# Tagged Questions

Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.

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### Is there a parametrization for the shape of space?

I was thinking about how the space is curved. And how do we know that the shape of space arround a singularity is something like that: So I was trying to make a similar parametrization of this kind ...
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### Numerical simulation of strong field gravitational vacuum solutions colliding

I am interested in the current state of knowledge of strong field General Relativity learned from numerical investigations of gravitational wave packets colliding with each other or black holes. If ...
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### Schwarzschild manifold

I am given the following metric $$ds^2 = \frac{dr^2}{1-2m/r} + r^2dS,$$ where $dS$ is the standard metric on the unit sphere $S^2$. I am told that this is isometric to $\mathbb{R}^3$ or (taking its ...
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### Black hole singularity from collapsing light vs dust

Consider two black holes, one formed from a spherical cloud of electromagnetic radiation, and one formed from a non-interacting dust solution. The stress energy tensor is traceless for ...
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### Meaning of $R=0$, $R_{ab}=0$. $R_{abcd}=0$

First let me state some definition The Einstein tensor is given by \begin{align} G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R \end{align} and note that \begin{align} G^{\mu}_{\phantom{\mu} \...
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### Identity $\epsilon_{abcd} R^{cd}_{\phantom{cd}mn} = \epsilon_{mncd} R^{cd}_{\phantom{cd}ab}$ in vacuum

starting from \begin{align} \epsilon_{\rho\lambda\xi \kappa} R^{\xi \kappa}_{\phantom{ab} \sigma\tau} + \epsilon_{\rho\sigma \xi \kappa} R^{\xi \kappa}_{\phantom{ab} \tau \lambda} + \epsilon_{\rho \...
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### In the orthonormal tetrad method, what is the relation between basis one forms and commutation coefficients?

If $\omega_i$ are dual basis one forms corresponding to an orthonormal tetrad basis $e_i$, and given that the commutation coefficients $C_{ij}^k$ are defined by [e_i,e_j]=C_{ij}^k ...
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### Non-Euclidean mechanics; is it useful?

Special relativity has the following single-particle Lagrangian: $$S = \int_{t_0}^{t_f}\sqrt {\langle \mathrm d\vec{s},\mathrm d\vec{s}\rangle}.$$ Clearly it is based on Euclidean norms; it is in ...
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### Is it known what the necessary and sufficient conditions are for the existence of a “3+1 split” (by means of a foliation) of a (Lorentzian) manifold?

When trying to do physics on a more general pseudo-Riemannian manifold we want to require that there is a foliation of this manifold into three-dimensional subspaces. By this I mean we would like to ...
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### A problem with ADM mass in the derivation of 1st law of black hole thermodynamics

The definition of ADM mass is $$M=\frac{1}{16\pi}\lim_{r\rightarrow\infty}\int \left(\frac{\partial h_{\mu\nu}}{\partial x^\mu}-\frac{\partial h_{\mu\mu}}{\partial x^\nu} \right)N^\nu dA$$ according ...
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### Components of dual vectors

(This is a close retelling of Wald, problem 2.4b. Not for homework; just curiosity and an increasingly alarming suspicion that I've never actually understood anything.) Let $Y_1 ... Y_n$ be a ...
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### Divergence of inverse of metric tensor

I know that the Levi-civita connection preserves the metric tensor. Is the divergence of the inverse of metric tensor zero, too?! I'm not so familiar with the divergence of the second ranked tensor. ...
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### Does nature really follow the heat equation?

I think the heat equation says that the first derivative of temperature with respect to time in a stationary solid varies as the negative of the second derivative of temperature with respect to ...
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### Chern-Simons theory

The Chern-Simons 3-form is given by $\omega_3={\rm Tr} \left[ A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right]$ where $A$ is a connection one-form in the adjoint representation of a non-Abelian gauge ...
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### Transformation matrices for basis and coordinate transformation in non-orthonormal coordinates

The transformation matrices for covariant and contravariant vectors are different but in orthonormal coordinate system numerical values in matrices turn out to be same although in mathematical proof ...
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### Computational advantages of various notations for electromagnetism [closed]

Most undergraduate electromagnetism classes and textbooks use vector notation to describe Maxwell's equations. However, there are other notations like differential geometry and geometric calculus ...
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### Intuitive meaning of Globally Hyperbolic

I am been studying differential geometry and spacetime and I keep coming across the term globally hyperbolic. I am having a hard time coming up with an intuitive understanding of this idea. What is an ...
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### Two ways of writing coordinate basis vectors confusion

In Schutz's A First Course in General Relativity (p122) he derives the polar coordinate basis vector$$\vec{e_{r}}=\frac{\partial x}{\partial r}\vec{e_{x}}+\frac{\partial y}{\partial r}\vec{e_{y}.}$$ ...
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### How should Christoffel symbols be written (in LaTeX)? [closed]

I'm writing a summary of a lecture on relativity, and we've recently introduced the Christoffel symbols. It seems that the upstairs indices are the "leftmost" and the downstairs indices are somewhat ...
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### A manifold question: Why smooth functions and what is a Jacobian?

My question is what does a Jacobian have to do with the change of coordinates (coordinate transformation). Why do we care about this notion to start with? Also, why should it be non-singular?
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### What does coordinate invariance mean?

I would like to really understand what the mathematical as well as Physical meaning of coordinate invariance is. I have pretended to know what this means, but upon thinking a little harder today, I am ...
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### What is the metric tensor for?

I am wondering how to use the metric tensor, in practice? I read the book and done the exercises in A student's guide to vectors and tensors by Dan Fleisch. The concept of a tensor and their ...
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### Is an event formally a 4-vector? [duplicate]

An event is a 4D point in spacetime. At every point in spacetime there is a tangent space. 4-vectors live in the tangent space. One can contract two events using a metric tensor. Is there a process ...
p. 21 in this paper (http://arxiv.org/abs/0704.0247) $V$ is Killing vector, where $V^2 = −4b\bar{b}$, which means it is timelike Killing vector. The authors say: From $V^2 = −4|b|^2$ and $V = ∂... 0answers 52 views ### Squashed spheres in general dimension The point of this question is to help me find references regarding squashed spheres in general dimension. I am interested in the general theory of squashing for arbitrary dimension. All of the ... 1answer 70 views ### Variation of a tensor Let a change of coordinates be given by$x^{\mu}\to x^{\mu '}=x^{\mu}+\varepsilon \xi^{\mu}(x)$with epsilon a small quantity. Given a tensor$T$we define$\delta T:=T'(x)-T(x)$. I guess this means$...
If we look at the killing equation for a vector field $X$ in $\mathbb{R}^{(p,q)}$ (or on an open subset thereof) in coordinates with constant diagonal pseudo-metric we get: X_{\mu,\nu}+X_{\nu,\mu}=...