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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17
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1answer
855 views

Causality and how it fits in with relativity

I was talking to my teacher the other day about Einstein's spacetime and there's one thing he couldn't explain about the nature of Cause. I may be being stupid or just unable to comprehend, thanks for ...
2
votes
1answer
89 views

Perfect fluid and Cauchy momentum equation

The stress-energy tensor of a perfect fluid is given by $$T^{\mu\nu}=\left(\rho+pc^{-2}\right)u^\mu u^\nu+pg^{\mu\nu}$$ The divergence of the stress-energy tensor is zero: $\nabla_\mu T^{\mu\nu}=0$. ...
1
vote
0answers
48 views

Why do we hyperbolas for distance? [on hold]

I'm confused about how distance is measured in spacetime. I've read a few texts that say that our normal distance equation doesn't apply because it violates causality and because it won't work for a ...
0
votes
1answer
47 views

Using metric tensor to contract

Can the metric tensor also contract the indices in the $$\epsilon^{\tau\lambda\mu\nu}~?$$ For example, if we have ...
0
votes
0answers
46 views

Contraction of Kronecker delta = 4 [duplicate]

This suggests, as a shortcut notation, the concept of lowering indices; from any vector we can construct a (0, 1) tensor defined by contraction with the metric: $$A_\nu ≡ g_{\mu\nu}A^\mu$$ so that ...
0
votes
1answer
50 views

Can't derive FRW Christoffel symbol [closed]

I'm trying to confirm that the $\Gamma^1_{01}$ Christoffel symbol of the FRW metric is $\dot{a}/a$. I have the FRW metric: $$ds^2=-dt^2+a(t)^2\left[\frac{dr^2}{1-kr^2}+r^2(d\theta^2+\sin^2\theta\ ...
3
votes
1answer
51 views

Does Birkhoff's theorem apply to rotating collapsing stars?

Birkhoff's theorem states that every spherically symmetric vacuum solution to $R_{\alpha\beta} = 0$ is static, which greatly assists in the solution to the Schwarzschild solution by eliminating time ...
2
votes
2answers
163 views

Relativity question about 4-velocity

Given a 4-velocity $u^0$, how do you find $u_0$? Do you use $u_{\alpha}u^{\alpha} = -1$?
1
vote
0answers
25 views

Tangent Vector Field from Metric

Question: Starting from an arbitrary spacetime metric, how does one obtain a tangent vector field? (We might need to assume certain geodesic congruences but my understanding is very limited.) Build ...
2
votes
2answers
61 views

Schwarzschild metric: motivations and applications in physics

I have a mathematical background and I have just derived the expression of the Schwarzschild metric. Now I was wondering what were the motivations and applications in physics of this metric. Any info ...
2
votes
2answers
105 views

“Derivation” of Minkowski metric?

Is there a deeper meaning behind the the Minkowski metric? Does it just come from the SR formulae? Or is there some deeper geometrical meaning, maybe in the context of GR?
0
votes
0answers
45 views

Proof of Schwarzschild metric construction (O'neill chap 13)

I am struggling with a few steps of the proof in O'neill book $\textit{Semi-Riemannian Geometry, with applications to Relativity}$ on the construction of Schwarzschild's metric (chap13, Lemma1). Is ...
1
vote
1answer
108 views

Differentiating the Lagrangian to find geodesic equations?

I'm stuck pretty much at the first hurdle trying to follow the derivation of the geodesic equations from the Lagrangian ...
1
vote
1answer
42 views

Singularities in the Reissner–Nordström metric

I am doing a presentation on black holes but I'm having trouble finding information on the Reissner–Nordström metric. From the metric ...
2
votes
4answers
104 views

Any tips on evaluating Riemann tensor?

I am calculating the Riemann tensor for the Schwarzschild solution. I've calculated all 9 non-vanishing Christoffel symbols already. Now I need to evaluate the Riemann tensor and I find no easy way to ...
5
votes
2answers
106 views

Lie derivative vs. covariant derivative in the context of Killing vectors

Let me start by saying that I understand the definitions of the Lie and covariant derivatives, and their fundamental differences (at least I think I do). However, when learning about Killing vectors I ...
0
votes
1answer
31 views

Determinant of the curved space scalar wave operator

I am reading a paper titled 'Analogue Gravity' (http://www.livingreviews.org/lrr-2011-3 or http://arxiv.org/abs/gr-qc/0505065) In the paper (page 15/159) they say this: $$\det(\sqrt{-g} g^{\mu \nu}) ...
0
votes
1answer
66 views

Varying wrt metric [closed]

I saw people write $\frac{\partial( F^{ab} F_{ab})}{\partial g^{ef}}$ as $\frac {\partial (g^{ca}g^{db}F_{cd}F_{ab})}{\partial g^{ef}}$ in a way that exposes the dependence on the metric. but ...
1
vote
1answer
30 views

Why does this allegedly Hermitian Kähler metric have non-zero diagonal terms?

In string theory, the Kähler potential of Kähler moduli (e.g. - the volume of a Calabi-Yau manifold) is given by (see, for instance, Becker, Becker, Schwarz: "String Theory and M Theory" p. 498) $$K ...
3
votes
2answers
94 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
24 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] $$ ...
2
votes
0answers
70 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
votes
2answers
51 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, ...
1
vote
1answer
164 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 ...
1
vote
1answer
67 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. ...
-2
votes
1answer
53 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
61 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
152 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
59 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
46 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 ...
4
votes
2answers
106 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
129 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
vote
0answers
60 views

Gauge invariant quantities [closed]

In the context of cosmological perturbation one write the most general perturbed metric as $$ ...
0
votes
3answers
80 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
80 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
91 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 ...
7
votes
3answers
802 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
510 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
175 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} & ...
-1
votes
1answer
46 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 ...
0
votes
1answer
16 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
92 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
129 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
3answers
144 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
49 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
87 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
66 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
154 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
94 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 ...
3
votes
1answer
100 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 ...