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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15
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6answers
986 views

Proving that interval preserving transformations are linear

In almost all proofs I've seen of the Lorentz transformations one starts on the assumption that the required transformations are linear. I'm wondering if there is a way to prove the linearity: Prove ...
14
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3answers
320 views

What is meant when it is said that the universe is homogeneous and isotropic?

It is sometimes said that the universe is homogeneous and isotropic. What is meant by each of these descriptions? Are they mutually exclusive, or does one require the other? And what implications rise ...
14
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2answers
1k views

Why is spacetime not Riemannian?

I apologize if this is a naïve question. I'm a mathematician with, essentially, no upper-level physics knowledge. From the little I've read, it seems that spacetime is Lorentzian. Unfortunately, the ...
14
votes
4answers
622 views

The Lagrangian as a metric

My question is, can the (classical) Lagrangian be thought of as a metric? That is, is there a meaningful sense in which we can think of the least-action path from the initial to the final ...
13
votes
1answer
481 views

What are the local covariant tensors one can form from the metric?

Normally in differential geometry, we assume that the only way to produce a tensorial quantity by differentiation is to (1) start with a tensor, and then (2) apply a covariant derivative (not a plain ...
12
votes
4answers
214 views

Difference between matrix representations of tensors and $\delta^{i}_{j}$ and $\delta_{ij}$?

My question basically is, is Kronecker delta $\delta_{ij}$ or $\delta^{i}_{j}$. Many tensor calculus books (including the one which I use) state it to be the latter, whereas I have also read many ...
11
votes
4answers
2k views

Why is the space-time interval squared?

The space-time interval equation is this: $$\Delta s^2=\Delta x^2+\Delta y^2+\Delta z^2-(c\Delta t)^2$$ Where, $\Delta x, \Delta y, \Delta z$ and $\Delta t$ represent the distances along various ...
10
votes
5answers
1k views

Minkowski Metric Signature

When I learned about the Minkowski Space and it's coordinates, it was explained such that the metric turns out to be $$ ds^{2} = -(cdx^{0})^{2} +(dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2} $$ where $ ...
10
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3answers
4k views

Why is the covariant derivative of the metric tensor zero?

I've consulted several books for the explanation of why $$\nabla _{\mu}g_{\alpha \beta} = 0,$$ and hence derive the relation between metric tensor and affine connection $\Gamma ^{\sigma}_{\mu ...
10
votes
3answers
600 views

Equation of motion of a photon in a given metric

I have this metric: $$ds^2=-dt^2+e^tdx^2$$ and I want to find the equation of motion (of x). for that i thought I have two options: using E.L. with the Lagrangian: $L=-\dot t ^2+e^t\dot x ^2 $. ...
10
votes
2answers
844 views

Symmetry of the $3\times 3$ Cauchy Stress Tensor

When presenting the stress tensor (say in a non-relativistic context), it is shown to be a tensor in the sense that it is a linear vector transformation: it operates on a vector $n$ (the normal to a ...
10
votes
1answer
409 views

Causal and Global structure of Penrose Diagrams

What kind of global and causal structures does a Penrose diagram reveal? How do I see (using a Penrose diagram) that two different spacetimes have a similar global and causal structure? Also, I ...
9
votes
2answers
501 views

Does Kaluza-Klein Theory Require an Additional Scalar Field?

I've seen the Kaluza-Klein metric presented in two different ways., cf. Refs. 1 and 2. In one, there is a constant as well as an additional scalar field introduced: $$\tilde{g}_{AB}=\begin{pmatrix} ...
9
votes
1answer
380 views

How does the Hubble parameter change with the age of the universe?

How does the Hubble parameter change with the age of the universe? This question was posted recently, and I had almost finished writing an answer when the question was deleted. Since it's a shame to ...
9
votes
2answers
880 views

Question about proper time in general relativity

I think I may have some fundamental misunderstanding about what $dt, dx$ are in general relativity. As I understand it, in special relativity, $ds^2=dt^2-dx^2$, we call this the length because it is ...
9
votes
1answer
338 views

Contracting Indices

Does anyone know how to get from (1) to (2) in the system $$ \begin{align} \mathrm{g}^{\mu\nu}_{,\rho}+ \mathrm{g}^{\sigma\nu}{{\Gamma}}^{\mu}_{\sigma\rho}+ ...
8
votes
4answers
1k views

Minkowski spacetime: Is there a signature (+,+,+,+)?

