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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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 ...
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1answer
119 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 ...
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1answer
52 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 ...
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4answers
146 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 ...
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4answers
381 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 ...
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1answer
37 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}) ...
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1answer
72 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 ...
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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 ...
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2answers
98 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 ...
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0answers
28 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] $$ ...
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84 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: ...
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2answers
53 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, ...
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1answer
215 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 Christoffel symbol of the second kind with the ...
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2answers
103 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. ...
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1answer
55 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 ...
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2answers
70 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 ...
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2answers
168 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 $ ...
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1answer
71 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 ...
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1answer
58 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 ...
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2answers
124 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 ...
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2answers
152 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 ...
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63 views

Gauge invariant quantities [closed]

In the context of cosmological perturbation one write the most general perturbed metric as $$ ...
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3answers
123 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 ...
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1answer
83 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 ...
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2answers
107 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 ...
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2answers
886 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, ...
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3answers
550 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 ...
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4answers
219 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} & ...
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1answer
56 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 ...
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1answer
18 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
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1answer
99 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
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2answers
184 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 ...
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3answers
150 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 ...
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0answers
56 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 $S^n$ ...
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1answer
92 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 ...
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1answer
67 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} ...
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3answers
159 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
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1answer
107 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 ...
5
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3answers
187 views

Is every spacetime metric physically realizable?

Is every spacetime metric physically realizable? I know that given any spacetime metric, you could work out a stress-energy tensor for each position that would result in that metric. However, I also ...
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3answers
239 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 ...
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2answers
91 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
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3answers
492 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 ...
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1answer
63 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
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1answer
71 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 ...
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4answers
369 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 ...
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116 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, ...
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1answer
55 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
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1answer
162 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 ...
2
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1answer
92 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} ...
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1answer
41 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, ...