Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [general-relativity]

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).

Mathematical topics within the theory or relativity include:

among others.

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Open problems in General Relativity

I would like to know if there are some open mathematical problems in General Relativity, that are important from the point of view of Physics. Is there something that still needs to be justified mathematically in order to have solid foundations?
Benjamin
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Intuition for curvature in Riemannian geometry

Studying the various notion of curvature, I have not been able to get the intuition and deeper understanding beyond their definitions. Let me first give the definitions I know. Throughout, I will consider a $m-$dimensional Riemannian manifold…
user166467
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Why is this a first integral? - particle near Schwarzschild black hole

Background I know that the Schwarzschild metric is: $$d s^{2}=c^{2}\left(1-\frac{2 \mu}{r}\right) d t^{2}-\left(1-\frac{2 \mu}{r}\right)^{-1} d r^{2}-r^{2} d \Omega^{2}$$ I know that if I divide by $d \lambda^2$, I obtain the…
zabop
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Interpreting the scalar curvature in a semi-Riemannian manifold

Background: Let $M$ be a smooth Riemannian manifold of dimension $n$ and scalar curvature $R$ (with respect to the Levi-Civita connection). Let $m \in M$ and let $B$ be the geodesic ball of radius $r$ centered at $m$. That is, $B = \{ Exp_m(v)\ |\…
marlow
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Stochastic interpretation of Einstein equations

Einstein's theory of gravitation, general relativity, is a purely geometric theory. In a recent question I wanted to know what the relation of Brownian motion to the Helmholtz equation is and got a very thorough answer from George Lowther. He…
Robert Filter
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Why Mathematicians and Physicists Approach Integration on Manifolds Differently?

I have been attempting to find an answer to this for a few weeks, and decided to finally ask. I'll ask the questions at the beginning and then give the necessary background below: (1) Are volume forms $dx^1\wedge\ldots\wedge dx^n$ $n$-forms and…
user218912
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What is the geometric interpretation of the Connection?

I am currently enrolled in a General Relativity course, and was taught about the connection but I can't really wrap my head around it qualitatively. All I can think of is that it must have something to do with the coordinate system that one uses to…
TheQuantumMan
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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, Instead, the curvature is said to be…
Markus
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Ellipticity of Ricci tensor, does it depend on coordinates?

Well, I am afraid this is a silly question because I know the answer must be 'yes, it does'. But I don't see why. I put the problem in context. The ricci tensor can be regarded as a differential operator acting of metrics (it is an expression that…
juan rojo
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Riemann tensor symmetries

The Riemann tensor has its component…
Ernesto Lopez Fune
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Prove that $\Gamma_{abc}=\frac{1}{2}\left(\partial_bg_{ac} + \partial_cg_{ab}-\partial_ag_{bc}\right)$

I am tasked with the following problem Use the equation $$\nabla_ag_{bc}=\partial_ag_{bc}-\Gamma_{cba}-\Gamma_{bca}=0\tag{1}$$ where $$\Gamma_{abc}=g_{ad}\Gamma^d_{bc}\tag{A}$$ and the (no torsion) condition $$\Gamma_{abc}=\Gamma_{acb}\tag{i}$$ to…
FutureCop
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Find sequential orthographic projections, linking three different manifolds of dimension $n=1,2,3$

Below is an image of the family of 2D curves for reference. The image is arbitrary. Still trying to formulate a concise question. I know that the geodesics for flat Euclidean Space are straight lines. But I want to take these curves and try to work…
geocalc33
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What does Differential Geometry lack in order to "become Relativity" - References

Given a regular curve $\gamma \colon I \to \mathbb{R}^n$, if we consider the variable $t \in I \subset \mathbb{R}$ as the time, then we have the usual interpretation of $\gamma'(t)$ as the (instantaneous) velocity vector at the position $\gamma(t)$…
Derso
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Matrix representations of tensors

I've been trying to teach myself general relativity, and I always get stuck at the same point: I don't really understand what the metric tensor is. Unless I'm incorrect, and please correct me if I'm wrong, I realize that it defines a geodesic…
nycguy92
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What is contracting a tensor actually doing?

I'm learning about tensors, and have a vague idea regarding what contracting a tensor means—but I'm still not sure of exactly what it's doing. Maybe someone here can put it in more intuitive terms. Say we have a $(k,l)$ tensor, $T^k_l$. This is some…
AmagicalFishy
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