting vector $$I(\bar{x})=\left<\|\bar{P}(t,\bar{x})\|\right>_{t}$$ $$\begin{split}\bar{P}(t,\bar{x})=&\frac{1}{\mu_{0}}\mathcal{R}e(\bar{E}(t,\bar{x}))\times\mathcal{R}e(\bar{B}(t,\bar{x}))\\\ =&\frac{1}{\omega}\bar{E}_{0x}^{2}\cos^{2}(\bar{k}\bar{x}-\omega t)\bar{k}+\frac{1}{\omega}\bar{E}_{0y}^{2}\cos^{2}(\bar{k}\bar{x}-\omega t+\delta(t))\bar{k} \end{split}$$ $$\Leftrightarrow I(\bar{x})=\frac{1}{2}c\epsilon_{0}\bar{E}_{0x}^{2}+c\epsilon_{0}\bar{E}_{0y}^{2}\left<\cos^{2}(\bar{k}\bar{x}-\omega t+\delta(t))\right>_{t}$$ Can we simplify this further? Is the remaining average also $1/2$?


Speaker cabinet to improve sound of mobile phone loudspeaker (music)

I want to experiment with an enclosure for my phone so the frequency response has a little more punch at the bottom end. I understand that something can't be created from nothing, but enclosures work for drivers so I can't see why not for the phone? What sort of cabinet design would do the job? It would be nice if it preserved a natural mid and treble as well. Even if this design just dampens medium and high frequencies that would be fine too. The response without any enclosure seems to have started to tail off at about 260Hz and is almost gone by 130Hz. I can just about hear 60Hz if I put the volume on full and my ear against the speaker. (Don't try this on your hifi at home kids) Thanks


Deviation from power law distribution of earthquakes

One of the most accepted frameworks for the relationship between the magnitude and frequency of an earthquake is that of the critical phenomena. In this framework, the magnitude of events must be distributed following a power law. However the Gutenberg-Richter law clearly shows a deviation at low magnitudes from the power law called a _roll-off_. How can the roll-off deviation of the GR prediction be explained?


Hilbert of quantum gravity: bulk $\otimes$ horizon

I was reading a paper dealing with the Hilbert of quantum gravity (or more precisely what should it look like considering what we know from QM and GR) ref: http://arxiv.org/abs/1205.2675 and the author writes the following: $${\cal{H}_M} = {\cal{H_{M,\,\textrm{bulk}}}}\otimes{\cal{H_{M,\,\textrm{horizon}}}}$$ for a specific manifold $\cal{M}$. I know very little about the holographic principle and the AdS-CFT correspondence but isn't it a redundant description? If there is a duality between the gravitational theory in the bulk and the CFT on the boundary, knowing one means knowing the other, so why can't we restrict ourselves to one of the Hilbert spaces? Moreover, the author writes, a couple of lines after this first element, that the two Hilbert space have same dimension ( $\textrm{exp}({\frac{{\cal{Area}}}{4}})$ ) so they are totally equivalent, as a complex Hilbert space is only defined by its dimension.


Rolling (without slipping) ball on a moving surface

I've been looking at examples of a ball rolling without slipping down an inclined surface. What happens if the incline angle changes as the ball is rolling? More precisely I've been trying to find equations for programming a simulated (2D) ball rolling inside a swinging bowl/arc. I thought I could still use the same equations for just having a ball rolling inside a still bowl (see below) and the changes in the incline angle (tangent at the contact point of the ball and the bowl) would take care of itself: For a ball rolling inside a bowl: The only torque acting on the ball is the frictional force: $τ=Iα=fr$, using the rolling without slipping condition $a=rα$ and the moment of inertia for a solid sphere, $I = \frac{2}5 mR^2$, we get $f=\frac{2}5ma$. The net force acting on the system is gravity and the force of friction, $F=ma=mgsinθ−f$ and therefore, $a=\frac{5}7gsinθ$ I am speculating that due to the surface itself moving (swinging on a circular path), it's the relative motion that contributes to the friction? But, I don't know how to include that. Can someone please help me?


Who used the concept of symmetries first?

Who "invented" the concept of symmetries? This article is quite extensive, but it blurs the history with the modern understanding. http://plato.stanford.edu/entries/symmetry-breaking/ Some of the concepts can be traced to Galileo and Newton, but I'm quite certain the modern notion is incompatible with their view of the world. Does the notion come from group theory specifically? Can the first mention be traced accurately? > Although the spatial and temporal invariance of mechanical laws was known > and used for a long time in physics, and the group of the global spacetime > symmetries for electrodynamics was completely derived by H. Poincaré [7] > before Einstein's famous 1905 paper setting out his special theory of > relativity, it was not until this work by Einstein that the status of > symmetries with respect to the laws was reversed.


Stiffness tensor

Let's have a stiffness tensor: $$ a^{ijkl}: a^{ijkl} = a^{jikl} = a^{klij} = a^{ijlk}. $$ It has a 21 independent components for an anisotropic body. How does body symmetry (cubic, hexagonal etc.) change the number of independent components of the tensor? For example, for cubiс symmetry it has three components. How to explain it? Update. Is the explanation a simple realization of idea $$ a_{ijkl}' = \beta_{im}a^{m}\beta_{jt}a^{t}\beta_{k f}a^{f}\beta_{ld}a^{d} = a_{ijkl}, $$ where $\beta_{\alpha \beta}$ is a components of a matrix $\beta$ for rotation around z-, x-, y-axis at the same time?


Why do we must know the Weyl tensor for 4-dimensional space-time?

I heard that we must know the Weyl tensor for fully describing the curvature of the 4-dimensional space-time (in space-time with less dimensions it vanishes, so I don't interesting in cases of less dimensions). So I have the question: what is physical (or geometrical) sense of the Weyl tensor and why don't we need only Riemann tensor for describing the curvature? Does it connected with gravitational waves directly?


How can we have massive states of strings and CFT on the string worldsheet at the same time?

Ok, so we can have conformal invariance on a string world sheet. However, it is well known that to preserve conformal symmetry we require states to be massless. So how is it that string theories incorporate CFT but allows massive states? Is it because the CFT is on the worldsheet and therefore applies to the worldsheet coordinate X (X is treated as the field) - however the physical states arise from the the creation/annihilation operators that we get from X? Therefore the CFT doesn't actually act on the states (massive or massless) but instead it acts on the field X.


