# Proof of the Komar formula for the mass of a spacetime

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1. Sep 4, 2014

### Luca_Mantani

1. The problem statement, all variables and given/known data
Hi, i'm italian but i hope to write in decent english
Here's my problem. I want to proof the Komar formula for the mass
$$M=-\frac{1}{8\pi}\int_S\epsilon_{abcd}\nabla^c\xi^d \, ,$$
where $\epsilon_{abcd}$ is the Levi-Civita tensor and $\xi^a$ is the timelike Killing vector of the spacetime.
I'm stuck at the beginning of the proof. I started from the equation:
$$F=\int_SN^b(\xi^a/V)\nabla_a\xi_bdA \, ,$$
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. $N^a$ is the unit normal to S which is orthogonal to $\xi^a$. I can't understand why F is also equal to
$$F=\frac{1}{2}\int_SN^{ab}\nabla_a\xi_bdA=-\frac{1}{2}\int_S\epsilon_{abcd}\nabla^c\xi^d$$

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

3. 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.