The variables used in general relativity to describe the shape of spacetime. If your question is about metric units, use the tag "units", and/or "si-units" if it is about the SI system specifically.

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General relativity applications other than gravity

Do the Einstein field equations successfully predict/describe physical processes other than gravitational ones?
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How to define the distance between two points in a conformal transformed space?

Consider a particular conformal transformation $x^\mu\rightarrow x'^\mu$, and the metric of a flat space transforms in the following way, $$\eta_{\mu\nu}\rightarrow g'_{\mu\nu}=\Lambda^2(x)\eta_{\mu\...
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1answer
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Metric with Harmonic Coefficient and Stress-Energy Tensor in General Relativity

I have two question: Is there any possible implies or interest to use in general relativity a metric whose coefficients are harmonic functions? What is the meaning (physical) if the stress-energy ...
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1answer
129 views

What is the metric of a constant electromagnetic (pure electric or pure magnetic) field?

For example, imagine a magnetic field $B_x$ directing in $\hat{x}$ direction filling all the space. What is its associated metric field? I can construct the electromagnetic stress-energy tensor for ...
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1answer
105 views

Derivation of Christoffel Symbols

So I am reading a book on relativity & differential geometry and in the text, they gave the Christoffel symbols in terms of the metric and its derivatives, but I wanted to derive it myself. ...
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Conformal invariance of Maxwell equation in presence of external current

It is known that pure electrodynamics in curved space-time is invariant under Weyl transformations $$ \tag{1} g_{\mu\nu} \to \Omega(x)g_{\mu\nu}, \quad F_{\mu \nu} \to \Omega^{-1}(x)F_{\mu\nu}. $$ ...
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1answer
51 views

Age of universe from Hubble's constant

Assume the Robertson-Walker metric: $$g = -d\tau^2 + a^2(\tau)\gamma$$ where $\gamma$ is the flat, spherical or hyperbolic spatial metric and $a$ is the scale factor. Wald seems to calculate the age ...
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1answer
117 views

What is the relation between the metric tensor and the graviton?

In Zee's quantum theory in a nutshell, at the end of chapter I.10, he states that the graviton is of course the particle associated with the field $g_{\mu\nu}$. My understanding of quantum ...
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3answers
247 views

Spacelike to timelike four vectors

First at all, let me just say that I'm not a Physicist, I study mathematics. So, I have this question. If you have a spacelike four vector, is there any transformation that could change it to be a ...
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2answers
225 views

Performing Wick Rotation to get Euclidean action of scalar field

I'm working with the signature $(+,-,-,-)$ and with a Minkowski space-stime Lagrangian $$ \mathcal{L}_M = \Psi^\dagger\left(i\partial_0 + \frac{\nabla^2}{2m}\right)\Psi $$ The Minkowski action is $$ ...
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2answers
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Is spacetime flat inside a spherical shell?

In a perfectly symmetrical spherical hollow shell, there is a null net gravitational force according to Newton, since in his theory the force is exactly inversely proportional to the square of the ...
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151 views

General expression of the redshift: explanation?

In some papers, authors put the following formula for the cosmological redshift $z$ : $1+z=\frac{\left(g_{\mu\nu}k^{\mu}u^{\nu}\right)_{S}}{\left(g_{\mu\nu}k^{\mu}u^{\nu}\right)_{O}}$ where : $S$ ...
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3answers
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How to prove the raising/lowering indices operation?

I've read this related question, though it didn't satisfy me; I hope this complements it. I know that if I contract a covariant tensor ${A_{\alpha\beta}}$ with a vector ${B^\beta}$, I get some other ...
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1answer
49 views

The form of the metric after a dimension is compactified

Upon the compactifiation of one spatial dimension, it is said (as though an axiom) that the 5 dimensional spacetime metric separates into a 4 dimensional metric, a vector, and a scalar, (4D gravity, ...
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1answer
90 views

The most general way to write flat space metric [closed]

What is the most general way to write flat space (in d=4 in particular), but still preserving some isometries? In particular I'm interested in the case with 2 isometries, basically by using explicitly ...
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1answer
160 views

Can special relativity be derived from the invariance of the interval?