Rotation axis of a rigid body

I am confused about a trivial concept. Let the rotation of a rigid body, say with one point fixed, be described by the equation $\vec{x}(t)=R(t)\vec{x}(0)$, with $R(0)=I$. Then, at each instant there is only one real eigenvector of $R(t)$ with eigenvalue 1 that we may call $\vec{v}(t)$ and which we may take to be normalized. That vector $\vec{v}(t)$ is what geometrically we would call the (instantaneous) axis of the rotation. Kinematically, however, the instantaneous axis of rotation is the line of points with vanishing instantaneous velocity $\dot{\vec{x}}(t)=\vec{\omega}(t)\times\vec{x}(t)=0$. That is the direction of $\vec{\omega}(t)$. As is obvious (for example from the Rodrigues formula), in general $\vec{v}(t)$ and $\vec{\omega}(t)$ are not parallel. So, why are there two axes of rotation, and does $\vec{v}(t)$ play any role in the kinematics/dynamics of the motion?


Diffraction Grating in spectrophotometer

In terms of spectrophotometer, we need to make the light monochromatic before passing it to sample. For that, we use diffraction grating. From the research, I found out that the grating reflects light into different wavelengths. Also, the grating moves in such a way that each wavelength has an opportunity to pass through slit to sample. According to one journal, the monochromatic light is needed to make sure every photon has same energy and wavelength, which means everyone of them, has an equal opportunity to hit the particle. Is there better explanation in this theory? Also, Am I right in thinking that all wavelengths scattered from grating will hit the particles but particles will only absorb certain wavelength?


units of measure

This question is rather historical one. kilometres can be defined in metres, metres in centimetres, centimetres in millimetres. There must be some elementary unit (like millimetre or smth.) which cannot be defined in smaller units. The question is : How does this elementary unit came into being?? e.g. How did scientists decide about exact distance between point A and point B which is considered millimetre??


What's the importance of conformal transformations in general relativity?

I tried to understand the importance of conformal transformations in general relativity, but I failed. I didn't see that conformal transformations help to simplify the metrics, and also I didn't see that some physical metric (i.e., metric which describes geometry of some physical system) with conformal scale factor. Can you give some examples when conformal transformations are physically useful?


Work and Area under a Curve relating to Hooke's Law

> If it takes work W to stretch a Hooke’s-law spring (F = kx) a distance d > from its unstressed length, determine the extra work required to stretch it > an additional distance d (Hint: draw a graph and give answer in terms of > W!). I don't understand why the answer is not 2W since Force is proportional to x, or even how to begin using a graph to disprove why the answer is not 2W. Any help would be greatly appreciated.


Nuclear waste reclamation

Is it possible to reclaim nuclear waste from commercial reactors for useful purposes, if not necessarily energy production?


Will I recieve a shock if I charged a capacitor then i touch the 2 poles in same time?

What happens if I charged a capacitor then i touch the 2 poles in same time?


Definition of Fine-Tuning

I've looked in and out the forum, and found no precise definition of the meaning of fine-tuning in physics. **QUESTION** Is it possible to give a precise definition of fine-tuning? Of course, I guess most of us understand the empirical meaning of the phrase... but it seem so ethereal, that's the reason behind my question.


How is multiplicity given by 2S+1?

Suppose there are two electrons in an atom with $s_1 = \frac{1}{2}$, $l_1 = 1$ and $s_2 = \frac{1}{2}$, $l_2 = 1$. Hence the total $S$ (of the atom) may be +1 or 0. And total $L$ is either $+2$, $+1$ or $0$. Now If we consider $$\begin{align} S=1,L=2 &\to 2S+1=3; J=3,2,1\\\ S=0,L=2 &\to 2S+1=1; J=2\\\ S=1,L=1 &\to 2S+1=3; J=2,1,0\\\ S=0,L=1 &\to 2S+1=1; J=1\\\ S=1,L=0 &\to 2S+1=3; J=1\end{align}$$ But the last one does not show that $2S+1$ is multiplicity as it has only one $J$ value. Where am I making a mistake?


Help in understand Magnetostatic Energy

$$E_{\mathrm{ms}} = \frac{1}{2}\mu_0 \int_V \mathbf{M} \cdot \mathbf{H}_{\mathrm{ms}} d^3 r$$ I can't understand this formula, what is the magnetostatic stored potential energy?! What does it show? Does it explain anything relative to magnetization?


Constraints of massive relativistic point particle in hamiltonian mechanics

I try to understand constructing of Hamiltonian mechanics with constraints. I decided to start with the simple case: free relativistic particle. I've constructed hamiltonian with constraint: $$S=-m\int d\tau \sqrt{\dot x_{\nu}\dot x^{\nu}}$$ $\phi=p_{\mu}p^{\mu}-m^2=0$ $-$ first class constraint. Then $$H=H_{0}+\lambda \phi=\lambda \phi.$$ **So, I want to show that I can obtain from this Hamiltonian the same equation of motion, as obtained from Lagrangian.** But the problem is that I'm not sure what to do with $\lambda=\lambda(q,p)$. I tried the following thing: $\dot x_{\mu}=\\{x_{\mu},\lambda \phi\\}=\\{x_{\mu},\lambda p^2\\}-m^2\\{x_{\mu},\lambda\\}=\lambda\\{x_{\mu},p^2\\}+p^2\\{x_{\mu},\lambda\\}-m^2\\{x_{\mu},\lambda\\}=2\lambda \eta_{\mu b} p^b+p^2\\{x_{\mu},\lambda\\}-m^2\\{x_{\mu},\lambda\\}=2\lambda \eta_{\mu b} p^b+p^2\frac{\partial \lambda}{\partial p^{\mu}}-m^2\frac{\partial \lambda}{\partial p^{\mu}}$ $\dot \lambda=\\{\lambda, \lambda \phi \\}=\\{\lambda,\lambda p^2\\}-m^2\\{\lambda,\lambda\\}=\lambda\\{\lambda,p^2\\}+p^2\\{\lambda,p^2\\}=2\lambda\eta_{ak}p^{a}\frac{\partial \lambda}{\partial x^{k}}$ $\dot p_{\mu}=\\{p_{\mu},\lambda p^{2}-m^2\lambda \\}=p^{2}\\{p_{\mu},\lambda\\}-m^2\\{p_{\mu},\lambda\\}=-p^{2}\frac{\partial \lambda}{\partial x^{\mu}}+m^2\frac{\partial \lambda}{\partial x^{\mu}}$ If we recall that $p^2-m^2=0$, then we get from the third equation: $\dot p=0$, and from the first: $\dot x_{\mu}=2\lambda\eta_{ak}p^{a}$. So we have 1) $\dot x_{\mu}=2\lambda\eta_{\mu b}p^{b}$ 2) $\dot \lambda=2\lambda\eta_{ak}p^{a}\frac{\partial \lambda}{\partial x^{k}}$ 3) $\dot p=0$ _But I dont know what to do next._ Can you help me?


