Questions related to the mathematical aspects of Einstein's theory of relativity. For the physics and its interpretations, please ask at the physics.SE. You may also consider the tags (differential-geometry) and (pde).

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Equivalence of definitions of harmonic (or wave) coordinates

In GR, one often uses harmonic (or wave) coordinates to simplify things. Now, one definition involves the coordinates themselves: $$ \Box_g x^{\alpha} = 0 $$ where $ \Box_g = g_{\mu \nu}\nabla^{\mu}...
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Extending vector fields defined on open sets

I'm interested in finding sufficient conditions for when a vector field $X$ defined on an open subset of a smooth manifold $M$ can be extended. It is clear that this can't always be done. For example $...
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27 views

Covariant Taylor series

I am reading the following lecture notes of Avramidi https://www.researchgate.net/publication/255565392_Analytic_and_geometric_methods_for_heat_kernel_applications_in_finance I want to understand ...
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1answer
36 views

What do you need to understand the Theory of Relativity?

If someone has studied calculus, what "instruments" or what fields does one still need to understand the formulas behind the 2 theories of relativity (special and general)? By understand I mean more ...
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26 views

Covariant derivative of parallel transport

I am learning Riemannian geometry and don't get why the following is true. We are on a Riemannian manifold with the Levi Cevita connection $\nabla$. Let $\mathcal{P}(x,x')$ be the parallel transport ...
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1answer
42 views

Prove is compact

I'm trying to solve this problem and I don't know how to start Let $M$ be a connected time-oriented Lorentz manifold of dimension $n$. Let $J^+(K)=\{q\in M: \text{there is a $p\in K$ with $p\leq q$}\}...
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1answer
46 views

Intuition behind definition of K-K asymptotic flatness

I am reading some notes on black holes, and am confused by this definition of Kaluza-Klein asymptotic flatness: If a spacetime $(M, \mathbf{g})$ contains a spacelike hypersurface $\mathscr{I}_{ext}...
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3answers
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Differential Geometry for General Relativity

I'm going to start self-studying General Relativity from Sean Caroll's Spacetime and Geometry: An Introduction to General Relativity. I'd like to have a textbook on Differential Geometry/Calculus on ...
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2answers
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What is the difference between intrinsic and extrinsic curvature?

In general relativity, energy bends spacetime. However, this doesn't mean that a fifth dimension for spacetime to "bend into" exists." That is, spacetime isn't embedded in a higher dimensional space, ...
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1answer
61 views

What is the simplest way to describe a mathematical space?

As a complete noob in mathematics, I was wondering, what is the simplest way to describe a (preferably 2-dimensional, becuase it will be simpler) non-euclidean space in mathematics. For example in ...
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26 views

Deriving E=mc^2 from hollow box with mass M and photon

I'm working on a problem to derive E=mc^2 using conservation of momentum and center of mass. We have a hollow block of length L and mass M. A photon passes through taking mass m and adding it to the ...
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1answer
43 views

Confused About Indices in Deriving Curvature

Asking about a step regarding indices in deriving the Curvature tensor from the geodesic equation. Starting from $$ \frac{d v^a}{du} = - \Gamma^a_{bc}v^b \frac{dx^c}{du}$$ we integrate $$v^a(u) = ...
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1answer
56 views

Schwarzschild metric, speed of ball as measured by observer who catches the ball, just before ball is caught?

The Schwarzschild metric, describing the exterior gravitational field of a planet of mass $M$ and radius $R$, is given by$$ds^2 = -(1 - 2M/r)\,dt^2 + (1 - 2M/r)^{-1}\,dr^2 + r^2(d\theta^2 + \sin^2\...
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0answers
34 views

Confusion with conclusion to positive mass theorem

I am trying to understand the positive mass theorem as it is presented in the survey paper by Corvino and Pollack http://arxiv.org/abs/1102.5050 I am fundamentally confused by the structure of their ...
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1answer
53 views

Gauge condition equivalent to condition that coordinate functions satisfy wave equation to first order

Let $\eta_{ab}$ be the metric of special relativity and let $x^\mu$ be global inertial coordinates of $\eta_{ab}$. Let $\gamma_{ab}$ be a small perturbation of $\eta_{ab}$. How do I see that the gauge ...
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1answer
29 views

Approximating a metric tensor

I am reading Carroll's book on General Relativity, but this is much more a math question than a physics question. We can approximate a metric tensor on a curved manifold by writing $g_{\mu\nu} = \eta_{...
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17 views

Surface element in GR

I have seen this in Padmanabhan's book about gravitation. How can I verify this: $d\Sigma_{mn}=\frac{1}{2!}\epsilon_{mnab}\frac{\partial(x^a,x^b)}{\partial(\theta,\varphi)}d\theta d\varphi=\epsilon_{...
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0answers
16 views

Linearity of transformation of coordinates.

