General relaivity Definition and 166 Threads

I can see that by the tensor transformation law of the Kronecker delta that ##\frac{\partial x^a}{\partial x^b}=\delta^a_b## And thus coordinates must be independent of each other. But is there a more straightforward and fundamental reason why we don’t consider dependent coordinates? Is it...

Let’s say to the average Astronomer conducting research; generalizing the research to “astronomy” ; which would be more useful for the “average” Astronomer on a day to day basis: Quantum Mechanics or General Relativity? Obviously most are, but which so more? Which would be referred to more?

The energy-momentum tensor of a free particle with mass ##m## moving along its worldline ##x^\mu (\tau )## is \begin{equation} T^{\mu\nu}(y^\sigma)=m\int d \tau \frac{\delta^{(4) }(y^\sigma-x^\sigma(\tau ))}{\sqrt{-g}}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}. \end{equation} Let contract...

I'm confused by this question, from minimal coupling shouldn't the answer simply be ## \nabla^a \nabla_a F_{bc} = 0 ##? Any help would be appreciated. EDIT: I should also point out ##F_{ab}## is the EM tensor.

https://arxiv.org/abs/gr-qc/0306101 I am now reading this attached paper. But i can not get energy result(2.8), and I calculated it and found it is zero. here is my process: firstly, i use Gauss law and rewrite the (2.6): ##E=\frac{1}{8 \pi} \iint \chi_{0}^{0 \beta} \mu_{\beta} d S## where µβ is...

I was studying linearized GR where we make the following coordinate transformation ## \tilde{x}^{a} = x^{a} + \epsilon y^{a}(x) ## This coordinate transformation is then meant to imply ## g_{ab}(x) = \tilde{g}_{ab}(x) + \epsilon \mathcal{L}_{Y} g_{ab} ## Would anyone be kind enough to explain...

Since in 2D the riemman curvature tensor has only one independent component, ## R = R_{ab} g^{ab} ## can be reversed to get the riemmann curvature tensor. Write ## R_{ab} = R g_{ab} ## Now ## R g_{ab} = R_{acbd} g^{cd}## Rewrite this as ## R_{acbd} = Rg_{ab} g_{cd} ## My issue is I'm not...

Hi everyone, this is my first post on PhysicsForums. Thank you so much in advance for your help! My question is the following. Let us suppose we have an event A in a curved spacetime which, for definiteness, is the spacetime curved by the bodies of the solar system. Adopting a coordinate system...

Does Komar mass include the energy of gravitational filed or just only the energy of matter. For instants, we calculate the Korma mass of Schwarzschild space-time and we get the result that is M, which is the mass of central star. Does that mean Komar mass only contain the energy of matter not...

Hi, I am writing a report on uniqueness theorems and I am at the section for non asymptotically flat spacetime. I know that if we request certain restrictions, there are the existence of certain uniqueness theorems, however for the most part there are (so far) not many and they are hard to find...

So I just learned that in general relativity Magnatic and Electric fields are dependent on the observer. What are some consequences as a result?

Hello everyone, Any object has a gravitational potential energy as a function of the distance from the Earth (R). Does this energy depend only on the rest mass of the object; or one must take into account it's relativistic mass? In other words, if we imagine two identical bullets on the top...

I am under the impression, there is no unique solutions to Einstein's field equations for a cosmological constant, or for higher dimensional spacetimes. Has anybody got a counter example for a solution including the cosmo constant to show there are multiple solutions, for example, i know of the...

I have tried twice now to calculate acceleration of gravity using the general relativistic equation of geodesic deviation and both times my solution is twice the correct answer. What am I doing wrong? As an example here is one problem: Calculate the acceleration of gravity g at the earth’s...

Homework Statement We are asked to show that: ## \frac{d^2x_\mu}{d\tau^2}= \frac{1}{2} \frac{dx^\nu}{d\tau} \frac{dx^{\rho}}{d\tau} \frac{\partial g_{\rho \nu}}{\partial x^{\mu}} ## ( please ignore the image in this section i cannot remove it for some reason ) Homework Equations The...

Whether it be through Hawking radiation, miniature black holes, or even white holes, is it possible that one day energy could be harnessed from black holes and used on earth?

Hi, a simple question related to the gravitational wave detection. The net effect of gravitational wave is basically the stretching of the space including all the measurements tools (meter sticks just to illustrate the concept) that could be used to detect it. I am aware of laser...

This is a problem from Tensor Calculus:Barry Spain on # 69 Prove that a space with Schwarzschild's metric is an Einstein space, but not a space of constant curvature. The metric as given in the book is $$d\sigma^2=-\bigg(1-\frac{2m}{c^2r}\bigg)^{-1}dr^2-r^2d\theta^2-r^2\sin^2 \theta...