Introductory texts for functionals and calculus of variation

I am going to learn some math about functionALs (like functional derivative, functional integration, functional Fourier transform) and calculus of variation. Just looking forward to any good introductory text for this topic. Any idea will be appreciated.


microcanonical distribution

We know that in an isolated system, the density matrix is the microcanonical distribution matrix. That this the possibility for all the states with energy in a certain interval is a constant? But how can I deduce this from the postulate of equal probability?


So, I'm not too physics savvy but I am curious to ask. Is there a finite amount of mass in the universe? or is there more and more being created from somewhere or something? If the universe is infinite, and there's a finite amount of mass, that just seems kinda weird I guess. Hopefully this isn't too dumb a question...


Adiabatic proccess and Carnot cycle in a photon gas

I am making a comparation between the photon gas and the ideal classic gas for my Thermodynamics class. The photon gas is defined by the equations: $$U=aVT^4 $$ $$P=\dfrac{1}{3}aT^4$$ I found this document: http://www.csupomona.edu/~hsleff/PhotonGasAJP.pdf which explain how to find some basic things, like enthalpy and entropy. It says that a great exercise is to compare the Carnot cycle of the photon gas with the Carnot cycle of the ideal gas. According to it, the efficiency is $\eta=1-\frac{T_2}{T_1}$ the same as the ideal gas. I think that this is really interesting for my comparation, so I'm trying to calculate the Carnot cycle efficiency for this gas. I have no problem with the isothermal process, which is solved in that document: $$W_{ab}=-\dfrac{1}{3}aT^4\Delta V$$ However, I'm not sure if my result of the adiabatic process is correct. Work is $W=\int PdV$. Now, I can use the photon gas adiabatic equation (see the document) $PV^{4/3}=k$, where $k$ is a constant, to substitute $P$ in work equation, and integrate to obtain: $$W_{bc} =\dfrac{3}{4}k\left( \dfrac{1}{V_b^3} - \dfrac{1}{V_c^3} \right)$$ I'm not sure if this result is correct. When I try to calculate the efficiency of the cycle, I have: $$\eta=\dfrac{|W_T|}{|Q_{ab}|}=\dfrac{|W_{ab}+W_{bc}+W_{cd}+W_{da}|}{|Q_{ab}|}$$ where $W_{ab}$,$W_{cd}$ are isothermal and $W_{bc}$,$W_{da}$ are adiabatic. The heat is also defined in the document as: $$Q_{ab}=\dfrac{4}{3}aT^4\Delta V$$ But with these values I can't obtain the correct expression for the efficiency, or I don't know how to reduce the efficiency expression to obtain what I want. Which is the correct way to calculate adiabatic work in a photon gas? And the Carnot cycle efficiency? Thank you all for your answers :D


Shaping a wire such that a bead sliding on it has exactly isochronous oscillations

Let a wire be shaped according to some even function $y=f(x)$, with $f'(0)=0$ and $f''(0)>0$, and let a bead of negligible size slide frictionlessly on the wire. Let the bead oscillate under the influence of gravity about $x=0$ with amplitude $A$ (i.e., between $x=-A$ and $x=+A$) and frequency $\omega(A)$. Clearly $\omega$ is nearly constant for small $A$; it differs from the frequency $\omega_o$ of simple harmonic motion by at most $O(A^2)$. By choosing $f$ to be a fourth-order polynomial, we could presumably adjust the wire's shape so as to eliminate the errors of order $A^2$ and make $\omega(A)$ constant up to $O(A^4)$. Possibly we could continue this process of approximation and make all the derivatives $d^n\omega/dA^n$ vanish up to some finite $n$, or maybe for all $n$. If the derivatives can be made to vanish for all $n$, then I think $\omega$ would have to be nonanalytic at $x=0$. It seems impossible that there is any $f$ such that isochrony holds for arbitrarily large $A$. No matter how steep you make the sides, the bead can't do any better than accelerating downward with acceleration $g$. Therefore I think the best $f$ you can find is probably one that blows up to infinity at $|x|$ equal to some $x_{max}$. On dimensional grounds, we would have to have $x_{max}=cL$, where $c$ is a unitless constant and $L=g/\omega_0^2$. So my multipart question is: (1) Is there a function $f$ that gives $d^n\omega/dA^n=0$ for all $n$? If so, ... (2) How is $f$ characterized, and what is $c$? (3) Is $\omega(A)$ analytic at $x=0$, and if so, what is its radius of convergence to its Taylor series in units of $L$?


What are the properties of the partially polarized light on refraction?

When a ray of ordinary light is passed on the surface of the water the reflected light will be completely polarized( vibrations in one plane). My question is what will be **plane of vibration in the partially polarized light** that undergoes refraction? **How many planes of vibration** will be there? Deep explanation focusing on the **planes of vibration** of the partially polarized light would be appreciated.


Partial derivatives in Lagrangian formalism

Suppose I have a function $f = xy$. A partial derivative of $f$ with respect to $x$ implies holding $y$ constant: $$ \frac{\partial f}{\partial x} = y $$ Does this mean that in order to evaluate this derivative, $y$ cannot depend on $x$? For example, if $y = x$ then $$ \frac{\partial f}{\partial x} = \frac{\partial (x^2)}{\partial x} = 2x $$ Which is inconsistent with the first calculation of $\frac{\partial f}{\partial x}$. If $y$ indeed cannot depend on $x$, then how does the Lagrangian formalism of classical mechanics make sense? The Lagrangian is a function of $q$ and $\dot{q}$, and when we evaluate $\frac{\partial L}{\partial \dot{q}}$ we 'ignore' all of the $q$ dependence, even though $\dot{q}$ is a function of $q$.


Why does a cup with 100 g water float when placed on another cup with 50 g of water?

Imagine we have cup A with 50 g of water and cup B (smaller in width than A) with 100 g of water. Now put cup B into cup A. If the width of both cups are of comparable size then the cup with 100 g of water floats. It does not touch the bottom of cup B. Now think about Archimedes law of flotation. It says that the weight of displaced liquid = weight of the floating object. However in this case the bottom cup has only 50 g of water. How can an object float without displacing water equal to its own weight? Am I not applying Archimedes principle correctly or because of both things beings of comparable size Archimedes principle does not apply?


why is there no ninth gluon?

