# Does the Komar mass density act like a four density?

The Komar mass is a means of measuring gravitational mass in spacetime. Via Wikipedia (https://en.wikipedia.org/wiki/Komar_mass) it is stated as (for a stationary metric):

$$m=\int\rho d\mathrm{vol}=\intop\sqrt{g_{00}}(2T_{\mu\nu}-Tg_{\mu\nu})e^{\mu}e^{\nu}d\mathrm{vol}$$

where $T_{\mu\nu}$ is the stress energy tensor, $g_{\mu\nu}$ is the stress energy tensor, $e^{\mu}$ is a unit time-like vector and $d\mathrm{vol}$ is the three-volume form. Considering only orientable metrics, we can write:

$$d\mathrm{vol}=\sqrt{g_{11}g_{22}g_{33}}e^{1}\wedge e^{2}\wedge e^{3}$$

Where, for simplicity we've considered a diagonalized metric. When we apply this to the above:

$$m=\intop\sqrt{g_{00}}(2T_{\mu\nu}-Tg_{\mu\nu})e^{\mu}e^{\nu}\sqrt{g_{11}g_{22}g_{33}}e^{1}\wedge e^{2}\wedge e^{3}$$

$$=\int(2T_{\mu\nu}-Tg_{\mu\nu})e^{\nu}\sqrt{g}d^{4}x$$

Which now appears as a full spacetime integral. Is this correct? I may have messed something simple up (such as a contraction somewhere), even then a variation of this appears valid. Tips would be greatly appreciated!

• Thank you. I'm pursuing a line of thought that: $$\rho^{2}=T^{\mu\nu}T_{\mu\nu}$$ Functions as an effective gravitational mass density, and it's naturally invariant. For that expression, there should be no static limitations. I was seeing how similar it is to the Komar mass. Jul 15, 2016 at 6:30
• Sorry, the above is factored such that $$\rho=e^{\nu}T_{\nu}^{\mu}e_{\mu}$$ which is not the same as the trace BTW: Due to it's symmetric nature, the metric is always diagonalizable Jul 15, 2016 at 6:37