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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3
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2answers
79 views

Can a hypothetical universe have more than 2 types of dimensions: spatial and temporal?

Our universe is often described as having 3 space-like dimensions and 1 time-like dimension. Can hypothetical universe exist with more than space- and time-like dimensions? If so how would these ...
1
vote
0answers
31 views

Killing vector field along geodesic [on hold]

I was trying to show that a Killing vector field satisfies the Jacobi Equation for a geodesic, just by assuming that \begin{equation} \nabla_\mu X_\nu + \nabla_\nu X_\mu=0 \end{equation} Indeed, if I ...
1
vote
0answers
18 views

FRW Metric maximally symmetric, derivation, $R=3K$ or $R=6K$ confusion, two different texts

I'm looking at Tod and Hughston Introduction to GR and writing the metric in the two forms; [1]$$ ds^{2}=dt^{2}-R^{2}(t)(\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+sin^{2}\theta d\phi^{2})) $$ [2] $$ ...
1
vote
0answers
53 views

Schwarzschild metric circular orbits and kepler's 3rd law

I have been looking at the Schwarzschild metric presented to me as the following within lectures: ...
0
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2answers
48 views

Is there any physical meaning for the inverse metric?

I've been wondering if we can attribute any physical meaning to the inverse metric. I mean when we talk about the metric itself, there are lots of insights we can have towards its role in spacetime, ...
0
votes
0answers
47 views

How do I obtain the metric tensor of a constant electric or magnetic field? [on hold]

Consider the empty space with a constant magnetic or electric field. From the Faraday or the Maxwell tensor that describes this situation, how do I obtain its metric tensor? I could try the Einstein ...
0
votes
0answers
31 views

Perturbations and Chronology condition [closed]

Let $(M,g)$ be the quotient of the 2-dimensional Minkowski space-time by the discrete group of isometries generated by the map $f(t, x) = (t + 1, x + 1)$. Show that $(M, g)$ satisfies the ...
1
vote
1answer
120 views

Prove Christoffel Symbol Identity

In a book I am reading, the following identity is claimed and then "left to the reader to prove." $g_{ij}$ is the metric tensor, and $\Gamma$ is the Christoff symbol of the second kind with the ...
0
votes
0answers
17 views

Atlas 2 For Mathematica. Need to calculate Riemann tensor, etc [migrated]

Is anyone familiar with Atlas 2 for Mathematica to calculate the Riemann Tensor, Ricci Tensor, and scalar I have a metric that I need to calculate these things for. Can anyone help? I'm not too up on ...
0
votes
0answers
28 views

How do you get the curvature tensor of the Schwarzschild Solution? [closed]

So, on the Wikipedia page on the derivation of the Schwarzschild solution, I get everything up to the part about the Ricci tensor. What were the components of the tensor that were used? Could ...
1
vote
1answer
59 views

Question about basic formalism of GR and the metric tensor

I really don't know much about GR, but I've come across a few rough sketches of its formalism in my DG books. I'm trying to piece it together to get a very basic intuition of what spacetime is in GR. ...
-3
votes
1answer
51 views

Does this identity that applies to the metric tensor also apply to the stress-energy tensor?

Okay so if the $g_{00}$ component of the metric is $-c^2$ and $g_{11}=g_{22}=g_{33}$ and all the other other components are zero, the question is simple, would similar identities apply to the ...
1
vote
2answers
54 views

Metric and the Lagrangian [duplicate]

Does the Lagrangian formalism require a metric on the configuration manifold $Q$ in order to define a Lagrangian $L$ on the tangent bundle $TQ$? Further, if we specify a metric on the tangent bundle ...
1
vote
2answers
144 views

$\nabla^{\mu}\nabla_{\mu}$ in general relativity [closed]

I am trying to work out $\square=\nabla^{\mu}\nabla_{\mu}$ in the metric $ ds^{2}=-A(r)dt^{2}+B(r)^{-1}dr^{2}+r^{2}d\Omega^{2} $$ My work: when applying $\square$ to a scalar $\phi$, then $ ...
2
votes
1answer
49 views

Rindler and Minkowski space future/past infinity

In my black holes course, we are looking at the Penrose diagram for 1+1 D Minkowski space. My notes don't specifically describe $i^{\pm}$ (future/past timelike infinity) but do say all timelike curves ...
0
votes
1answer
41 views

Metric in Lagrangian and the minimum total potential energy principle

I was wondering why physical systems "like" to go to the minimum of potential energy and I found this question, that tries to justify the minumum total potential energy principle. I was also reading ...
5
votes
2answers
92 views

Once I calculate the Riemann curvature tensor, what do I do with it?