A teacher of mine told me once that there were no ninth gluon because such a one should be white and interact infinitely far, and no one has been observed. Is there also a theoretical reason?


Can we transfer the charge on a plate of a capacitor elsewhere?

I know it is hard , but can we transfer the charge on a capacitor plate elsewhere?


How can an object's instantaneous speed be zero and it's instantaneous acceleration be nonzero?

I'm studying for my upcoming physics course and ran across this concept - I'd love an explanation.


Calculating axle load in cargo container (pics are included)

I'm current developing cargo loader software, but i have a little challenge with calculating load for each axle when a cargo is placed on container. I know that all axles will be affected (the nearest ones are the most affected ones of course), and I want to find approximately the value of each load. Here's example scenario in pictures: ![enter image description here](http://i.stack.imgur.com/LCqjF.jpg) I tried to find loads using balance rules taking each point as a pivot and calculating equation based on that position as below (EDIT: on the following figure in case 4 the 5th axle is pivot, I forgot to show it on figure): ![enter image description here](http://i.stack.imgur.com/3AP9g.jpg) ![enter image description here](http://i.stack.imgur.com/v0D88.jpg) But I think I have problem with above solution as I could not find F3, F4, F5 in special examples (putting "d" distance values from pivot - P.S in this last picture all "d"s are different distances from pivot in that case - so d3 in the first equation is not equal to d3 in second equation as these both have different distances to different pivots) - for easy calculation you can use Cramer's rule online calculator here. I tried to solve this problem with moment but I could not done as I'm not good in physics :( I could not find relations between different forces in that case, I had only F1*d1 + F2*d2 + F3*d3 = F4*d4 + F5*d5 and F1 + F2 + F3 + F4 + F5 = mg ===> i got 5 unknowns but only 2 equations :( So, i need your help! Any suggestions will be appreciated!


Static Friction work and Energy

I have this problem: A 4.0 kg block is given an initial speed of 8.0 m/s at the bottom of a 20° incline. The frictional force that retards its motion is 15.0 N. (a) If the block is directed up the incline, how far does it move before stopping? (b)Will it slide back down the incline? I've managed to get part (a) which was 4.51m but I'm not sure how to start part (b).. The coefficient of Kinetic Friction was 0.407 Any ideas?


How does a transverse wave propogate in a medium?

I have been told that trasverse wave propogates by the oscillation of medium particles in direction perpendicular to propogation. Consider a wave on a taught string (x-y plane). What is the mechanism of movement of the same sinosudial function along the string? Suppose we start moving the particle on one end sinosudialy the how come the next particle also moves sinosudialy? How does elastic force between them does this work? If we displace the particle, there is some elastic force developed which is at some angle with horizontal so the next particle should move at some angle and its vertical displacement should be less than the previous one. So how come in a propogating wave all particle rise to the same height?


Can I make a cold air trap (thermal equilibrium)

Thermodynamic buoyancy. I have an air intake for combustion air entering into a basement furnace room. During cold weather, the air enters unchecked (no damper on pipe is allowed). I have this 5 foot vertical pipe, suspended downwards within a 2 foot high box, terminating 12" from the bottom. The cold air spills over the top of the box (combustion appliances are off at this point). How do I create a "cold air trap" like a p-trap (thermal equilibrium)? Would raising the height of the box help?


How is spacetime depicted in quantum field theory? Is space and time completely separate, and time is just nature of law as in Newtonian mechanics?


What happened to David John Candlin?

This is an ultra-soft question about relatively recent history. While reading some of Mandelstam's papers, I noticed that he cites David John Candlin consistenly whenever he does anything with Grassman path-integral. Everyone else cites Berezin. So I read Candlin's 1956 paper, and I was stunned to find a complete and correct description of anticommuting variables, presented more lucidly than anywhere else, with a clear definition of Grassman integration, and a proof that it reproduces the Fermionic quantum field. This is clearly the original source of all the Grassman methods. I was stunned that the inventor of this method is quietly buried away. I wrote the Wikipedia page on the guy, but I couldn't find out anything beyond the sketchy stuff I found on an old Princeton staff listing. The fellow doesn't google very well at all. Here are the questions:   * Is he still alive? (Hello? Are you there?)   * Did he become the experimental physicist David John Candlin in the late 1970s/early 1980s? Or is this someone else with the same name?   * Did he get any credit for his discovery? I mean, this is one of the central tools of modern physics, it is used every day by every theorist, and the inventor is never mentioned. It's 50% of the path integral. Why the silence?


Richtmyer Meshkov instability in MHD

In magnetohydrodynamics, the Richtmyer Meshkov instability is found to get suppressed by application of longitudinal magnetic field. Exactly what happens at the interface? Why instability gets suppressed? (How one can get the physical intuition of what is happening?)


Differentiation operator with respect to observable acting as a function of the observable?

In his _Principles of Quantum Mechanics_ Dirac writes: $$\int \langle \phi \frac{d}{dq}|q'\rangle dq' \psi(q')=\int \phi(q') dq' \frac{d\psi(q')}{dq'}.$$ To me it is rather strange, and it seems as if he was treating the operator $\frac{d}{dq}$ as a function of the observable canonical coordinate $q$, beacuse for functions of observables he gave the definition: $$f(\xi)|\xi'\rangle=f(\xi')|\xi'\rangle,$$ where $|\xi'\rangle$ is an eigenket of the observable $\xi$. Using this analogy, taking $f(q)=\frac{d}{dq}$ one could write $$\frac{d}{dq}|q'\rangle=\frac{d}{dq'}|q'\rangle$$ but then we would also need to differentiate $|q'\rangle$, since it is a function in $q'$. So what happens here? Please explain! Also, as a side question: why does $$\int \langle \phi \frac{d}{dq}|q'\rangle dq' \psi(q')=-\int \frac{d\phi(q')}{dq'} dq' \psi(q')$$ imply $$\langle \phi \frac{d}{dq}|q'\rangle=-\frac{d\phi(q')}{dq'}~?$$ Simply because the results of the integrations equal, that doesn't mean that their arguments also equal. The book can be accessed here. The formulas are on page 90, using the books original numbering.


Does a moving object curve space-time as its velocity increases?

We always hear how gravity bends space-time; why shouldn't velocity? Consider a spaceship traveling through space at a reasonable fraction of the speed of light. If this spaceship, according to special relativity, gains mass as a factor of _y_ as it approaches _c_ , then its gravitational field should increase in strength as well. Hence, space-time should warp. Note: Changes in space-time, gravity and mass should only be measureable by an outside observer with a different velocity. Those inside of the ship moving with it would not be able to measure the change in these properties.


