Mass/energy in general relativity

[PeterDonis]

the surface integral formulation of Komar mass is unaffected, in principle, by the interior - keep the metric on the surface and out to infinity the same, changing the interior at will, and the Komar mass is unchanged.

Yes, I agree with this.

Your own analysis can be taken to support this - there is really no valid integration volume for the komar mass.

I agree that just looking at the volume integral, without trying to work out if/how it is related to the surface integral, makes it seem obvious that, formally, the volume integral should be zero for any vacuum spacetime--and since this clearly doesn't make sense for Schwarzschild spacetime, that would seem to indicate that the volume integral is invalid for this spacetime. I also agree that trying to finesse this for the maximally extended spacetime by taking limits in order to deal with the fact that the Killing vector field is null at the origin of Kruskal coordinates might not be valid.

What confuses me somewhat is the fact that the source you linked to, which is specifically about black holes, doesn't appear to mention any of this when it writes down the Komar volume and surface integrals, and then just blithely says that the Komar energy of Schwarzschild spacetime is $M$. That makes me wonder whether there is some subtle point that I'm missing, that makes it OK to take the volume integral in Schwarzschild spacetime and somehow has it work out to be $M$ instead of zero.

in the eternal BH spacetime (rather than a post collapse BH), one is required to perform the Komar surface integral for both exteriors in opposite sense, getting M and -M, for total of zero.

I had thought of this as well--in fact it would actually make a kind of sense if the two exterior regions were connected, since the "direction" of the timelike KVF is opposite in the two exterior regions.