# 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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### Extended Born relativity, Nambu 3-form and ternary (n-ary) symmetry

Background: Classical Mechanics is based on the Poincare-Cartan two-form $$\omega_2=dx\wedge dp$$ where $p=\dot{x}$. Quantum mechanics is secretly a subtle modification of this. By the other hand, ...
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### Classical mechanics: Generating function of lagrangian submanifold

I have a short question regarding the geometrical interpretation of the Hamilton-Jacobi-equation. One has the geometric version of $H \circ dS = E$ as an lagrangian submanifold $L=im(dS)$, which is ...
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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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### Trajectories in Rindler space with zero net time dilation

I've discovered a family of curves in Rindler space that have zero net time dilation. However I struggle to see why this should be so, i.e. what the physical significance of these curves is. My ...
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### How does one determine if a spacetime is globally hyperbolic?

A spacetime $M$ is said to be globally hyperbolic if it is strongly causal and if the sets $J^+(p)\cap J^-(q)$, for all $p,q\in M$, are compact. (For more information, see the Wiki article on causal ...
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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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### How the Poisson bracket transform when we change coordinates?

I'm studying the book Geometric Mechanics by Darryl D. Holm and there's one exercise in the book I'm not quite getting what has to be done. The same discussion the author makes in the book is made on ...
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### Trajectories in AdS

On page 2 of this paper (http://arxiv.org/abs/1106.6073), Maldacena explains (and has a very nice picture) showing the trajectories that a timelike and null particle would take in AdS space. Of ...
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### Angle sum of triangle in Schwarzschild solution

Curvature of space is often intuitively explained as angles of a triangle not adding up to 180 degrees. My questions concerns that. Suppose you have a perfectly spherical star of uniform density - so ...
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### Geometric interpretation of quantum Yang-Mills field

In most books\articles review geometric interpretation of classical Yang-Mills field in terms of principal bundle, connections...etc. What are geometric interpretation of quantum Yang-Mills field? ...
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### How many Killing spinors exist on $S^5$?

So, I know that on $S^n$, a spinor of the form $$\Sigma^\pm = \frac{1 \pm i\gamma^\alpha z_\alpha}{\sqrt{1+z^2}}\Sigma_0$$ where $\Sigma_0$ is a constant spinor, is a Killing spinor on $S^n$ ...
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### Chocolate dynamics

Now I have found a possible model on how to describe chocolate when it is chewed. It has to do with geometrical transformations when a curve $\gamma$ intersects a manifold $M$. The chocolate is ...
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### Are there relativistic theories with spacetime modelled on $\mathbb C^4$ rather than real Minkowski space $\mathbb R^4$?

Does anybody know of references to theories where relativity & spacetime is modelled on a (complex/KÃ¤hler) manifold which is locally diffeomorphic to $\mathbb C^4$ rather than $\mathbb R^4$, hence ...
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### Level quantization of 7d $SO(N)$ Chern-Simons action

In 3d, one can write down the $SO(N)$ Chern-Simons action to be $$S(A)=\frac{k}{192\pi}\int_{M}\text{Tr}(A d A +\frac{2}{3}A^3),$$ where $A$ is an $SO(N)$ connection. The level quantization can be ...
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### Tricks for Computing Riemann Curvature Tensor with Levi-Civita connection

I am new to differential geometry, so far it seems to me that computing the Riemann tensor tends to be a rather tedious task, I wanted to know whether there are some tricks that I am missing. In ...
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### Lie derivative of Dirac Delta

In the setting of general relativity, I came across a source term of the wave equation of the following form: $$\frac{1}{\sqrt{q}}\,\delta^{(3)}(p-\gamma(t))$$ where $p\in M$ is a point in our 4d ...
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### Is it possible to build up holography in a closed manifold, i.e., in a manifold with a mathematical boundary?

I was wondering about the AdS/CFT correspondence basics. It is constructed on the idea of conformal compactification, in which a open manifold $M$ is homeomorphic related to a closed one $N$ through a ...
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