I am considering the Schwarzschild metric. I have calculated my Christoffel symbols and am able to calculate the Riemann tensor (I think). In short, I have done a bunch of work to find this thing ...
1
vote
2answers
117 views

Sign convention for the Minkowski metric $\eta_{\mu\nu}$

In special relativity, one is confronted with a quadratic form called proper time, which is $c^2t^2-(x^2+y^2+z^2)$, $t$ being time and $x,y,z$ being the space coordinates. One usually introduces a ...
1
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0answers
58 views

Gauge invariant quantities [closed]

In the context of cosmological perturbation one write the most general perturbed metric as $$ ...
0
votes
3answers
59 views

Definition of non-degenerate metric tensor

We know that a metric has a property which is called non-degeneracy. I was searching for what does that mean and saw it associated with the fact that $det(g_{\mu\nu})\neq0$. How does this relate to ...
0
votes
1answer
75 views

How does the Lorentz boost change if we introduce transformation to the minkowski metric

Let's say we have the Lorentz boost given by the $ \Lambda^\mu_\nu$ in the Minkowski metric $diag\{1,-1,-1,-1\}$. Now if I do a transformation on the Minkowski metric such that the new metric is ...
0
votes
2answers
77 views

Off-diagonal terms in metric for 4D space-time [closed]

Consider a delta between two events in 4D space-time written as a 4-vector, $x^\mu=(dt, dR)$. The time $dt$ is a scalar difference in time. The 3-vector $dR$ points some direction in space. One ...
6
votes
3answers
746 views

Gödel's solutions to Einstein's relativity equations and their consequences

Gödel gave certain solutions to Einstein's relativity equations that involved a rotating universe or something unusual like that; that predicted stable wormholes could exist and therefore time travel, ...
11
votes
3answers
480 views

Space is expanding so what is time doing? [duplicate]

Space is expanding and as we know space and time are intrinsically linked to be now known as spacetime. What is happening to time during expansion? Is there more time, longer time or is the time part ...
2
votes
4answers
131 views

Normal Vectors to these Hypersurfaces on a Lorentzian Manifold

With respect to the coordinates $(x^{0},x^{1},x^{2},x^{3})=(v,r,\theta,\phi)$, we have the following components of the metric tensor: $\begin{bmatrix} g_{00} & g_{01} & g_{02} & ...
0
votes
1answer
42 views

Weyl scalar calculation

I'm trying to compute Weyl scalars, but don't really understand the formulae for them, in the sense I don't understand how to compute them. Let's take ...
1
vote
1answer
15 views

Commutativity and symmetric property in tensor manipulation

I have been trying to express $\eta^{\mu\nu}$ in terms of $\eta_{\mu\nu}$ and I have stumble upon the following relation: $\eta^{\mu\nu} = \eta^{\mu\alpha}\eta^{\nu\beta}\eta_{\alpha\beta}$ I can ...
3
votes
1answer
84 views

Curvature of Light around a Black Hole [duplicate]

I am in a computer graphics class at my university and for my final project, I have chosen to create a program which renders a simple non-rotating black hole and models the curvature of light around ...
2
votes
2answers
91 views

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 ...
1
vote
0answers
33 views

Wave equation given a metric [closed]

Can you explain me how I can obtain a wave equation given a metric? For example, if I have this metric $$g_{\mu\nu}=diag(-e^{2a},e^{2b},e^{2b},e^{2b})$$ where $a=a(t)$ and $b=b(t)$, how can I ...
1
vote
3answers
140 views

Determining whether a space is really three or two dimensional? [closed]

A space purports to be three dimensional with the metric $$dl^2=dx^2+dy^2+dz^2-\left(\frac{3}{13}dx+\frac{4}{13}dy+\frac{12}{13}dz\right)^2$$ How can I show that it actually represents a two ...
4
votes
0answers
46 views

How many Killing spinors exist on $S^5$?

So, I know that on $S^n$, a spinor of the form $$ \Sigma^\pm = \frac{1 \pm i\gamma^\alpha z_\alpha}{\sqrt{1+z^2}}\Sigma_0$$ where $\Sigma_0$ is a constant spinor, is a Killing spinor on $S^n$ ...
1
vote
1answer
84 views

How can I use Einstein's field equations? [duplicate]

Every time I try to find the answer to this question I get redirected to different pages that ultimately do not end up answering my question. I have some understanding of Riemannian geometry but have ...
1
vote
1answer
65 views

Does anyone recognize the line element $ds^2 = ( 1 - \frac{2m}{r} )dt^2 + 2 dt dr$?

