Homework Help: Mass of Schwarzchild geometry

1. May 4, 2012

mongolianbeef

1. The problem statement, all variables and given/known data
Find the Mass of the Schwarzchild geometry by calculating,

$\frac{1}{4\pi}$$\int_{S}n^{\alpha}\sigma_{\beta}\nabla_{\alpha}\xi^{\beta}dA$

in a Schwarzchild spacetime and for S a large sphere of coordinate radius R. Here ζ is the Killing vector corresponding to time translation invariance, and because we are integrating in a 4D spacetime we need two normal vectors n$^{\alpha}$ and $\sigma_{\beta}$, which are both normalized. n$^{\alpha}$ is timelike because it is normal to the choice of constant t surface and $\sigma_{\beta}$ is spacelike being normal to the choice of a constant r surface. Also, don't forget that dA includes factors from the metric.

2. Relevant equations

3. The attempt at a solution
Expanding the covariant derivative, the equation inside the integral becomes $n^{\alpha}\sigma_{\beta}\left(\frac{\partial\xi^{beta}}{\partial x^{\alpha}}+\Gamma^{\beta}_{\alpha\gamma}\xi^{gamma}\right)$

Since the only component of the Killing vector is t, and the derivative of the Killing vector is 0, and n$^{\alpha}$ only has t components and $\sigma_{\beta}$ has only r components, the equation reduces to $n^{t}\sigma_{r}\Gamma^{t}_{rt}\xi^{t}$, which reduces to $\Gamma^{t}_{rt}=\frac{M}{r^{2}}\left(1-\frac{2M}{r}\right)^{-1}$

I'm not sure if this is the write integrand to evaluate, and also I'm not sure what dA is composed of in terms of the metric