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Mass of Schwarzchild geometry

  1. May 4, 2012 #1
    1. The problem statement, all variables and given/known data
    Find the Mass of the Schwarzchild geometry by calculating,

    [itex]\frac{1}{4\pi}[/itex][itex]\int_{S}n^{\alpha}\sigma_{\beta}\nabla_{\alpha}\xi^{\beta}dA[/itex]

    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[itex]^{\alpha}[/itex] and [itex]\sigma_{\beta}[/itex], which are both normalized. n[itex]^{\alpha}[/itex] is timelike because it is normal to the choice of constant t surface and [itex]\sigma_{\beta}[/itex] 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 [itex]n^{\alpha}\sigma_{\beta}\left(\frac{\partial\xi^{beta}}{\partial x^{\alpha}}+\Gamma^{\beta}_{\alpha\gamma}\xi^{gamma}\right)[/itex]

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

    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
     
  2. jcsd
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