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!
Answer
Komar mass is only well defined, i.e. Invariant, in a stationary spacetime, i.e., one admitting a timelike Killing vector. Your derivation seems to be good only for a static spacetime, i.e., no rotations, so a Kerr metric for instance would not be admitted. Not only that, you seemed to assume a diagonalized metric also in the space coordinates, not sure if that constrains it further (it may not, not sure).
The Wikipedia reference does show an invariant version of the mass formula for only stationary (and no other assumption) spacetime.
I'd not not know if your derivation, once you got to an invariant formation by assuming lots of zeros, is still right for a general stationary spacetime. you could assume more zeros, like T = 0, and that would certainly not prove the equation was right then In general
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