As far as I know, the classical approach to special relativity is to take Einstein's postulates as the starting point of the logical sequence, then to derive the Lorentz transformations from them, and ...
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2answers
254 views

Coordinate Singularity in Metric

Suppose I have some metric $$ds^2=g(t)dt^2+\frac{1}{r}dr^2$$ which has a singularity at $r=0$. However, if I make the coordinate transformation $u=\frac{1}{r}$, then I get: $$ds^2=g(t)dt^2+r^3 du^...
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1answer
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Symmetries of AdS$_3$, $SO(2,2)$ and $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$

Basically, I want to know how one can see the $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ symmetry of AdS$_3$ explicitly. AdS$_3$ can be defined as hyperboloid in $\mathbb{R}^{2,2}$ as $$ X_{-1}^2+X_0^...
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1answer
94 views

What is the additional gravitational term from general relativity given by?

Carroll gives the potential energy in general relativity by $$ V(r)=\frac{1}{2}\epsilon-\epsilon\frac{G\,M}{r}+\frac{L^{2}}{2r^{2}}-\frac{G M L^{2}}{r^{3}} $$ My first question is does $V(r)$ have ...
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1answer
43 views

Symmetry group of FLRW metric

$$ g = dt^2 - a^2(t) (dx^2+dy^2+dz^2) = dt^2-a^2(t)(dr^2+r^2d\Omega^2)$$ So this is my metric. What is the symmetry group of it? I think that my Killing vectors are 3 translation vectors: $$K_i = \...
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2answers
642 views

Killing Vectors in Schwarzschild Metric

Given the Schwarzschild metric with $(-,+,+,+)$ signature, $$\text ds^2=-\left(1-\frac{2M}{r}\right)dt^2+\left(1-\frac{2M}{r}\right)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2)$$ the lack of ...
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1answer
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Inverse metric in Newtonian limit of GR

I am reading Carroll's book. So looking at the Newtonian limit we write $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$ where $h_{\mu\nu}$ is some small perturbation. He says that because $g^{\mu\nu}g_{\nu\...
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3answers
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Ricci scalar for a diagonal metric tensor

I was wondering if there is a general formula for calculating Ricci scalar for any diagonal $n\times n$ metric tensor?
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Extrinsic curvature for cylinder [migrated]

Suppose we have the following metric describing a cylinder: $$ds^2=ud\rho^2+\rho^2d\phi^2$$ where $u$ is a function of $\rho$. We know the definition of the extrinsic curvature that is, $$K_{ij}=-\...
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A question from “The meaning of the relativity, by A.Einstein” - Lorentz transformations [duplicate]

Let $K$ and $\bar K$ be two cartesian co-orditate systems in $\mathbb{R}^3$. The element: $$s^2=(\Delta x^1)^2+(\Delta x^2)^2+(\Delta x^3)^2$$ is an invariant in all co-ordinate system. I want prove ...
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2answers
240 views

Different signatures

I was working out the christoffel symbols, once where the metric that I am using has (+---) signature and another time where it has (-+++) signature because two books had different signatures and I ...
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0answers
40 views

Vector addition over orthogonal group [migrated]

I am working on a fun problem. The problem is to solve the following equation: $O_1x_1+O_2x_2=x_1+x_2$, where $x_1, x_2 \in \mathbb{R}^2$ are known and $O_1,O_2 \in \mathbb{O}(2)$ are unknowns. [$\...
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1answer
45 views

How to measure time in presence of a strong gravitational field? [duplicate]

I need an operative definition of "measuring time in general relativity" that takes in consideration also the presence of strong gravitational fields between me and clock, able to deviate the light ...
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1answer
34 views

Worldsheet metric & event horizon

Given a certain metric $g_{\alpha \beta}$ (not necessarily diagonal) in which $g_{\tau \tau}=0$ for a certain function, is there any way of determining if there is a singularity in that point, or if ...
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1answer
68 views

“Measure of time in general relativity” [duplicate]

Suppose to be in an arbitrary gravitational field and you are moving in it arbitrarily with a clock in your hand. In this general situation I ask: if I read the positions of the hands of the clock, ...
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1answer
54 views

Poincare Group (Wald, Chapter 4 Page 59)

In Wald's text on general relativity, he mentions that in special relativity, many different global inertial coordinate systems are possible and can be put into one-to-one correspondence with elements ...
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35 views

Nature of the singularity in the Taub-NUT metric

Consider the Taub-NUT metric $$ds^2=-V(dt+2N(1-\cos\theta)d\phi)^2+\frac{1}{V}(dr^2)+(r^2+N^2)(d\theta^2+\sin^2\theta{}d\phi^2),$$ where $$V=\frac{(r-r_+)(r-r_-)}{(r^2+N^2)} \qquad r_{\pm}=M\pm \...
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1answer
46 views

EFE and Local Minkowski

Suppose we view the Einstein Field Equations (EFE) in the context of a boundary value problem with a given stress-energy tensor and boundary conditions. The problem is solved by finding a pseudo-...
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1answer
27 views

Ground state metric?

