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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27
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4answers
5k views

Are black holes perfect spheroids?

What I know about black holes (correct me if I'm wrong) is that they are the most compact objects in the universe that have been discovered. Due to all that gravity, wouldn't black holes be a perfect ...
2
votes
2answers
66 views

Can we exit the event horizon of merging black holes?

I have an intuitive scenario. Consider we have a spaceship just below the event horizon of a BH, which is merging with another black hole. Finally, the singularities merge and we have a single black ...
2
votes
1answer
149 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 ...
4
votes
1answer
174 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. ...
23
votes
5answers
4k views

Conformal transformation/ Weyl scaling are they two different things? Confused!

I see that the weyl transformation is $g_{ab} \to \Omega(x)g_{ab}$ under which Ricci scalar is not invariant. I am a bit puzzled when conformal transformation is defined as those coordinate ...
-4
votes
0answers
37 views

Parenthetical tensor notation [on hold]

Just out of curiosity, what does it mean to be a type $(n,m)$ tensor? For instance in the context of special and general relativity, the Minkowski metric $\eta$ is considered a type $(0,2)$ tensor. I ...
24
votes
1answer
2k views

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 ...
2
votes
2answers
271 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^...
0
votes
1answer
63 views

Is there energy output when mass moves between two spacetimes [closed]

Is there energy output when mass $m$ moves between two spacetimes? Say, it starts in a flat spacetime and then falls into a black hole (other examples don't come to mind, but this doesn't mean they ...
2
votes
4answers
2k views

Calculating the Riemann tensor for a 3-Sphere

I have worked out all the connection symbols for the 3-sphere using calculus of variations, cf. this Phys.SE post. So to find the Riemann tensor I am trying to find all the nonzero components of: \...
1
vote
2answers
102 views

Stationary v/s Static

Blau, in his GR book, says that a stationary and spherically symmetric metric is automatically static. He says this easily follows from the fact that for a stationary metric, and in spherical symmetry,...
6
votes
2answers
244 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 $$ ...
2
votes
0answers
55 views

Solving the Friedmann Equation [closed]

Through substituting for values of $\rho$ and $k$, I have: $$H^2=\left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi G C}{3a^4} + \frac{\Lambda c^2}{3}$$ $a=a(t)$, and $a(t=0)=0$. Note that $C$ is a ...
1
vote
3answers
57 views

What is the measure of distance in higher dimensions?

In our world we are using kilometers to measure distance. What measurement is used to measure distance in higher dimensions?
1
vote
1answer
36 views

A question about metric tensor and their minors and cofactors in general relativity

In Einstein's book- 'the meaning of relativity', he says- The equation 55 mentioned is this one- I don't understand what the equation (62) means or how it can be proved. I know that the metric tensor ...
3
votes
1answer
54 views

Effective Potential in General Relativity

I would like to clarify a concept about the Effective Potential in General Relativity when the kinetic energy term is not unitary. Suppose (in spherical coordinates) one has a generic line element of ...
0
votes
2answers
46 views

Given a metric of a torus can we measure it's thickness?

What I mean is, if you have a metric of a torus $T_2$, and you want to distinguish between say very thin stringy tori and very thick tori with the same surface area, is there a nice formula for this ...
6
votes
3answers
129 views

Two Robertson-Walker observers, at what time will a light signal be received?

Here is a question I have that is inspired by this question here. The spacetime metric of a radiation-filled, spatially flat ($k = 0$) Robertson-Walker universe is given by$$ds^2 = - dT^2 + T[dx^2 + ...
0
votes
0answers
9 views

How to calculate the lensing amplification of the weak field images in schwartzchild metric?

In Strong field limit of black hole gravitational lensing, Bozza et al state (without proof) that the amplification of each weak field image is given by $$ \mu_{wfi}=\frac{1}{\beta}\sqrt{\frac{2\,D_{...
3
votes
1answer
114 views

Geometrical point of view of the harmonic constraints ($\Delta g_{ij}=0$) in General Relativity

What does it mean, from the geometrical point of view, use (in General Relativity) of the constraints on the metric tensor's coefficients such that $\Delta g_{ij}=0$? (where $\Delta$ is the Beltrami-...
5
votes
1answer
518 views

What is the Schwarzschild metric with proper radial distance?

Reading the marvellous book "The Membrane Paradigm" I stumbled upon a suggested change of variable that I'm not able to deal with. Starting with the usual Schwarzschild metric for the spatial 3-...
6
votes
1answer
252 views

Geometric formulation of the equivalence principle

Let $(M,g)$ be a $4$-dimensional Lorentzian manifold. It is well know that given $(U,\psi=(x^1,\ldots,x^4))$ local chart around some $p\in M$, it is posible to find a change of coordinates given by $(...
0
votes
2answers
116 views

How to calculate the event horizon and the cosmological radius in a metric?

From reading about general relativity, the event horizon and the cosmological radius are the radius when $f(r)=0$, in the metric $$ ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2} $$ However, ...
0
votes
0answers
102 views

Intuition behind deriving the FRW metric

I am studying the FRW metric and am looking at a motivation for it here. The motivation seems to use four spatial dimensions. Why do we need the fourth spatial dimension in this? This doesn't seem ...
1
vote
1answer
72 views

Does the use of $\gamma=\left(1-v^{2}/c^{2}\right)^{-1/2}$ automatically assume a (+ - - - ) metric? [closed]

In Special Relativity, does the use of $\gamma=\left(1-v^{2}/c^{2}\right)^{-1/2}$ automatically assume a (+ - - - ) metric convention? For introductory textbooks, the Lorentz factor is is always ...
15
votes
4answers
719 views

Curvature of Hilbert space

That may appear as a dumb question, but: Does Hilbert space have curvature, or is it a flat space? How and why?
1
vote
0answers
28 views

How To Arrive At Ground State Metric of Kaluza-Klein Theory

The ground state metric, after an extra dimension of space is compactified (to a circle) in Einsteinian gravity, is the metric which corresponds to the R_4 × S_1 geometry of the separated dimensions. ...
1
vote
2answers
138 views

(Hyper)Surface of Simultaneity

How can I determine the surfaces of simultaneity if I know the metric? In particular, what are the surfaces of simultaneity for rotating disk with Langevin metric: $$ ds^2=-(1-\omega^2r^2)dt^2+2r^2\...
4
votes
1answer
256 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) ...
1
vote
1answer
66 views

Lorentz Transformations in Minkowski space

If $\Lambda$ represents the Lorentz transformation matrix, then the transformation of contravariant components $x^\mu$ is given by $$x'^\mu=\Lambda^{\mu}{}_{\nu} x^\nu$$ and that of the covariant ...
1
vote
2answers
88 views

General relativity applications other than gravity

Do the Einstein field equations successfully predict/describe physical processes other than gravitational ones?
5
votes
1answer
54 views

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\...
3
votes
1answer
95 views

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 ...
2
votes
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 ...
2
votes
1answer
128 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 ...
1
vote
3answers
257 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 ...
3
votes
2answers
153 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 : $S$ ...
1
vote
3answers
734 views

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 ...
1
vote
1answer
53 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, ...
3
votes
1answer
95 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 ...
7
votes
1answer
162 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 ...
8
votes
1answer
285 views

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^...
1
vote
1answer
97 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 ...
0
votes
1answer
46 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 = \...
2
votes
2answers
699 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 ...
1
vote
1answer
51 views

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\...
2
votes
3answers
3k views

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?
0
votes
0answers
43 views

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 ...
2
votes
2answers
255 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 ...
0
votes
1answer
46 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 ...