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).

learn more… | top users | synonyms

0
votes
0answers
45 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 ...
2
votes
0answers
33 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: ...
1
vote
1answer
28 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} = ...
6
votes
1answer
30 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 ...
0
votes
1answer
40 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 ...
3
votes
1answer
36 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 ...
0
votes
2answers
35 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 ...
0
votes
1answer
73 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, ...
0
votes
0answers
21 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) $R^a_{\,b[cd;e]}=0$. ...
4
votes
2answers
59 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: ...
2
votes
1answer
35 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 ...
0
votes
1answer
24 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 ...
0
votes
0answers
18 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 ...
5
votes
1answer
44 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?
2
votes
1answer
27 views

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 ...
4
votes
1answer
48 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 ...
10
votes
1answer
110 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 ...
6
votes
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 ...
1
vote
0answers
45 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 ...
1
vote
1answer
33 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: $$ ...
3
votes
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 ...
0
votes
1answer
51 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 ...
2
votes
0answers
37 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 ...
2
votes
0answers
43 views

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 ...
0
votes
1answer
24 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 $$\eta_{\mu \nu} (\Delta x^\mu)(\Delta x^\nu)$$ ...
2
votes
0answers
21 views

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 ...
3
votes
1answer
62 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. ...
2
votes
2answers
107 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 ...
1
vote
1answer
35 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 ...
0
votes
1answer
33 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
votes
1answer
45 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 ...
1
vote
0answers
35 views

Is it possible for distinct geodesics to be equivalent over a finite segment?

Is it possible for two geodesics $\gamma_1, \gamma_2$ to be identical within a finite interval without being identical outside the interval? IOW: $\gamma_1(t) = \gamma_2(t)$ for $t \in (A,B)$ but ...
3
votes
2answers
63 views

Covariant derivative identity

I am trying to prove the following identity for contravariant vectors $X$ and $Y$ (this appears in exercise 6.7 of D'Inverno): $\nabla_{X}(fY) = (Xf)Y + f \nabla_X Y$. I have a way of proving it but ...
2
votes
2answers
108 views

Divergence in Riemannian Geometry (General Relativity)

I'm taking a course in General Relativity and I'm having some problems with the notation. I know that Einstein's tensor verifies $\nabla_aG^{ab}=0$. In physics textbooks this consequence of Bianchi ...
3
votes
1answer
26 views

Extrinsic Curvature of Surface of Codimension > 1

We can define the extrinsic curvature of a codimension-one surface as $$K_{ab} = q_a^{\phantom{a}c} q_b^{\phantom{b}d} \nabla_c n_d,$$ where $n^d$ is the normal vector to the hypersurface and ...
4
votes
1answer
72 views

Physical meaning of Hawking's Singularity theorem

I'm studying O'Neill's "Semi-Riemannian Geometry with applications to Relativity". I know that the following theorems are related to the Big Bang, but I don't understand how. Let $M$ be a ...
0
votes
0answers
46 views

How does one determine whether a coordinate basis is orthogonal or not?

Apologies for what is perhaps a very basic question, but I have been studying differential geometry with a view to gain a deeper understanding of general relativity and I have hit a stumbling block. ...
1
vote
0answers
20 views

Weyl Transformations and Group actions

I have the following question. Let $(M,g_{ab})$ be a Riemannian manifold $M$ with metric $g$, and with an action of a Lie group $G$. Moreover, the Riemannian metric $g_{ab}$ is taken to be invariant ...
0
votes
1answer
80 views

Book recommendation for rigorous multilinear algebra , tensor analysis, manifolds.

I am looking for recommendation on books about multilinear algebra, tensor analysis, manifolds theory, basically everything to be able to understand basic concepts of general relativity. I am ...
0
votes
1answer
22 views

Differentiate with respect to $x^{a}(t)$

having a little difficulty with this conceptually. Can someone quickly walk through this differentiation please? $K=\frac{1}{2}g_{ab}\dot{x}^{a}\dot{x}^{b}$ Find $\frac{dK}{dx^{a}}$ In this case, ...
1
vote
1answer
51 views

Invert tensor expression involving Levi-civita symbol

I have to prove that $$ \omega_{\mu \nu} = \epsilon_{\mu \nu \lambda \kappa} \omega^\lambda u^\kappa $$ given the relations: $$ u_k u^k = -1 $$ $$ \omega_{\mu \nu} u^\nu = 0 $$ $$ \omega^\mu = ...
2
votes
0answers
48 views

Why is the transformation law for tangent space coordinate basis vectors “easily seen” using the chain rule?

In Spacetime & Geometry, Carrol immediately posits that, given that you have some tangent space vector (in the coordinate basis representation): ...
3
votes
1answer
43 views

Where does the minus sign come from in this expression?

I'm trying to solve a problem from a course on General Relativity. Consider the change of coordinates $x^\mu \to x'^\mu = x^\mu + \varepsilon \xi^{\mu} (x)$ and for a tensor $T$ define $\delta T(x) = ...
0
votes
0answers
66 views

Definitions of riemann curvature tensor

Could someone please, in a concrete and non-circular way, explain what is mean by terms such as $ \nabla_u\nabla_v w\\ \nabla_{\nabla_u}v \text{ whoops, this one should probably be }\nabla_{\nabla_u ...
1
vote
0answers
46 views

Basis of differential one-form confusion

I'm trying to understand one-forms in the context of general relativity. Lee (Introduction to Smooth Manifolds) says that at a point $p$ and with a vector field $X$ we define a covector field ...
1
vote
0answers
33 views

Finding diffeomorphism given vector fields

Given a vector field how do you find the associated diffeomorphisms? Say I am given a vector field in Minkowski space, $\textrm{d}s^2 = -\textrm{d}t^2 + \textrm{d}x^2 + \textrm{d}y^2 + \textrm{d}z^2$, ...
0
votes
0answers
25 views

Integral curves and null geodesics

Let be $(M^{n+1},g)$ a spacetime (Lorentz manifold, connexe and time-oriented), $n\ge 2$, and $S\subset M$ a null hypersurface (codim $S=1$ and the restriction of $g$ to each tangent space $T_p ...
0
votes
0answers
59 views

Finding null geodesic that intersects a point and a time-like geodesic

I'm trying to find the right way to calculate the point of intersection of the null hypersurface emitted by a point $S$ event in a spacetime manifold with metric $g_{\mu \nu}$, with another time-like ...
1
vote
0answers
54 views

Weyl transformation of geodesic distance

Consider a Riemannian manifold $M$ with a metric $g$. For two points $x,y \in M$ the geodesic distance $d(x,y)$ is defined in the usual way. I would like to know if there is a formula expressing how ...
1
vote
0answers
26 views

Integral curves in null hypersurfaces

Let be $(M^{n+1},g)$ a spacetime (Lorentz manifold, connexe and time-oriented), $n\ge 2$, and $S\subset M$ a null hypersurface (codim $S=1$ and the restriction of $g$ to each tangent space $T_p ...