Does a moving star have a larger gravitational pull?

A moving star's relativistic mass is larger than its rest mass. Is its gravitational pull larger? What about its inertial mass? Does it have larger inertial mass, keeping in mind that inertial mass should equal gravitational mass according to the equivalence principle. I suppose this quote from wikipedia is relavant: > However, it turns out that it is impossible to find an objective general > definition for the concept of invariant mass in general relativity. At the > core of the problem is the non-linearity of the Einstein field equations, > which makes it impossible to write the gravitational field energy as part of > the Stress–energy tensor in a way that is invariant for all observers. For a > given observer, this can be achieved by the Stress–energy–momentum > pseudotensor.[21]


How to determine viscous dampening coefficient of spring?

I'm trying to determine the viscous dampening coefficient of a spring $c$. Read about it on Wikipedia here. The two equations which I have are: $f=-cv$ and $ma+cv = -kx$ I know the spring constant $k=5$, the mass is $50\text{ }\mathrm{g}$ and the initial amplitude of the spring is $10\text{ }\mathrm{cm}$.


What is the fallacy in this infinite motion machine?

![alt text](http://i.stack.imgur.com/IKFiu.jpg) I realize this isn't possible, but I can't see why not, especially if you change the model a little bit so that the balls simply travel through a tube of water on the way up, rather than exactly this model. Please be clear and detailed. I've heard explanations like "the balls wouldn't move" but that doesn't do it for me - I really do not see why the balls on the right would not be pulled/pushed up, and the rest of the chain wouldn't continue on.


Why does this perpetuum mobile not work?(Gases and Densities)

I recently came up with the following concept. It is very simple, and may have been thought of before. A picture says more than a thousand words, so here is it explained in a picture: ![Conveyor belt using densities of gases to keep it in motion](http://i.stack.imgur.com/DQhdL.png) _Note that water was used to make the example easier to understand. Another gas (denser than Gas A and B) could be used instead, resulting in less friction than when using water._    At first glance, it seems that this machine could run for an indefinite amount of time. But I do not deem it possible to break the law of energy conservation. However, I have a hard time finding out what kind of force would cause this machine to slow down and stop.


Force and Energy of interaction of conducting sphere and point charge

> _A completely isolated neutral conducting sphere of radius $R$ is kept such > that its center is at a distance of $r\left( >R\right)$ from a point charge > $+Q$._ How can I find the force of interaction of the induced charges and the point charge, or at least the energy? I can't use "method of images" because the sphere is not grounded. **Note:** The actual question has a sphere already charged with $+Q$ charge and it asks for the $r$ at which the point charge is in equilibrium. I thought of breaking the force down into a superposition of two forces, one from the $+Q$ of sphere and the other from the induced charges. If there is some other way to solve this, I would like to know that too.


Best EM/Photon rocket using avalable tech?

Given 10-1000 Watts of electrical power (and no other consumables), what is the current best way to turn it into thrust? Just running it through a heater on an insulating pad would result in an IR thruster, but has bad focus. A laser has good focus but only for a small percentage of the energy.


What does it mean to increase volume by X decibels?

I am trying to decipher what decibels are: http://en.wikipedia.org/wiki/Decibel It seems to be a log ratio of audio amplitude multiplied by a constant. I am confused by what this means though. If my original volume is X, what does say increasing the volume by Y decibels mean? Does it mean `New Volume = 10 log ( Y / X )`?


Significance of an operator with a negative variance

Suppose we have a quantum state, well described by its time-independent wave function $\Psi$. And we have a well-defined Hermitian (self-adjoint) operator $\hat{A}$. We successfully evaluate the expectation value of the operator $\hat{A}$. Next we derive the general formula for the higher moments of $\hat{A}$ (i.e. the expectation value of $\hat{A^n}$ for $n=2,3,4…$). Finally we scale the operator $\hat{A}$ appropriately, in order to make the result dimensionless and to remove a possible growth factor (of type $C^n$) in the moments. We obtain: $$ <\hat{A^n}> = Cn + D $$ for $n=1,2,3,...$ and where $C$ and $D$ are constants. Let us now define a new operator $\hat{B}$ as follows: $$ \hat{B} = \hat{A^{n+1}} - \hat{A^n} $$ We can easily verify that the first and second moment of B are given by: $$ < \hat{B} > = C $$ $$ < \hat{B^2} > = 0 $$ Therefore the variance of operator $\hat{B}$ is negative! In violation of statistical laws. Should we conclude from this example that the results derived for the moments of $\hat{A}$ must be flawed? Or should we conclude that the new operator $\hat{B}$ is not a proper operator after all and therefore its strange properties are insignificant with respect to questions about the validity of $\hat{A}$? $$\begin{align} <\hat{B}> &= <\hat{A^{n+1}}> \- <\hat{A^n}> \\\ &= C(n+1) + D - Cn - D = C \end{align}$$ $$\begin{align} <\hat{B^2}> &= <\hat{A^{2n+2}}> \- 2<\hat{A^{2n+1}}> \+ <\hat{A^{2n}}> \\\ &= C(2n+2) + D - 2C(2n+1) - 2D + C(2n) + D \\\ &= 0 \end{align}$$ $$\begin{align} \text{Variance of }\hat{B} &= <\hat{B^2}> \- \left(<\hat{B}>\right)^2 \\\ &= -C^2 \end{align}$$


Newtonian physics

According to the book I am referring while working with non inertial frame of reference we have to apply a pseudo force equal to mass x acceleration in the opposite direction of the acceleration of the non inertial frame of reference with respect to the inertial frame of reference. But which frame is to considered to be absolutely inertial? Please don't say earth because Earth is not perfectly inertial its just inertial to good approximation. This can be justified while calculating the apparent weight of object due to earth's rotation.


What determines which frames are inertial frames?

I understand that you can (in principle) measure whether "free particles" (no forces) experience accelerations in order to tell whether a frame is inertial. But fundamentally, what determines which frames are inertial (i.e. what principle selects in which frames free particles will not appear to accelerate)? I've been told that the cosmic microwaves determine the ultimate rest frame of the universe, but that doesn't make sense to me, since one can still ask why that frame is an inertial frame. Also, I understand that there are no real inertial frames in general relativity, but it seems like there certainly exists approximate inertial frames and we can ask why those frames are approximately inertial and not others. For example, in the frame of a person riding a merry go round, free particles appear to greatly accelerate; while in the frame of someone standing next to the merry go round there are no such great accelerations. Why does the guy (or gal) on the merry go round see free particles accelerating while the other guy doesn't. And if you're gonna tell me that it's "the rest of the stuff in the universe" that determines whether the person on the merry go round sees free particles accelerate, I'll ask how you know that all that stuff is not spinning. I hope this question sort of makes sense, it's been bothering me for a while and my study of relativity (most of special relativity and just the outline of general relativity) hasn't really clarified it for me much.


