General relativity Definition and 999 Threads

In the book General Relativity for Mathematicians by Sachs and Wu, an observer is defined as a timelike future pointing worldline and a reference frame is defined as a timelike, future pointing vector field Z. In that sense a reference frame is a collection of observers, since its integral lines...

It is said that GR in the weak field limit it produces Newtons familiar law, so why can't GR produce other formulas for "strong field" which I guess it means at short distances.

I've read Collier's book on General Relativity and consulted parts of Schutz, Hartle and Carroll. In the terms they use, i have yet to gain anything resembling an intuitive understanding of parallel transport. In fact, it seems to me it is usually presented backwards, saying that the geodesic...

Consider the Einstein-Maxwell action (setting units ##G_{N}=1##), $$S = \frac{1}{16\pi}\int d^{4}x\sqrt{-g}\ (R-F^{\mu\nu}F_{\mu\nu})$$ where $$F_{\mu\nu} = \nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu} = \partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}.$$ This describes gravity coupled to...

Dark Matter was never found, what Modified Gravity theories should we check now?

In general relativity, does momentum conservation mean conservation of 4-momentum or 3-momentum

Hello everyone. I was reading Einsteins 1916 original paper on GR, the "The foundation of the general theory of relativity". There are some derivation that he did but I didn't quite understand. It would be nice if someone can give me some direction or some guidance on it. Here is the link to...

Hello there, suppose we take ##M## to denote the spacetime manifold. Suppose also that ## ds^2 = g_{\mu \nu} dx^\mu dx^\nu##. I have some confusions with regards to the metric and the line elements. My main confusion is at which points in the manifold are ## ds^2## defined? Is it correct that...

I want to prove that ##E = -g_{\mu \nu}u^\mu p^\nu## is the energy measured by an observer with velocity ##u^\mu## of an object with momentum ##p^\mu##. My reasoning is that in special relativity we know that ##\gamma m = E##. We can transform to coordinates where ##u'^\mu = (1,\vec{0})##. Since...

One way to see that spacetime is curved is to try and draw a "rectangle" in spacetime (see the figure in the Feynman lectures, ch 42.7): If I wait for 100 seconds and then move upwards on earth, I end up at a different point in spacetime than when I first move upwards and then wait for 100...

Hello, I am new to this forum, but have read a lot of posts and it seems really cool. I am 38 and have a B.S. in mathematics (from many years ago). I work in insurance, so I am pretty far removed from acedemia now. I have kept up my math studies as a hobby. After studying a lot of theoretical...

Homework Statement Two plane gravitational waves with TT (transverse-traceless) amplitudes, ##A^{\mu\nu}## and ##B^{\mu\nu}##, are said to have orthogonal polarizations if ##(A^{\mu\nu})^*B_{\mu\nu}=0##, where ##(A^{\mu\nu})^*## is the complex conjugate of ##A^{\mu\nu}##. Show that a 45 degree...

Given the metric of the gravitational field of a central gravitational body: ds2 = -ev(r)dt2 + eμ(r)dr2 + r2 (dθ2 + sin2θdΦ2) And the Chritofell connection components: Find the Riemannian curvature tensor component R0110 (which is non-zero). I believe the answer uses the Ricci tensor...

Homework Statement Attached Homework EquationsThe Attempt at a Solution So the question says 'some point'. So just a single point of space-time to be isotropic is enough for this identity hold? I don't quite understand by what is meant by 'these vectors give preferred directions'. Can...

I've thought of a new way (at least I never read it anywhere) of counting the independent components of the Riemann tensor, but I am not sure whether my arguments are valid, so I would like to ask whether my argument is sound or total bonkers. The Riemann tensor gives the deviation of a vector A...

Homework Statement I have several problems that ask me to prove that some quantity "transforms like a tensor" For example: "Suppose that for each choice of contravariant vector (a vector) A^nu(x), the quantities B_mu(x) are defined at teach point through a linear relationship of the form...