Hello everyone, By considering the effects of the gravitational time dilation the speed of the inner stars must be higher for the local observer than for the external one. So why the gravitational time dilation can not potentially explain the galaxy rotation curve? I already read that the...

Depending on the source, I'll often see EFE written as either covariantly: $$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = 8 \pi GT_{\mu\nu}$$ or contravariantly $$R^{\alpha\beta} - \frac{1}{2}Rg^{\alpha\beta} = 8 \pi GT^{\alpha\beta}$$ Physically, historically, and/or pragmatically, is there a...

Hi, Do we obtain the same escape velocity equation: Ve = sqrt(2GM/r) using both Newtonian and Relativistic approach?

Consider ##X## and ##Y## two vector fields on ##M ##. Fix ##x## a point in ##M## , and consider the integral curve of ##X## passing through ##x## . This integral curve is given by the local flow of ##X## , denoted ##\phi _ { t } ( p ) .## Now consider $$t \mapsto a _ { t } \left( \phi _ { t } (...

Hi, the trace of the extrinsic tensor K(Myu,Nu) in Alcubierre metric, relative to the foliation normal vector is non zero on the front x=rs+R and rear x=rs-R "walls". This is the normal volume component Tr =k(1,1)+k(2,2)+k(3,3) and should affect the Ricci tensor in Lorentzian/Minkowsky 4D. How...

In an ideal world with uniform gravity field, what does its space-time curvature look like? Is it non-zero? If not, how a free particle would be accelerated with the point view of space-time curvature? By uniform gravity field, I mean a gravity field with same value, same direction everywhere...

the problems/challenges that you have to face daily are mostly related to code issues with the physics itself? Is there room to improve our knowledge of fundamental physics while working on it? Do you enjoy doing it? why? I'm asking this because I'm considering working on numerical relativity...

Hello, I graduated from Penn State in 1986 with a B.S. in Physics and a minor in Electrical Engineering. My interests are in Special and General Relativity. I also very interested in finding ways to teach and explain physics to the general public. Joe Bigler

Hey! I will start my third year on the theoretical physics program. I have taken an introduction course in particle physics, just the basics, not much math. (quark and Feynman diagrams the forces and interaction , CRM matrix and cabibbo angle etc. ) Now I'm choosing between relativistic...

I'm trying to understand the BMS formalism in General Relativity and I'm in doubt with the so-called Bondi Coordinates. In the paper Lectures on the Infrared Structure of Gravity and Gauge Theories Andrew Strominger points out in section 5.1 the following: In the previous sections, flat...

What do people think of Fulvio Melia's argument for the necessity of "zero active mass" in FRW cosmologies? (i.e. the overall equation of state must be ##\rho+3p=0## at all times) Here is a link to an interesting lecture video: Here is a recent paper: https://arxiv.org/abs/1807.07587

Where in this though-experiment do I get it wrong? Even though no mass can travel faster then c, maybe information can? And I'm not talking about quantum entanglement etc. Consider a pipe, filled with balls that are very tightly arranged. If I push the outermost ball on one side of the pipe...

Some models of gravity, inspired by the main theme of spacetime fabric of Classical GR, treat the metric of the manifold and the connection as independent entities. I want to study this theory further but I am unable to find any paper on this, on ariXiv atleast. I will be very thankful if...

I am trying to calculate the Ricci tensor in terms of small perturbation hμν over arbitrary background metric gμν whit the restriction \left| \dfrac{h_{\mu\nu}}{g_{\mu\nu}} \right| << 1 Following Michele Maggiore Gravitational Waves vol 1 I correctly expressed the Chirstoffel symbol in terms...

Hi 1-)If an object's total velocity through space-time(four-velocity)is c, for example even we stand still we move with velocity c (through time) and if mass slows down time, can we say mass also increase our velocity in space? 2-) Is Four-velocity magnitude constant in General Relativity...

Hi I have 2 questions. There are 2 planets and one clock on each of them. One of them has a bigger gravitational field strength. And two clock have same distance from the core. 1-) Does time dilation occur between two? Which clock ticks slower? 2-) If time dilation occurs, which formula...

How do the components of the stress-energy tensor act on gravity regarding a) the FRW-universe? b) a solid ball? In a FRW-universe ##\rho + 3P## determines the second derivative of the scale factor. So, there are no non-diagonal components. Just theoretically, if the perfect fluid was...

I am watching these lecture series by Fredric Schuller. [Curvature and torsion on principal bundles - Lec 24 - Frederic Schuller][1] @minute 34:00 In this part he discusses the Lie algebra valued one and two forms on the principal bundle that are pulled back to the base manifold. He shows...

