- #1

Tertius

- 58

- 9

GR has very limited situations in which a total mass-energy can be defined. The Komar mass, for example, requires the presence of a timelike killing vector field and an asymptotically flat spacetime. Basically, if the metric change with time or it's spacelike curvature does not flatten out asymptotically, you cannot define the Komar mass integral. From the derivation, if you can find a timelike killing vector you can define a Noether conserved quantity in the timelike direction, because it is continuously symmetric. Then, the spatial curvature changes are accounted for in the proper volume element, and the integration bounds are defined at infinity where it is asymptotically flat.

Obviously the FRW metric does not have a timelike killing vector field, so this fails immediately.

We can, however, compute the trace of the energy momentum tensor. This is commonly interpreted as the invariant mass energy density. If we then take the trace of the energy momentum tensor and integrate it over a certain spacelike region, with the proper volume element correction (from the spacelike surface metric ##\gamma##), we have

$$\int T^{\mu\nu} g_{\mu\nu} \sqrt{\gamma} dx^3$$

This will be a function of time. I would expect the result to be the invariant mass-energy of that region of space as a function of time. Is this actually observer (defined as a timelike worldline) independent? If we take the spacelike portion to be flat, does this integration actually work? (it doesn't work at infinity because the answer would be infinite mass)

Obviously the FRW metric does not have a timelike killing vector field, so this fails immediately.

We can, however, compute the trace of the energy momentum tensor. This is commonly interpreted as the invariant mass energy density. If we then take the trace of the energy momentum tensor and integrate it over a certain spacelike region, with the proper volume element correction (from the spacelike surface metric ##\gamma##), we have

$$\int T^{\mu\nu} g_{\mu\nu} \sqrt{\gamma} dx^3$$

This will be a function of time. I would expect the result to be the invariant mass-energy of that region of space as a function of time. Is this actually observer (defined as a timelike worldline) independent? If we take the spacelike portion to be flat, does this integration actually work? (it doesn't work at infinity because the answer would be infinite mass)

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