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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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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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: ...
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1answer
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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
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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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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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M is set with O as topology defined on it alongwith this smooth atlas and connection is given. [on hold]

Chart transition maps are given and connection coefficient functions with respect to polar chart are to computed.this is actually a tutorial problem of online light and gravity course at WE-Heraeus ...
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1answer
38 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
41 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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speed of light in general relativity [migrated]

in general relativity, the speed of visible light is defined as a constant. But visible light is only a small part of the electromagnetism field. So why?
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108 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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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 ...
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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
31 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
49 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
49 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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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
21 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
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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
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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
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 ...
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1answer
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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 ...
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1answer
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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 ...
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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 ...
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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 ...
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2answers
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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 ...
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1answer
23 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 ...
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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 ...
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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. ...
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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 ...
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1answer
75 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 ...
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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, ...
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1answer
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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 = ...
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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): ...
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1answer
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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) = ...
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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 ...
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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 ...
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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, $\textrm{d}s^2 = -\textrm{d}t^2 + \textrm{d}x^2 + \textrm{d}y^2 + \textrm{d}z^2$, ...
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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 ...
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56 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 ...
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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 ...
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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 ...
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What is an isometry in terms of the metric tensor, not distances?

I get that an isometry preserves distances, but I'm a bit confused about how to determine if something is an isometry using the metric only. Consider in 1D for simplicity: $$ ds^2 = \frac{dx^2}{x^2} ...
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Find how a vector changes from parallel transport around a spherical triangle?

So I'm trying to find the components of a unit vector, initially on the sphere's equator parallel to the $\phi=0$ line, after parallel transport to $\phi=\phi_0$, then to the pole and then back to the ...
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1answer
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Calculating the line element after a change of coordinates

1. The problem statement, all variables and given/known data Consider $\mathbb{R}^3$ in standard Cartesian co-ordinates, and the surface $S^2$ embedded within it defined by $(x^2+y^2+z^2)|_{S^2}=1$. ...
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Higher order derivatives than Riemann Tensor

Does anyone know of any meaningful tensors that are related to the derivative of the riemann tensor? i.e. in the following picture we can consider Given arbitrary Pseudoriemannian manifold and metric ...
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The crhonological future set of $a\in M$, $I^{+}(a)$, is open in $M$.

I'm studying some topics about Lorentz Geometry and I have a little problem in the proof of a Causality Condition theorem. Let be $(M^{n+1}, g)$ a spacetime (Lorentz manifold, connected and ...
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1answer
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Einstein-Cartan Theory vs Metric Affine Gravitation Theory

Can anyone point out the real difference between Einstein-Cartan Theory and Metric Affine Gravitation Theory? Both of them rely on a pseudoriemannian metric $g$ and generalised affine connection ...