Suppose a transformation of coordinates given by: $$\bar x^\nu =\bar x^\nu(x^1,x^2,x^2,x^4)\quad\text{for }\nu =1,2,3,4 $$ such that: $$\left\{\begin{array}{lll} (dx^1)^2+(d x^2)^2+(d z^3)^2=c^2(d t)^...
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12 views

Different between space of reference and system of co-ordinates

In the book "The meaning of the relativity" by A. Einstein, it is referring to two different concepts: space of reference and system of coordinates. What it is the difference? It says: "we cannot ...
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67 views

Intuitive, short explanation of differential forms and exterior calculus

Are there any introductory lecture notes on differential forms and exterior calculus, preferably aimed at physics students studying General Relativity and Black holes? I have some familiarity with GR ...
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0answers
40 views

Solving a 1D integral with system of equations for retarded electromagnetic fields

I need to solve the following integral to calculate the effect of retarded electromagnetic fields on a test charge: $\int\limits_0^\zeta\frac{(\psi-(1+x)\sin(\psi+\alpha))(\frac{\psi^2}{2\beta^2(1+x)}...
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1answer
29 views

Co-ordinate transformation of metric

In a past exam paper that I am using to prepare for my upcoming finals, I have encountered the following question (paraphrased): Given the metric: $$\mathrm{d}s^{2} = -c^{2}\:\mathrm{d}t^{2}+\left(...
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1answer
43 views

Orthogonal geodesics to hypersurfaces

Say we have a Riemannian manifold $(M, g)$ with vector field $Y$, obeying: $g(Y, Y) = 1$; and the $1$-form $\varphi(X) = g(X, Y)$ is $d$-closed, $d\varphi = 0$. I know that the integral curves of $...
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1answer
56 views

Why is the Einstein Static Universe an infinite cylinder?

The Einstein static universe metric is $$ds^2=-dt^2 + d\chi^2 + \sin(\chi)^2d\Omega^2$$ where $-\infty<t<\infty$ , $0<\chi<\pi$ and $d\Omega^2$ is the metric on a $S^2$. It describes the ...
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52 views

Integral curves of vector field are geodesics

Say we have a Riemannian manifold $(M, g)$ with vector field $X$ obeying the following: $g(X, X) = 1$; and the $1$-form $\varphi(Y) = g(Y, X)$ is $d$-closed, $d\varphi = 0$. Does it necessarily ...
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2answers
39 views

Index attached to derivative operator

So in "General Relativity", Wald introduces a derivative operator $\nabla$ on a smooth manifold $M$ that sends $(k, l)$ tensors to $(k, l + 1)$ tensors. One of their properties he says is that if $f \...
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1answer
79 views

Confusion with abstract tensor notation

I am currently going through the abstract tensor notation in Wald's "General Relativity". I understand the purpose of it, but I need help understanding some of the conventions and definitions. So, ...
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1answer
46 views

Intuition about antisymmetrizing tensor equations

I was looking at the symmetries of the Riemann tensor, and tried to prove a couple of properties, namely If $\nabla$ is torsion-free, then: (i) $R^a_{\,[bcd]}=0$, and (ii) $R^a_{\,b[cd;e]}=0$. ...
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2answers
79 views

Proving the Ricci identity

I'm trying to prove the Ricci identity Let $Z^a$ be a vector field, $R^a_{\,bcd}$ the Riemann curvature tensor and $\nabla$ a torsion-free connection. Then: $\nabla_c\nabla_dZ^a-\nabla_d\nabla_cZ^...
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1answer
41 views

How to make parallel transport around a circle invariant for all vectors for a given metric?

I came across this question: "A 2d Riemannian manifold has metric $$ ds^2=dr^2+f(r)^2d\phi^2 $$ where $\phi$ is identified periodically with period $2\pi$. Determine the necessary and sufficient ...
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1answer
31 views

On a proof that the metric volume form is parallel wrt to the Levi-Civita connection

In the context of (semi-)Riemannian geometry, the following fact is well-known: if a (semi-)Riemannian manifold $(M,g)$ is oriented, then the unique volume form $\epsilon = \mathrm{vol}_g$, induced by ...
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25 views

trace of einstein equation - general relativity

I know quite well what the trace of a matrix is; however, I am not quite sure I understand the meaning of the 'trace' concept when applied to tensors. I would be very grateful to you if: How can i ...
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1answer
49 views

Scalar curvature of metric? [closed]

The metric components in a two-dimensional spacetime are given in terms of the coordinates $(t, x)$ by$$ds^2 = f(x)\,dt^2 + dx^2.$$What is the scalar curvature, $R$, of this metric?
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1answer
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Identity surrounding Killing vector field on a spacetime $\nabla_a \nabla_b w_c = -{R_{bca}}^d w_d$

Let $w^a$ be a Killing vector field on a spacetime $(M, g_{ab})$, i.e., $w^a$ satisfies $\nabla_{(a}w_{b)} = 0$. I hypothesize that$$\nabla_a \nabla_b w_c = -{R_{bca}}^d w_d,$$but I am not sure how I ...
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1answer
56 views

“Flow lines” of “dust” are geodesics?