I've stumbled upon the line element $ds^2 = ( 1 - \frac{2m}{r} )dt^2 + 2 dt dr$. Obviously the corresponding metric tensor has components: $\begin{bmatrix} g_{tt} & g_{tr} \\ g_{rt} & g_{rr} ...
8
votes
3answers
147 views

Why doesn't $ds^2 = 0$ imply two distinct points $p$ and $p'$ on a manifold are the same point?

Let's suppose I have a spacetime manifold $M$. Let $p$ be a point on my manifold. Now I move from $p$ to some other point $p'$. Presumably I should have moved some "distance" right? How can I speak of ...
2
votes
1answer
85 views

How can I use Einstein's field equations to find the metric tensor? [duplicate]

I have watched and read a lot on the topic of General Relativity and the geometry behind it. I am confident that I can derive an approximation of the the stress-energy-momentum tensor with just the ...
1
vote
0answers
21 views

Decomposition into symmetric and antisymmetric form [closed]

(a) Given a second-rank tensor Tμν, often viewed as an $N \times N$ matrix (for a space of dimension $N$), show by explicit construction that one can always decompose $T_{\mu\nu}$ into a symmetric ...
3
votes
1answer
98 views

Is any spacetime metric physically realizable?

Given a spacetime metric, you could work out a stress-energy tensor for each position that would result in that metric. I know building a wormhole requires negative energy densities, which are ...
-1
votes
3answers
185 views

Finding the appropriate coordinate transformation given two metrics

Given the two-dimensional metric $$ds^2=-r^2dt^2+dr^2$$ How can I find a coordinate transformation such that this metric reduces to the two-dimensional Minkowski metric? I know that ...
1
vote
2answers
79 views

What are $\mu$ and $\nu$ in $g_{\mu\nu}$ metric?

What are $\mu$ and $\nu$ in $g_{\mu\nu}$ metric? Consider the metric $g_{\mu\nu} = \begin{pmatrix} 1 & 0 &0 \\ 0 & r^2 & 0\\ 0 & 0 & r^2\sin^2\theta \end{pmatrix}$
4
votes
3answers
448 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 ...
0
votes
1answer
53 views

Will a stress-energy tensor have the same identities as it's metric?

Say I have a metric tensor where $$g_{00} = -c^{2}\ and $$ $$g_{01}=g_{02}=g_{03}=0$$ and $$g_{12}=g_{13}=g_{23}$$ and $$g_{11}=g_{22}=g_{33}$$ My question is straightforward: would the same or ...
4
votes
1answer
60 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 ...
1
vote
4answers
200 views

What is Minkowski spacetime?

I was browsing through an article on spacetime when I caught the words Minkowski Spacetime. A Wikipedia search brought me an article too complex for me to totally understand. So what is Minkowski ...
1
vote
0answers
52 views

Boyer–Lindquist coordinates

In the Kerr solution to the vacuum Einstein Equation written in Boyer–Lindquist coordinates. Because it is not spherical polar coordinates, $r$ ranges from 0 to infinity does not cover all the space, ...
0
votes
1answer
42 views

Energy-Momentum Tensor with mixed indices

I know that $T_{\mu\nu}$ is the Energy-Momentum Tensor and $T=g^{\mu\nu}T_{\mu\nu}$, but does anyone know what $T^{\nu}_{\mu}$ is and how its calculated?
2
votes
1answer
95 views

Inverse Metric Tensor

First the setup: Let $\mathcal M$ be a $2$-dimensional manifold. Let $U_P$ be some open neighbourhood of a point $P \in \mathcal M$. Let $\mathcal F : U_P \rightarrow \mathbb R \times \mathbb R$ be ...
3
votes
1answer
71 views

Fastest way to find the curvature terms from a given metric [closed]

I want to find the spherically symmetric, static solutions to Einstein's equations $$ R_{\mu \nu} - \frac{1}{2}Rg_{\mu \nu} = 0 $$ in four dimensions using the metric $$ g_{\mu \nu}dx^{\mu}dx^{\nu} ...
0
votes
1answer
38 views

Given any metric, how to find the straight line path between two points? [closed]

Say we are given a two-dimensional metric $$ds^2=f_1(x)dx^2+f_2(x)dy^2,$$ for any kind of function. How do we calculate the distance along a straight line path (not the shortest possibly) between, ...
13
votes
2answers
2k 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 ...