Unit conversion help (school)

Can someone please help me explaining how to convert units in problems like the one below? I've done many things, but I am very unsure of all my methods. Could you tell me what steps should I take? $$5.66\ \frac{\mathrm{hm}}{\mathrm{min.}} \frac{\hspace{2cm}}{}\frac{\mathrm{km}}{\mathrm{hr.}}$$ Thank you.


Heating and Recooling of an Object

Consider a piece of metal of length $L$ and linear thermal expansion coefficient $\alpha$. We eat the metal $\Delta T$ degrees, causing the metal to increase to length $$ L' = L + L \alpha \Delta T$$ Now, cool the object back to the original temperature. This causes the metal to decrease in length to $$L'' = L' - L'\alpha\Delta T \\\ = L + L \alpha \Delta T - (L + L \alpha \Delta T)\alpha \Delta T \\\ =L (1 - \alpha^2 \Delta T^2)$$ My intuition would lead me to believe that heating up and then recooling an object would cause it to return to the same size it began at (if it did not, then bridges which repeatedly warmed and cooled would continually shrink) but this is not true according to my mathematics, which indicates that warming and recooling an object leaves it slightly smaller than it was before. **Do objects really not return to their original size when recooled? If so, then why do objects which warm and cool on a daily basis not slowly shrink?** I suspect that this may be related to $\alpha$ varying over the range of temperatures, but I didn't think that was a significant effect for solids.


How is contact resistivity defined for a Schottky contact, or the Schottky barrier height for an ohmic contact?

Based on the transfer length method (TLM), one can accurately calculate the contact resistivity for an ohmic contact, by evaluating the absolute resistance measured through the test structure and plotting it as a function of the gap spacing between the two ohmic contacts. By extrapolation, the contact resistance and transfer length (and thus, the contact resistivity) can be calculated. However, what if a measurement of the contact resistivity of a Schottky contact was desired? In this case, the forward biased current is non-linear (does not follow Ohm's law), and thus the absolute resistance measured is a function of voltage. Is there another way to measure the contact resistivity in this case? On the flip side of the coin, I have only seen the Schottky barrier height calculated for Schottky contacts. However, some ohmic contacts (e.g. tunneling ohmic contacts) still have a positive Schottky barrier height. How is the height measured in this case?


Tensor algebra doubt

Is it possible to take a tensor to the other side of the equation, and the tensor becomes its inverse(i.e contravariant becomes covariant and vice versa)? It is a stupid question, but It confuses me. For example, if $A_{ij} = B_{ij},$ Can I write $A_{ij}B^{ij} = \delta_{j}^{i}$, (though I think it should be $A^2$) Or is it only valid for the metric tensor? Also, is there a difference in the matrix representations of a tensor's contravariant and covariant form, or only the transformation rules differ?


QM without complex numbers

I am trying to understand how complex numbers made their way into QM. Can we have a theory of the same physics without complex numbers? If so, is the theory using complex numbers easier?


'Pseudo-Relativistic' behavior in Graphene

I've read that electrons in Graphene behave 'pseudo-relativistically'; what does this mean? how do they behave differently from electrons in other materials?


Hawking radiation at the mouth of Schwarzschild wormholes

I was researching a question for another post and it occurred to me that you might expect to see Hawking radiation at the mouth of wormholes. * * * Given the mechanism of Hawking radiation at the event horizon of black holes: virtual partial pairs forming at the edge and being separated by the event horizon; would virtual particles at the edge of wormholes likewise be divided? As I understand it, the trip is one way, which should cause the spontaneous formation of electrons and separation of positrons. * * *   1. Am I on the right track or is this a ridiculous assumption?   2. I don't see this in the literature on Hawking radiation or wormholes. Can you point me to any resources?   3. If this wouldn't occur, why not?


Why a mono-atomic crystal layer (2D) can't be stable?

According to Peierls and Landau, 2D crystals were thermodynamically unstable. They can't exist! Of course, this theory was disapproved in 2004 (example: graphene). What is the general definition of stability of a general system? What is the thermodynamics' stability?


"Speed" of Gravity and Speed of Light

Some threads here touching speed of gravity made me think about that. This lead to some questions.   * The speed of gravity was not measured until today (at least there are no undebated papers to that effect).   * It seems common knowledge/belief among physicists that the speed of gravity is the same as the speed of light.  And this is my question: Is that speed of light = speed of gravity a result of GR or is that fed into the theory? Or is there some evidence from other sources than GR for this? Does the same speed imply some deep-lying relation between gravity and electromagnetism?


Is it sure that gravitons are massless bosons?

> **Possible Duplicate:**   >  "Speed" of Gravity and Speed of Light I'm wondering if gravitational waves have the same speed of light? They must if gravity is mediated by a graviton spin 2 massless boson. One can argue that since gravity is a long range force, then its boson must have zero mass. But neutrinos travel through the universe too and despite their tiny mass yet have non zero mass. So, I'm wondering if there is a deeper justification for believing that gravity's speed =c?


Self-adjoint extensions with 'teletransporting' boundary conditions

When choosing a self-adjoint extension of a Hamiltonian, in general one can obtain domains in which (i) the probabilities teleport* between points on the boundary and (ii) boundary conditions locally conserve probabilities. The ones which locally conserve probability currents somehow seems nicer to me. But this is not at all an argument especially since tunneling is allowed in quantum mechanics. Is there any fundamental physical reasoning one can use to discard teleporting boundary conditions? Thanks in advance for any useful inputs. *I have used terminology from discussion about a related question : Physical interpretation of different selfadjoint extensions


Where to find the current positions and velocities of the planets?

I've written a program which simulates the motions of planets and other bodies. I'd like to run it on our own solar system, but to do so I need to know the current positions (preferably in heliocentric coordinates) of the planets as well as their current velocities. Is there a website where I can find this? I've found all the positions of the planets here, and I can find their average orbital speed fairly easily, but for some planets (e.g. Mercury) the orbital speed varies a fair amount.


why is orbital moment quenched while atoms forming solid

atom has well defined spin(up and down) and orbital(s,p,d,etc) momentum, but when forming crystals, why the spin degree continues to be good quantum number while orbital momentum is quenched?