The stress-energy tensor representing "dust" takes the form$$T_{ab} = \rho u_au_b$$where $u^a$ is a unit timelike vector field, i.e., $u^au_a = -1$. Does it necessarily follow that in any solution to ...
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1answer
116 views

Identity in general relativity, not sure if true or not

Let $(M, g_{ab})$ be a spacetime and define a new metric, $\tilde{g}_{ab}$, on $M$ by $\tilde{g}_{ab} = \Omega^2 g_{ab}$, where $\Omega$ is a smooth, positive function. Let $\nabla_a$ denote the ...
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1answer
26 views

Does it necessarily follow that the integral curves of $k^a$ are null geodesics?

Let $f$ be a function on a spacetime $(M, g_{ab})$ whose gradient, $k_a = \nabla_a f$, ie everywhere null, i.e., $k_ak^a = 0$ throughout $M$. Does it necessarily follow that the integral curves of $k^...
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0answers
47 views

Motivation for Non-Euclidean geometry: relativity

I'm looking for references to motivate the study of non-Euclidean geometry. In particular I would like something about relativity. I do not want texts to learn non-Euclidean geometry, only ...
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1answer
35 views

Arclength formula under the metric tensor on polar coordinates

I was reading: https://en.wikipedia.org/wiki/Metric_tensor#Examples Is it correct that in the polar coordinate example, just after the euclidean metric example, that distance is measured as: $$ \...
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1answer
50 views

Kerr spacetime not symmetric?

I always see a term $dt \, d \phi$ in the Kerr-spacetime. Now assuming this means $dt \otimes d \phi$ this means that the Kerr spacetime is NOT(!) symmetric which is somehow non-sense. So do ...
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1answer
56 views

What is a Killing tensor?

Wikipedia gives the definition of a Killing tensor. Unfortunately, I don't know how to interpret the parentheses (it is also not explicitly explained in the link) and was therefore wondering whether ...
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0answers
39 views

Does the metric in the Theory of Relativity actually satisfy the definition of a metric?

Allow me to give a brief introduction to the topic, which has to do with physics; my question will still be a mathematical one. I think my question is aimed at people with a background both in physics ...
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Calculation of extrinsic curvature

I asked this question first on physics.SE but I got no complete answer so I thought maybe someone here could help. I'm trying to understand how to derive the extrinsic curvature (in order to ...
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1answer
30 views

Einstein Summation Convention Minkowski Metric

Picked up a book on General Relativity for Mathematicians, but I'm a bit unclear on some of the tensor notation. For example, the Minkowski Metric $$\eta_{\mu \nu} (\Delta x^\mu)(\Delta x^\nu)$$ ...
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Lifting the Einstein-Hilbert action into the frame bundle

If we have a four dimensional real spacetime $(M,g)$, with $g$ being a $(-+++)$ signature Lorentz-metric, and $\{\theta^0,\theta^1,\theta^2,\theta^3\}$ is a local orthornormal coframe defined in some ...
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1answer
67 views

Calculating Christoffel symbols using variational geodesic equation

Given the line element $$ds^2 = e^v dt^2 - e^{\lambda} dr^2 - r^2 d \theta^2 - r^2 \sin^2 \theta d \phi^2$$ we wish to compute the Christoffel symbols $\Gamma^{a}_{bc}$ using the geodesic equation. ...
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2answers
112 views

Where should the Lorentz transformations fit into this?

I am trying to figure out how to "see" things in relativity via a toy model. With a pinhole camera I'd like to capture a relativistic scene consisting of a vertical marked stick which is moving ...
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1answer
37 views

Components of Maxwell tensor under Lorentz boost transformation

The following is taken from exercise 12.4 in D'Inverno. We wish to compute the transformation properties of the electric field and magnetic induction under a Lorentz boost. Given the following boost ...
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1answer
34 views

Metric defining an sphere

I want to find for which cases this metric can define an sphere: $$\frac{1}{P^2}\left(\mathrm d\theta^2+\sin^2 \theta\; \mathrm d\phi^2\right)$$ where $P=\sin^2 \theta+K\cos^2 \theta$, with $K$ the ...
2
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1answer
49 views

Trying to understand parallel transport

I've been trying to go through an example for parallel transport but I cannot quite follow the solution. A surface (paraboloid) is given by the parametric equation $r(ρ, φ)$ = $ρ \cos(φ)\hat{i}$ + $ρ ...