Homework Statement Question attached My method was going to be: set ##r=R## and solve for ##n(R)## set ##r=2GM## and solve for ##n(2GM)## I was then going to integrate proper time ##s## over these values of ##r##: ##\int\limits^{n=cos^{-1}(\frac{4GM}{R}-1)}_{n=cos^{-1}(1)=0} s(n) dn ###...

Forgive me if this question is a bit amateurish but i am no physicist. I know in general terms that GR and QM aren't compatible with one another, but my question is...Do they even need to be? can it not be a handoff scenario? why can't GR govern what it is supposed to govern and QM govern what...

Homework Statement I'm a new one on general relativity there are two prolems: first how to certify the proper length of light is zero second ,how to certify proper length of light ,do not change in all inertial frame Homework Equations for second question, probally Lorentz transform, The...

Working through Schutz "First course in general relativity" + Carroll, Hartle and Collier, with some help from Wikipedia and older posts on this forum. I am confused about the gradient one-form and whether or not it is normal to a surface. In the words of Wikipedia (gradient): If f is...

I don't know much about general relativity, but I'm curious if Zeno somehow foresaw the reallty of the universe many many years before Einstein did(without any rigour ofc, but still the idea holds?). Understanding the Zeno's arrow paradox, stating that an arrow is motionless at a certain moment...

If general relativity in the formal sense constrains all velocities to the speed of light as a maximum, how would superluminal group velocities exceeding speeds of light (at their superpositions) be evaluated in mainstream physics? Would this be a case of General Relativity and Physics...

In the context of General Relativity spacetime is a four-dimensional Lorentzian manifold M with metric tensor g, its Levi-Civita connection \nabla and a time orientation vector field T \in \Gamma(TM). In this context I've seem the following three definitions: A coordinate system is a chart...

I just started reading general theory of relativity. I have some elementary questions. Not an english speaker so bear with me. I am reading the thought experiment which describes path of a photon in a free falling lift. For an observer inside the lift, the photon path is a straight line. But...

How do I calculate the gravitational mass of a cylinder of compressed gas, including the effects of pressure? By gravitational mass, I mean what I would measure on an ideal mass balance. (I know that the pressure is negligibly small in a realistic container, but I want to have a conceptual...

Have members of the community had the experience of being taught GR both from a mathematical and physics perspective? I am a trained mathematician ( whatever that means - I still struggle with integral equations :) ) but I have always been drawn to applied mathematical physics subjects and much...

This isn't homework, nor is it an exercise problem; merely a question about a diagram. Re: B.Schutz book "A First Course in General Relativity" 2nd Edition, (Asian print version), page 5, Figure 1.1 "A spacetime diagram in natural units". From section 1.4 Spacetime diagrams: A world line is...

Hello there, I've been considering the geodesic equations of motion for a test particle in Schwarzschild geometry for some time now. Similar to what we can do with the Kepler problem I would like to be able to numerically integrate the equations of motion. I'm quite interested to see how...

First I don't have extensive knowledge about gravity beyond General Relativity, so please forgive my ignorance about this subject. I have confusion about the relation between GR and QM and I just want a general picture so that I can connect the dots. My questions: 1- Why do we need quantum...

As per the above figure coil A is situated inside a magnetic field caused by small accelerating objects B which in turn are causing the larmor frequency in coil Object C or train of objects C have quadruple movement and are rotating around the coil A giving out gravitational waves. Objects C...

Hello everyone, here I come with a question about inertial frames as defined in General Relativity, and how to prove that the general definition is consistent with the particular case of Special Relativity. So to contextualize, I have found that one can define inertial frames in General...

Ignoring for the moment the plausibility of this scenario, what would happen? A black hole is the ultimate gravity well, right? In some circles, they're even considered tears in the fabric of the universe. So what would happen to that fabric if you moved the black hole? Would it behave like...

I desire to read General relativity by Wald. I am a student who is self teaching, and I know tensor calculus by Pavel grinfield's introduction to tensor analysis and calculus of moving objects. The book states things from advanced calculus. Do I have more math to learn before hand?