In history there was an attempt to reach (+, +, +, +) by replacing "ct" with "ict", still employed today in form of the "Wick rotation". Wick rotation supposes that time is imaginary. I wonder if ...
8
votes
4answers
465 views

How do you tell if a metric is curved?

I was reading up on the Kerr metric (from Sean Carroll's book) and something that he said confused me. To start with, the Kerr metric is pretty messy, but importantly, it contains two constants - ...
7
votes
4answers
538 views

Can a metric in General Relativity, Supergravity, String Theory, etc., be asymmetric?

Why is it that all problems I encountered until now have metrics that when represented in a matrix form turn out to be symmetric? Aren't there asymmetric matrices representing some metrics?
6
votes
5answers
144 views

Is $ds^2$ just a number or is it actually a quantity squared?

I originally thought $ds^2$ was the square of some number we call the spacetime interval. I thought this because Taylor and Wheeler treat it like the square of a quantity in their book Spacetime ...
6
votes
2answers
1k views

Infinitesimal Lorentz transformation is antisymmetric

The Minkowski metric transforms under Lorentz transformations as \begin{align*}\eta_{\rho\sigma} = \eta_{\mu\nu}\Lambda^\mu_{\ \ \ \rho} \Lambda^\nu_{\ \ \ \sigma} \end{align*} I want to show that ...
6
votes
2answers
338 views

Einstein tensor in Friedmann equations : where is the missing $c^2$?

I would like to demonstrate the several forms of the Friedmann equations WITH the $c^2$ factors. Everything is fine ... apart that I have a missing $c^2$ factor somewhere. In all the following $\rho$ ...
6
votes
3answers
257 views

Why does Minkowski space provide an accurate description of flat spacetime?

What is the chain of reasoning (beginning, of course, from observations about the universe) that leads one to predict that Minkowski space provides an accurate description of space-time in the ...
6
votes
2answers
111 views

Why does the $L_2$ norm give the shortest path between 2 points?

Why not the $L_1$ or $L_3$ distances? Is there some deep reason why the universe (at least at human scales) looks pretty much Euclidean? Could we imagine a different universe where a different ...
6
votes
2answers
226 views

What's the basic premise of General Relativity?

What is the basic assumption(s) required to explore general relativity? For example, if one merely assumes that the speed of light $c$ is the same for all observers, and the laws of physics are the ...
6
votes
1answer
519 views

energy momentum tensor and covariant derivative

In field theory, the energy momentum defined as the functional derivative wrt the metric $T_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta g^{\mu\nu}}$ (up to a sign depending on ...
6
votes
0answers
90 views

Can some components of metric be Finslerian while the others be Riemannian?

A Finsler metric reduces to a Riemann metric in case it loses its dependence on velocities. Now, my question is this: Can we have a Finsler metric in which some components of the metric have velocity ...
5
votes
4answers
675 views

How did “no prior geometry” father 50 years of confusion?

I've come across this quote attributed to Misner, Thorne & Wheeler from their book, Gravitation: Mathematics was not sufficiently refined in 1917 to cleave apart the demands for "no prior ...
5
votes
2answers
144 views

Metric tensor in SRT

I just read on this webpage that we have (click me) $g_{\alpha \beta} = g_{\alpha}^{\beta} = g^{\alpha \beta}.$ Now, although I understand that the first and the last one are equal, I don't think ...
5
votes
1answer
133 views

Are third derivatives of metric perturbations zero?

I'm working on a problem related to fluid perturbations of stars. I'm following this paper. They start with the Einstein equation: $$G_{\alpha \beta} = 8 \pi G T_{\alpha \beta}$$ and then perturb the ...
5
votes
1answer
378 views

When a variation of a tensor is not a tensor?

In a comment about variation of metric tensor it was shown that $$\delta g_{\mu\nu}=-g_{\mu\rho}g_{\nu\,\sigma}\delta g^{\rho\,\sigma}$$ which is contrary to the usual rule of lowering indeces of a ...
5
votes
1answer
73 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 $$ ...
5
votes
1answer
135 views

Extent of coordinate freedom to set metric components along a spacetime path

If we describe spacetime with a Lorentzian manifold, it is always possible to choose a coordinate system such that at any particular point $x^\alpha$, the components of the metric are: $$ ...
5
votes
0answers
168 views

Penrose diagram for spacetime which flows to $AdS_{2}$ at infinity

Consider I have the following 2 dimensional spacetime $(t,z)$: $$ds^2=\frac{4}{z^{2}}\left(1+\frac{1}{z}\right)^{-1}(-dt^{2}+dz^{2}).\tag{1}$$ When $z\rightarrow \infty$ we have ...
5
votes
0answers
126 views

Is it correct to sum over either index of the metric the same way?

