# 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).

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### Confused by the path I am asked to follow in order to solve the killing equation on S2.

I am asked to find the Killing vector fields on $S^2$ where the line element is given by $ds^2=d\theta\otimes d\theta +\sin^2\theta d\phi\otimes d\phi$. I know how to solve this problem by considering ...
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### Understanding the Geometry Spawned from Quotient Spaces $GL^+(4,R)/SO(3,1)$, $GL^+(4,R)/Spin(3,1)$, and $GL^+(4,R)/Spin^c(3,1)$

I'm working on a theoretical framework where I explore different quotient spaces formed with GL$^+$(4,R) and various groups. Specifically, I'm interested in the types of geometry that arise from the ...
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### Is there an easy proof of $d \star S = s \cdot \epsilon$ for two forms S

In https://arxiv.org/pdf/1906.08616.pdf eq. 3.51 the following identity is proposed to hold for two-form $S$ on a manifold with metric $g_{\mu \nu}$. $d \star S = s \cdot \epsilon$, where $d$ is the ...
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### Naive questions by a beginner in general relativity.

I want to ask some very naive questions in general relativity. I have the background of PDE and few Riemannian geometry. After Schwazchild gave a solution, people study its singularity and predict the ...
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### How to know if a set of equations are Lorentz Invariant Spinors

I'm currently busy with a course in QFT and am completely baffled by Spinors. In particular there are two parts, that while I mostly understand the theory, struggle to show mathematically (especially ...
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### A strange notation in Geroch's proof of the positive-energy theorem

The following passage came from a 1973 article authored by Robert Georch and titled "ENERGY EXTRACTION". The author tries to prove the positive-energy theorem. I would like to ask two ...
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Let $g_{\mu\nu}$ be the metric tensor of a spacetime manifold. Instead of expressing $g_{\mu\nu}$ directly, I wish to express it as a product of two other tensors, say $p_{\mu\alpha}$ and $q^{\... 10 votes 1 answer 2k views ### Need help with planning self study for learning differential/Riemannian geometry and General Relativity rigorously. I would like to learn Mathematics for understanding GR, Differential Geometry, Riemannian Geometry and related research papers rigorously. I would like to carve out a clear path to understand these ... 1 vote 0 answers 56 views ###$L^\infty$decay for the Klein–Gordon equation By Fourier transform, the flow of the Klein–Gordon wave flow is$e^{it\sqrt{1-\Delta}}$. That is, if we have initial data$\phi(0,x)=\phi_0(x)$, then the solution will be given by$\phi(t,x)=e^{it\...
The Vaidya geometry written in ingoing null coordinate $v$ is given by the metric, \begin{equation} ds^2 = -\left(1 - \frac{2 m(v)}{r} \right) dv^2 + 2dvdr + r^2 (d\theta^2+ \sin^2{\theta} d\phi^2) \...