Motion of two gravitational bodies?

I'm doing a basic realtime simulation of two bodies, but the orbits are unstable for some reason. This code is run at every timestep:               rsq = (a.x - b.x)*(a.x - b.x) + (a.y - b.y)*(a.y - b.y)          a.vx = a.vx - (a.x - b.x) * g * b.m * (1/rsq) * dt      a.vy = a.vy - (a.y - b.y) * g * b.m * (1/rsq) * dt          b.vx = b.vx - (b.x - a.x) * g * a.m * (1/rsq) * dt     b.vy = b.vy - (b.y - a.y) * g * a.m * (1/rsq) * dt          a.x = a.x + a.vx * dt     a.y = a.y + a.vy * dt          b.x = b.x + b.vx * dt     b.y = b.y + b.vy * dt      Where m is mass, x&y are position, vx&vy are velocity, g is the gravitational constant, and dt is the elapsed time. My bodies orbit eachother but the orbit is unstable and acts as though R is to an exponent other than 2. Have I missed something in my understanding of this problem or somewhere in my code?


Does mass affect velocity when travelling through frictionless medium?

I found the following question on an standardized test, and was debating with some friends what the answer would be: A car of mass **M** is travelling with a constant velocity through a plane in which friction is non-existent. An object of mass **m** (`m = M/3)` that is falling perpendicularly to the car lands inside of it. How will the velocity of the car be affected? This illustration can help explain the problem. ![Problem Illustration](http://i.stack.imgur.com/ZYLtp.jpg) My initial thought was that the velocity would be the same, given that friction is non-existent and that the momentum of the falling object is perpendicular to that of the car. However, some friends suggested that, since the mass of the car increases, the velocity should decrease.


What does the sky look like to human eyes from orbit?

There are numerous pictures, obviously, of the blackness of space from the shuttle, the space station, and even the moon. But they all suffer from being from the perspective of a camera, which is not sensitive enough to pick up the stars in the background when compared to the bright foreground objects (the limb of the Earth, the station, moon, etc). I've seen some photos that show a few of the brightest stars, but nothing special. Are there any photos or eye witness accounts from astronauts of what it looks like to a human with night-adjusted vision? If I were in an orbit similar to the space station and looked away from the Earth, would I be able to see more stars than I ever could on Earth, or would it only be marginally better than the best terrestrial night viewing?


SU(N) Yang-Mills $gg \to ggg$ scattering at tree level

When talking about the spinor-helicity formalism in his new textbook on quantum field theory, Matthew D. Schwartz claims as a highly nontrivial example, it is quite easy to use the Parke-Taylor formula to calculate the $gg\to ggg$ scattering cross section at tree level by hand, which is also one of the problems in the book. (Problem is found on page 560, problem 27.6) Can anyone tell me how to do this exactly? It looks like we cannot avoid summing $24^2$ terms, and in the paper, http://www.sciencedirect.com/science/article/pii/0550321386902300, the cross section of $gg\rightarrow gggg$ spans several pages! This makes me expect the cross section for $gg\rightarrow ggg$ is also complicated. Is there any smart way to do it?


What is the reason that relativistic corrections for hydrogen atom work?

Here I cite part from Sidney Coleman's lectures on Quantum Field Theory: > It is a phenomenal ﬂuke that relativistic kinematic corrections for the > Hydrogen atom work. If the Dirac equation is used, without considering > multi-particle intermediate states, corrections of $O \big(\frac{v}{c}\big)$ > can be obtained. This is a ﬂuke caused by some unusually low electrodynamic > matrix elements. What is the fluke about? Also, how can one justify the usage of Pauli- Schrodinger type equations that comes from first quantization of Dirac's equation? Schrodinger's equation is universal postulate valid for any quantum theory, and is equation for wave functionals in field theory. Could one go from non-relativistic QED field theory and then justify the usage of Pauli equation in which $\psi$ is interpreted as "wave function" in certain kinematical conditions (approximation)?


How do astronomers measure the distance to a star or other celestial object?

How do scientists measure the distance between objects in space? For example, Alpha Centauri is 4.3 light years away.


Pauli exclusion principle and Entangled pairs

It is true for fermions in the same potential that the total wavefunction of two particles must be antisymmetric with respect to exchange of electrons. Which means the spin wavefunction is given by $\chi=\frac{1}{\sqrt{2}}[\chi_+ (1)\chi_- (2)-\chi_+ (2)\chi_- (1)] $ which looks very much like the bell state, $\beta_{11}=\frac{1}{\sqrt{2}}[ |01\rangle - |10 \rangle]$. So, can we call those fermions, entangled states, as long as they are within the potential or there is something fundamentally special about entangled states (e.g. difference in measurement statistics) which makes them more unique? Apologies if the question is too simple for the level of this website. However, apparently it has made a lot of confusion for many people!


Do photons have acceleration?

Photons travels with the largest speed in our universe, the speed of light. Do photons have acceleration?


Does a photon instantaneously gain $c$ speed when emitted from an electron?

An excited electron looses energy in the form of radiations. The radiation constitutes photons which move at a speed $c$. But, is the process of conversion of the energy of the electron into the kinetic energy of the photon instantaneous. Is there a a simple way to visualize this process rather than math?


Is there any acceleration of light?

I find it hard to believe that photons always travel with 3 x 10^8 m/s just from the start. But there must be some acceleration of light. Maybe huge or taking place in picoseconds. So what maybe the acceleration of light or photons?


Does light initially accelerate?

Light travels with a speed of $3\times10^8{m\over s}$. My question is that was the light initially accelerating or it archived the speed in an instance? If it was accelerating then why it did not accelerate beyond that?


When photons are emitted, do they accelerate to reach the speed of light?

Photons are considered mass-less particle with a specific velocity but according to the electromagnetic theory, a photon is considered to have both energy and momentum. So what happen when they are emitted, do they accelerate to reach the speed of light or is it somehow instantaneous?


Does light photons have jerk?

While searching in web regarding whether rate of change of acceleration is possible or not; I came across the concept of jerk. I want to know whether photons which can be accelerated can also have jerk or not?