I don't know if the following is correct, i want to compute the following derivative $$\frac{\partial }{\partial (\partial_{\mu}A_{\nu})}\left(\partial^{\alpha}A^{\beta}\partial_{\alpha}A_{\beta} ...
4
votes
3answers
424 views

Why is the scalar product of four-velocity with itself -1?

My GR book Hartle says the scalar product of four-velocity with itself $-1$? Consider the definition of four velocity $\mathbf{u} = \frac{dx^{\alpha}}{d\tau}$. Suppose I take the scalar product of ...
4
votes
2answers
223 views

Difference between slanted indices on a tensor

In my class, there is no distinction made between, $$ C_{ab}{}^{b} $$ and $$ C^{b}{}_{ab}. $$ All I know, and read about so far, is the distinction of covariant and contravariant, form/vector, etc. ...
4
votes
3answers
112 views

Metric signature conventions: minus sign for $x^a$ or $x_a$?

Say I use the metric signature $(-+++)$. Then $\partial_a=(\partial_0,\partial_i)=(-\partial^0,\partial^i)$, but $\partial^a=(\partial^0,\partial^i)=(-\partial_0,\partial_i)$. The same goes for $p^a$ ...
4
votes
2answers
162 views

Why is the phase space a symplectic manifold rather than a manifold with a metric?

Why does phase space require a symplectic geometry rather than a metric? Is there some scenario where a metric is unable to describe the notion of length in phase space, specifically in relation to ...
4
votes
1answer
193 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 ...
4
votes
2answers
504 views

Metric tensor in special and general relativity

I'm having trouble understanding the metric tensor in general relativity. What I've understood so far has come from my course lecture notes used in conjunction with "The Road to Reality" by Roger ...
4
votes
3answers
113 views

Is a metric tensor field the same thing as $ds² = -dt² + dx²+ dy² + dz²$?

I am having trouble understanding the nature of the metric tensor field on spacetime manifolds. In particular, a Riemannian manifold $(M,g)$ is defined as a real smooth manifold $M$ equipped with an ...
4
votes
4answers
198 views

What makes a coordinate curved?

Bear with me while I try to explain exactly what the question is. The question Can a curvature in time (and not space) cause acceleration? is imagining a coordinate system in which the curvature is ...
4
votes
2answers
141 views

Getting the Lagrangian from the action in curved spacetime

Suppose I have this action: $$ S = \int \mathrm d^4 x\sqrt{-g}\times \text{something}$$ where $g$ is the determinant of the metric. Should I take the Lagrangian to be: $$ \mathcal L = \sqrt{-g} ...
4
votes
1answer
156 views

Non-stationary spacetime

What is an example for a spacetime that is non-stationary that is considered as a description of something in nature? So far all the spacetimes I encounted have always been stationary ...
4
votes
1answer
118 views

Curved paths through spacetime when standing still?

I have heard that falling objects fall at the same rate irrespective of their mass. They are 'following straight line paths through curved spacetime'. Does this mean that objects that accelerate in ...
4
votes
1answer
244 views

Computing Curvature via Cartan Formalism

Given a metric $g_{\mu \nu}$, one can select an orthonormal basis $\omega^{\hat{a}}$ such that, $$ds^2= \omega^{\hat{t}}\otimes\omega^{\hat{t}} - \omega^{\hat{x}} \otimes \omega^{\hat{x}} - ...$$ By ...
4
votes
1answer
142 views

Some hints for special case of metric tensor in GR

Let's have metric $$ ds^2 = dt^2 - dx^2 - dy^2 - dz^2 - 2f(t - z, x, y)(dt - dz)^2. $$ I need to prove that it is an exact solution for Einstein equations in vacuum for $\partial_{x}^{2}f + ...
4
votes
1answer
189 views

Intervals as infinitesimals of same order (Landau & Lifshitz)

I don't understand the following statement in Landau & Lifshitz, Classical Theory of Fields, p.5: $ds$ and $ds'$ are infinitesimals of same order. [...] It follows that $ds^2$ and $ds'^2$ must ...
4
votes
1answer
51 views

Computation of $T^{\mu\nu}$ from Lagrangian density $\mathscr{L} $

I am trying to understand how upper and lower indices are connected when computing the energy-momentum tensor. In particular, I found the simple problem where the Lagrangian density is given as ...