Hello everyone, I'll go straight to the question. The gravitational time dilation is equal to tearth = tspace*sqrt(1 - rs/r), with rs = 2GM/c2. However, the formula for speed of light in gravitational field is equal to v = c(1 - rs/r). My intuition tells me that these two formulas must be the...

Hi there guys, I was wondering does anyone have a layman's explanation of the GCRS as defined in the title. I am confused as to whether this is an inertial or non inertial system. In text modern reference books such as this (chapter 10, section 10.3.2) they define rotating/non rotating...

I was wondering where does the 1/2 factor come from in the Euler-Lagrange equation, that is: L = \sqrt{g_{\mu \nu} \dot{x}^\mu \dot{x}^\nu} implies that \partial_\mu L = \pm \frac{1}{2} (\partial_\mu g_{\mu \nu} \dot{x}^\mu \dot{x}^\nu ) I'm not sure I entirely understand where it comes...

Homework Statement Given some 2D line element, ## ds^2 = -dt^2 +x^2 dx^2 ##, find the Christoffel Symbols, ## \Gamma_{\beta \gamma}^{\alpha} ##. Homework Equations ## \Gamma_{\beta \gamma}^{\alpha} = \frac {1}{2} g^{\delta \alpha} (\frac{\partial g_{\alpha \beta}}{\partial x^\gamma} +...

Hello everyone, "Does a charged particle radiate in free-fall?". I read many threads on this subject and I was surprised to find out that there is no unanimous "Yes or No" answer to this question. Here is an interesting answer from researchgate.net: The question is widely discussed in the...

Lets consider some kind of metrics: \begin{equation} ds^2 = dt^2 - \frac{dr^2}{1-\frac{2M}{r}} - r^2(d\theta^2 + \sin^2\theta d\phi^2). \end{equation} here ##r = l/2\pi## is the radial coordinte like in Schwarzschild metrics. As far as I know, this metrics is describe the whormhole. Lets put...

Proposition: Consider an ##n + 1##-dimensional metric with the following product structure: $$ g=\underbrace{g_{rr}(t,r)\mathrm{d}r^2+2g_{rt}(t,r)\mathrm{d}t\mathrm{d}r+g_{tt}(t,r)\mathrm{d}t^2}_{:=^2g}+\underbrace{h_{AB}(t,r,x^A)\mathrm{d}x^A\mathrm{d}x^B}_{:=h} $$ where ##h## is a Riemannian...

Hi, let ##\gamma (\lambda, s)## be a family of geodesics, where ##s## is the parameter and ##\lambda## distinguishes between geodesics. Let furthermore ##Z^\nu = \partial_\lambda \gamma^\nu ## be a vector field and ##\nabla_\alpha Z^\mu := \partial_\alpha Z^\mu + \Gamma^\mu_{\:\: \nu \gamma}...

I've been playing around a bit with the Kerr orbit program I wrote, and have been thinking about ways to set the initial conditions. One thing I'd like to be able to do is specify a launch from some event in terms that would be convenient for an observer at that event with some given...

On Riemannian manifolds ##\mathcal{M}## the covariant derivative can be used for parallel transport by using the Levi-Civita connection. That is Let ##\gamma(s)## be a smooth curve, and ##l_0 \in T_p\mathcal{M}## the tangent vector at ##\gamma(s_0)=p##. Then we can parallel transport ##l_0##...

I have some questions about the curvature of space (NB not of spacetime) near a planet like Earth. Unambiguously defining space curvature requires choice of a coordinate system, so I choose the Swarzschild system. Here are my questions: Would constant-time hypersurfaces under the Swarzschild...

Suppose we have Einstein equation for *Universe free of matter* in form \begin{equation} G_{ik} = \chi T_{ik}, \end{equation} where the cosmological constant $\Lambda$ is transferred to the RHS of equation and written in the form of stress–energy tensor of Dark Energy...

Hi! I will soon begin my third year at the theoretical physics program. I have done a bunch of classical & Lagrangian mechanics, SP, atomic physics, electromagnetism, and basic particle physics. Is it a good idea to study general relativity and quantum field theory with this knowledge, what...

Homework Statement Derive the relativistic Euler equation by contracting the conservation law $$\partial _\mu {T^{\mu \nu}} =0$$ with the projection tensor $${P^{\sigma}}_\nu = {\delta^{\sigma}}_\nu + U^{\sigma} U_{\nu}$$ for a perfect fluid. Homework Equations $$\partial _\mu {T^{\mu \nu}} =...