In kaluza-klein theory, there's a notion of a "ground state metric" after compactification. What is the meaning of the term "ground state metric"?
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1answer
92 views

How much Gravity is required to stop time?

Clocks free of gravitational influence run faster than those experiencing gravity. Is it possible for gravitational influence to bring time to a stop? Additionally can acceleration affect clocks in ...
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897 views

Orbits around the Photon sphere of a black hole (Schwarzschild coordinates)

This is a follow-up question to the answer given at What is the exact gravitational force between two masses including relativistic effects?. Unfortunately the author hasn't been online for a few ...
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1answer
71 views

Spacetime background of Quantum mechanics [closed]

Why is it said that the Schrodinger equation suggests a fixed, non-dynamical background spacetime, with time as an external parameter? How does this interpretation come about from the Schrodinger ...
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1answer
39 views

Why are dimensions regarded as square/perpendicular?

Starting from the second dimension, the dimensions are basically represented by a square, cube, tesseract, and so on. I don't know if this is a stupid question or not, but is there an obvious or less-...
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2answers
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Do any two points in Minkowski spacetime determine a unique line?

Any two points in a Euclidean space determine a unique line, but I wasn't sure if this result generalized to Minkowski spacetime given that the latter is not a Euclidean 4-space, but is, instead, a ...
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0answers
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Scalar Curvature of a Conformally Flat Metric

Suppose that you have a metric $g_{\mu\nu}=\phi^2\eta_{\mu\nu}$ for some function $\phi$. There is a standard formula for what the scalar curvature $R$ looks like in terms of $\phi$, which is given by ...
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4answers
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Space-like and time-like: where do the names come from?

Space-like separated events are events that, in a well-chosen reference frame, can take place at the same time but never happen at the same location. On the other hand for time-like events, one can ...
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2answers
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What is the difference between time and space in general relativity?

I know that similar questions have been asked before, I will try to be specific. In special relativity time is the coordinate with minus sign in metric tensor. In general relativity the components of ...
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0answers
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Coordinate time difference between emiting and detecting a photon in bent spacetime

Consider an arbitrary non-trivial metric $g_{ij}$ - like the Schwarzschild metric. Now, consider two observers $A$ and $B$, staying at fixed radii $R_A$ and $R_B$, respectively, with $R_A > R_B$. ...
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1answer
230 views

How can I mathematically describe the parallel transport in the Roman soldiers example?

I've been trying to understand parallel transport. Many of the descriptions present a mathematical version: $\nabla_V X = 0$. And/or they present an example involving soldiers (usually Roman) ...
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1answer
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Are the Schwarzschild metric and the Geodesic Equation relevant in the context of the Earth? [closed]

The geodesic equation used in general relativity is the following: $$ {\mathrm d^2 x^\mu \over \mathrm ds^2} =- \Gamma^\mu {}_{\alpha \beta}{\mathrm d x^\alpha \over\mathrm ds}{\mathrm d x^\beta \...
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1answer
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Signature of $f: \Lambda^2(\mathbb{R}^4) \times \Lambda^2(\mathbb{R}^4) \to \mathbb{R}$, $f(\omega, \omega') = \omega \wedge \omega'$ [closed]

Define$$f: \Lambda^2(\mathbb{R}^4) \times \Lambda^2(\mathbb{R}^4) \to \Lambda^4(\mathbb{R}^4) \cong \mathbb{R}, \quad f(\omega, \omega') = \omega \wedge \omega'.$$ What is the signature of $f$? ...
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What spacelike, timelike and lightlike really mean?

Suppose we have two events $(x_1,y_1,z_1,t_1)$ and $(x_2,y_2,z_2,t_2)$, then we can define $$\Delta s^2 = -(c\Delta t)^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$$ which is called the spacetime ...