Questions tagged [general-relativity]

Question 1

I have been thinking of a black hole with a radius if 46.5 billion light years and negligible angular momentum(this is the radius of the observable Universe) According to the black hole calculator ...

Question 2

The answer to this question proves the following: \begin{equation} \partial_\sigma \sqrt{-\det{\mathrm{g}}} = \frac12 \sqrt{-\det{\mathrm{g}}} \;g^{\alpha\beta}\partial_\sigma g_{\alpha\beta} \qquad (...

Question 3

I am trying to find an intuitive interpretation to the Riemann tensor in the presence of torsion. If we look at the index definition: \begin{equation} x^by^cR^a{}_{ bcd}z^d=x^by^c\left[\nabla_b,\...

Question 4

I am studying O'Neill's Semi-Riemannian Geometry book and I'm struggling to understand a specific statement (that seems important). Lemma 33 of chapter 5 (pg 146) says: I understand the proof of the ...

Question 5

I want to prove the following property of vectors in Lorentzian Geometry. Let $p \in M$ and $f: T_{p} M \rightarrow \mathbb {R}$ where $f(X)=T(p)(X,.,.,.,X)$ be multilinear evaluation at p. Assume ...

Question 6

Following Nakahara GTP, page 288 following the example of calculating Riemann tensor of a sphere with standard metric $g=d\theta\otimes d\theta+\sin^2\theta d\phi\otimes d\phi$, with vielbeins $$e^1{}...

Question 7

I am reading Spacetime and Geometry: An introduction to General Relativity by Carroll. In chapter 2, he pointed out that if $W_{\nu}$ is the covariant component of a 1-form, and $\partial_{\mu}=\frac{\...

Question 8

Christoffel Symbols Recently I've been self studying or at least "trying" to study General Relativity. I'm taking Honors Algebra 2 at my school (New in secondary school). With somewhat ...

Question 9

These days I am reading Wald's General Relativity https://icourse.club/uploads/files/b10bbb171326589762e3f2784ba527888988f8f5.pdf. In equation 3.2.31 shown below, \begin{align*} \nabla^a G_{ab}=0 \end{...

Question 10

The Question: In the 2-space with line element $$ds^{2}=\frac{dr^{2}+r^{2}d\theta^{2}}{r^{2}-a^{2}}-\frac{r^{2}dr^{2}}{(r^{2}-a^{2})^{2}},$$ where $r > a$, show that the differential equation for ...

Question 11

I'm trying to understand the variation of the Ricci tensor in the derivation of the Einstein field equations, and I have encountered a confusion regarding the notation and interpretation of the term $\...

Question 12

I am currently reading this paper by Bartnik that defines the ADM mass on asymptotically flat manifolds. Throughout the paper, we use a specific kind of weighted Sobolev Spaces. The weight on $\mathbb{...

Question 13

This is a tricky counting problem that I've been struggling with. The goal is to determine the number of independent components of a 5-indices tensor $T^{abcde}$ that has the following symmetries: i) ...

Question 14

A toy model for a black hole is given by the 2D Schwarzschild metric, \begin{equation} ds^2 = -\left(1-\frac{2GM}{r}\right)dt^{2} + \left(1-\frac{2GM}{r}\right)^{-1}dr^{2}, \end{equation} where $ ...

Question 15

In Philippe G. Ciarlet's book 'An introduction to differential geometry', He gives the integrability conditions of the differential equations like this: $$ \partial_{i} F_{lj}=L^p_{ij} F_{lp},\,\,\,F_{...

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Questions tagged [general-relativity]

A Galaxy Falling into a Black Hole with a 46.5 Billion Light Year Radius [migrated]

Covariant derivatives of tensor densities

Riemann Tensor with Torsion Intuition

Can a lightlike curve leave the causal cone?

Prove that every future directed causal vector is a limit of future directed timelike vectors

Riemann Tensor on Unit sphere using Differential forms

Switch the order of second order partial derivatives

How to solve a Christoffel Symbol? [duplicate]

What do $\nabla^a$ and $\partial^a$ mean in general relativity and differential geometry?

Mapping of Geodesic Equation onto a Euclidean Plane

Is the notation of $\nabla_k(\delta\Gamma^k_{ij})$ in the variation of the Einstein-Hilbert action mathematically valid?

Proving interior elliptic regularity estimates on Weighted Sobolev Spaces (in the context of asymptotically flat manifolds)

Number of independent components of a 5-index tensor satisfying certain symmetries

Obstructions to analytic continuation arising from black hole topology

Integrability condition of metric compatible equation

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