Proof of the Komar formula for the mass of a spacetime

  • #1

Homework Statement


Hi, i'm italian but i hope to write in decent english :wink:
Here's my problem. I want to proof the Komar formula for the mass
[tex]M=-\frac{1}{8\pi}\int_S\epsilon_{abcd}\nabla^c\xi^d \, ,[/tex]
where ##\epsilon_{abcd}## is the Levi-Civita tensor and [itex]\xi^a[/itex] is the timelike Killing vector of the spacetime.
I'm stuck at the beginning of the proof. I started from the equation:
[tex]F=\int_SN^b(\xi^a/V)\nabla_a\xi_bdA \, ,[/tex]
where F is the total outward force that must be exerted by a distant observer to keep in place a unit surface mass density distributed over S. [itex]N^a[/itex] is the unit normal to S which is orthogonal to [itex]\xi^a[/itex]. I can't understand why F is also equal to
[tex]F=\frac{1}{2}\int_SN^{ab}\nabla_a\xi_bdA=-\frac{1}{2}\int_S\epsilon_{abcd}\nabla^c\xi^d[/tex]

Homework Equations


[tex]\nabla_a\xi_b=\nabla_{[a}\xi_{b]}[/tex]
$$N^{ab}=\frac{2}{V}\xi^{[a}N^{b]}$$
$$\epsilon_{abcd}=-6N_{[ab}\epsilon_{cd]} $$
where ##\epsilon_{cd}## is the volume element on S.

The Attempt at a Solution


Comparing the first identity, i thought that
$$\frac{1}{2}N^{ab}=N^{b}\xi^{a}/V$$
but i can't understand why. With regards to the second identity i have no idea how to start proving it.
 

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