Deflection of magnetic compass sorrounded by alternating poles

What happens to the deflection of the magnetic compass if it is surrounded by south poles and north poles in alternating direction of a magnet around it in a circular pattern. Will it be deflected in any particular direction ? ![enter image description here](http://i.stack.imgur.com/MHqqH.jpg)


rectlinear motion with constant acceleration

Friends, this is a numerical homework problem. I tried my best to solve it but my answer is not matching with the one given at the back of the text book. Please help me out: A motor car moving at a speed of 72 km/h can come to a stop in 3 seconds, while a truck can come to a stop in 5 seconds. On a highway, the car is positioned behind the truck, both moving at 72 km/h. The truck gives a signal that it is going to stop at emergency. At what distance the car should be from the truck so that it doesn't collide with the truck. The typical human response time is 0.5 sec. My logic and answer: since car can decelerate to a stop much faster than the truck, it only need to worry about human response time which is 0.5sec. car would cover 10m in 0.5seconds at a speed of 72 km/h. so it just need to be 10m behind the truck minimum. but the answer in the book is 1.25 m How is this possible?


Why the pressure of atmosphere doesn't crush you when you e.g. walk outside? I mean the density of air is $1.26 kg/m^3$, so with $100 km$ above us, it exerts much pressure on you when you walk outside.


Kicking a soccer ball

I wonder which type of kicking may cause the ball to go a larger distance? One way is kicking the ball when it is at rest and another is kicking a moving ball in the opposite direction. If the ball goes farther in the second way,then what is the reason despite the fact that a moving ball will exert extra force to the player's leg?


How to calculate Riemann and Ricci tensors for a sphere?

Let's have the metric for a sphere: $$ dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right). $$ I tried to calculate Riemann or Ricci tensor's components, but I got problems with it. The Ricci curvature must be $$ R_{ij}=\frac{2}{R^{2}}g_{ij}. $$ But when I use definition of Ricci tensor, I can't turn the expression into the expression for the metric tensor Maybe, there are siome hints, which can help?


How to increase grain size in iron

Cut, polished and etched iron meteorites have grain size about 1000 times greater than air quenched iron. The reason for the large grain size is the very slow cooling of the core of a planet or protoplanet. Check web for cross section picture Widmanstätten pattern What methods can be used to grow large iron crystals on earth. What is the largest iron grain size achieved on earth? How about zone refining?


How can a clock work if the uncertainty principle is true?

If the uncertainty principle and Copenhagen Interpretation are true, then how can a clock tick? Supposedly particles can do all sorts of things when not measured, then how can they be formed into gears that make the clock tick when not measured?


Uncertain if invoking uncertainty principle for wave function is handwaving

Why doesn't the electron collapse onto the proton in a hydrogen atom? One explanation seems to be given by the Heisenberg uncertainty principle, which follows from the purely physical assertion that the wave function in the position representation is the Fourier transform of the wave function in the momentum (or more precisely the wave number) representation. A different explanation is obtained by solving (or inspecting) the Schrӧdinger equation. To what extent are these explanations independent? To what extent is the latter more fundamental? Is the latter explanation more correct, or more complete? The hydrogen atom is just an example, a more general form of the question would be to what extent it is correct to use the Heisenberg uncertainty principle to answer this kind of questions, as is often done in popular science texts? Elaborating a bit on what I mean to ask, the fact that the Fourier transform of the position wave function is the momentum wave function seems to allow us to draw conclusions about the behaviour of a system from this principle alone. However, when loosely drawing these conclusions, we tend to invoke our classical idea of momentum, whereas the dynamics of the system are really governed by the Schrӧdinger equation.


Friction problem having two blocks on separate inclined planes connected by a string

This is a question I found in my old high-school textbook(I'm revising the topics for a course).Two blocks A and B of masses $1kg$ and $2kg$ are placed on a fixed triangular wedge by a massless in-extensible string as shown. ![enter image description here](http://i.stack.imgur.com/vkP06.png) The pulley is massless and friction-less. The coefficient of friction between block A and wedge is $\mu _1 = 2/3$ and that between B and wedge is $\mu _2 = 1/3$ (let's neglect the difference between static and kinetic friction).The inclines are at $45 ^\circ $ with the horizontal.We need to find the frictional force exerted on each block,and the tension in the string. I worked out the problem,and I thoroughly checked my steps.Since the maximum resistive(friction) force is higher than the components of weights of the bodies along the inclines,there should be a state of equilibrium.I got the equations: $$T= \frac {10}{\sqrt2}+f_A \space .........(1)$$ $$T+f_B=\frac {20}{\sqrt 2} \space .........(2)$$ $$=> f_A +f_B = \frac {10}{\sqrt 2} \space ....(3); f_A-f_B=2T-\frac{30}{\sqrt 2} \space ....(4)$$ $$max(f_A)=max(f_B)=\frac{10\sqrt2}{3}\space .........(5)$$ (Considering $g=10 m/s^2$) Where $f_A$ and $f_B$ are frictional forces experienced by blocks $ A$ and $B$. There seems to be no way to get exact values of $f_A$ and $f_B$.Still,the book says(only numerical answers are given in the end,no complete solution) that $$f_A=\frac {10}{3\sqrt2} , f_B=\frac{10\sqrt2}{3} and \space T= \frac{40}{3\sqrt2} $$ All I can see by looking at the book's answers is that the author simply guessed that let there exist limiting(max) friction between $B$ and the wedge,and he might have put the value $f_B= max(f_B)$ and solved.In that case,why not the friction between $A$ and the wedge be limiting and the answers be $$f_A=\frac{10\sqrt2}{3} , f_B=\frac {10}{3\sqrt2} and \space T= \frac{50}{3\sqrt2} $$ But when we look at eq$(3)$,we see that neither of the frictions needs to be limiting,their sum needs to satisfy $\frac{10}{\sqrt2}$. Any way to justify the book's answers?


If gravitation causes constant acceleration why moon does not fall into earth?

If moon travels with constant speed in one direction and earth gravitation causes constant acceleration in perpendicular direction why moon does not eventually fall into earth? I mean if gravitation causes moon to fall faster each second (10m/s2) shouldn't after time velocity toward earth be big enough to cause it to fall ?


Why does the atmosphere rotate along with the earth?

![Now im isolating Atmosphere](http://i.stack.imgur.com/FkZ4H.jpg) I was reading somewhere about a really cheap way of travelling: using balloons to get ourselves away from the surface of the earth. The idea held that because the earth rotates, we should be able to land in a different place after some time. As we all know, this doesn't happen. Someone said that the reason why this doesn't happen is because the atmosphere (air, clouds etc.) also revolves around the earth (with the same angular velocity as the earth's rotation). Since we are also part of the atmosphere, our position doesn't change relatively. Well, I'm not convinced with that answer. Why does the atmosphere rotate along with earth? Gravitational force is towards the centre of the earth, but I don't see how it's making the atmosphere rotate.