Tagged Questions

2
votes
2answers
87 views

Generally covariant Klein-Gordon equation

Consider a 4-dimensional smooth manifold $M$ on which there is a Lorentzian metric $g_{ab}$ and a function $\phi$ satisfying the following two equations (in abstract index notation): \begin{equation} ...
1
vote
1answer
81 views

Bullseye-shaped interference pattern in seminar-room chair

During a break n a seminar today, I noticed that the chairs in front of me all had slightly transparent black mesh fabric. The backs of the chairs were in the shape of a hyperbolic paraboloid. The ...
2
votes
1answer
95 views

Physics related question on Divergence Theorem for “general” smooth manifolds

I'm a physics major so bear with me here on the math. This is related to a problem from the textbook General Relativity - Wald. In classical electromagnetism say we have a vector field $V$ defined on ...
0
votes
0answers
58 views

Poisson bracket identities/properties [duplicate]

Possible Duplicate: Invertible antisymmetric matrix and identities How does $${\partial\over \partial \xi_i}M_{jk}+{\partial\over \partial \xi_j}M_{ki}+{\partial\over \partial ...
4
votes
3answers
363 views

Physical interpretation of the Lie Bracket

I've come accross this physical interpretation for $ [X,Y] $ which I don't understand : Follow $X$ for some time $\epsilon$; Follow $Y$ for $\epsilon$; Follow -X for $\epsilon$; Follow -Y for ...
4
votes
1answer
172 views

Is a twisted de Rham cohomology always the same as the untwisted one?

I am a physicist studying now some supersymmetric sigma models. My question can, however, be reformulated in a purely mathematical language: A twisted de Rham complex involves $d_W = d + dW \wedge $ ...
5
votes
4answers
379 views

What does the symbol $\operatorname{Tr}$ in the Yang-Mills action mean?

I find that many authors write the Yang-Mills action as follows: $$\mathcal{J}= \int \operatorname{Tr}(F \wedge \star F).$$ I have yet to find a formal description of the symbol $\operatorname{Tr}$ ...
3
votes
1answer
269 views

Covariant Derivatives and the Cross Product

I've recently read a paper that used a covariant derivative product rule for cross products. It was something like $\nabla_v (A \times B) = (\nabla_v A) \times B + A \times (\nabla_v B)$. Here, $A, ...
1
vote
0answers
103 views

Help on Einstein Summation

I am not sure how to interpret the following expression with regard to the Einstein summation convention \begin{equation} g^{ab}(\partial_c \Gamma^c_{ab} - \partial_b \Gamma^c_{ac}) \end{equation} ...
2
votes
0answers
68 views

A lower positive bound on the number of closed orbits with given energy for a mechanical system

Let be given a mechanical system with configuration manifold $M,$ potential energy $V$ and kinetic energy $K$ corresponding to a riemannian metric on $M.$ Its dynamics is determined by the ...
3
votes
2answers
156 views

boundary defining functions

Suppose that $M$ is the interior of a compact manifold with boundary $\partial M$. I'm often faced with so called boundary defining functions - or just 'defining functions'. That are by definition ...
15
votes
1answer
535 views

Stochastic interpretation of Einstein Equations

Einsteins 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 ...