It is commonly said that if you lift an object above the Earth it gains potential energy equal to mgh (m=mass, g=gravitational acceleration, h=height), suggesting that the potential energy is in the lifted mass. This cannot be. Consider the case of two perfectly rigid spheres, isolated in...

Is the Hubble's law(recessional velocity linearly proportional to distance) valid for all cases even when the spacetime is curved? Is there a nonlinear model for Friedman models or it's always linearly proportional?

Classical mechanics: a minimal standard course by Sergei Winitzki. It is not probably going to help you if you already did not know the subject but is a great refresher nonetheless. He also includes a differential equations refresher that I found invaluable. Looks like he has a ton of other...

Dear All, I am trying to find some good books that provide a comprehensive one-stop education for special and general relativity, with concise coverage of key fundamentals of the maths involved. It is intended for self study, and I do not have strong fundamentals on advanced physics / maths...

Homework Statement I want to calculate the on shell action for the case of a AdS Schwarzschild black hole. Homework Equations Following the case of the flat Schwarzschild black hole I tried to add a counter-term of the type of a Gibbons-Hawking boundary term for flat Minkowski spacetime...

Has anyone heard of "Bauer's Paradox" in Einstein's working out his equations for General Relativity? The description I came across was extremely fuzzy (something about an expression for a certain tensor adding up to be zero when it shouldn't, or vice-versa), so I am looking for a better...

I am thinking about taking GR as an elective,but I have no plans on going into cosmology or any related field.I wanted to know if when applying for graduate school,whether it is a positive for them to see you have taken a "high level" course,such as GR. I know that the best thing is to have...

Let us assume that a spacetime has a global time-displacement symmetry described by a timelike Killing vector field ##\xi^\mu=(1,0,0,0)##. Further assume that we have a particle with four-momentum ##P^\mu=m\ V^\mu##. It is well-known that a stationary observer with four-velocity ##U^\mu##...

I've been studying a bit of differential geometry in order to try and gain a deeper understanding of the mathematics of general relativity (GR). As you may guess from this, I am approaching this subject from a physicist's perspective so I apologise in advance for any lack of rigour. As I...

Last year I finished the undergraduate course in Mathematical Physics. This year, more precisely in March, I'm going to start the graduate course to acquire a master's degree in Physics. Now, for this course I must choose a research topic and find an advisor. This is being a little bit...

I have seen the derivation for Unruh radiation for a massless, non-interacting scalar field (Carroll). Are there interesting differences that arise for more realistic standard model cases. For example, what does QCD look like for an accelerating observer? Any papers that detail this would be...

Why is i that two cones connected at their vertices is not a manifold? I know that it has to do with the intersection point, but I don't know why. At that point, the manifold should look like R or R2?

Homework Statement I have ##R^{u}_{o b u } F^{o}_a - R^{u}_{oau}F^{o}_{b}## and I want to show that this equal to ##2R_{o[aF^{o}_b]}## where ## [ ] ## denotes antisymmetrization , and ##F_{uv} ## is a anitymstric tensor Homework Equations Since ##F_{uv} ##is antisymetric the...

As far as I can tell, in GR, the Chirstoffel symbol in the expression of the Connection is analogous to the vector potential, A, in the definition of the Covariant Derivative. The Chirstoffel symbol compensates for changes in curvature and helps define what it means for a tensor to remain...

Homework Statement Calculate the circumference of the circle θ = θ0 (a constant) in the spatial geometry \begin{eqnarray*} dS^2 = a^2(d\theta^2 + sin^2\theta cos^2\theta d\phi^2) \end{eqnarray*} Hence, (by finding R(z)) sketch the cross section of the surface embedded in three dimensions via...

Homework Statement A stationary, axisymmetric, spacetime has two Killing vector fields [ξt, ξφ] corresponding to translation along t or φ directions. A particle of unit mass moving in this spacetime has a four-velocity u = γ[ξt + Ωξφ]. (i) Explain why we can interpret this as a particle moving...

Hello, does anyone know if a solution to EFEs (Einstein's Field Equations) has been found that's analogous to Helmholtz's vortex ring solutions in fluid dynamics?