cesiumfrog said:
Do you have an example where local conservation of stress-energy implies local non-conservation of energy and/or momentum?
The example in JuanCasado's post works here. Just take a uniform gas of photons. The number density in a local co-moving region stays the same (since it's uniform, the number going into the local co-moving region equals the number exiting it), but if space is expanding, then the photons are getting stretched along with space, and so the energy density in a local co-moving region drops with time.
Quick note on what co-moving means: the size of the region you're looking at expands along with the universe.
How does this work in the context of the conservation of the stress-energy tensor? Well, with a uniform radiation fluid, there are no off-diagonal elements. There's just energy density in the time-time component, and pressure along the space-space components. Since the pressure of a photon gas is 1/3rd its energy density, and since it's uniform in all directions, the diagonal elements of the space-space part are all rho/3, where rho is the energy density of the photons. You can express the conservation of stress-energy in the following form:
\dot{\rho} = -3H\left(\rho + p\right)
Since p = \rho/3, and expanding the derivatives with respect to time:
\frac{d\rho}{dt} = -4H\rho
We can change all of our derivatives with respect to time to derivatives with respect to a by setting:
\frac{d}{dt} = \frac{da}{dt}\frac{d}{da} = a H \frac{d}{da}
So that we have:
aH\frac{d\rho}{da} = -4H\rho
\frac{d\rho}{da} = -\frac{4}{a}\rho
\frac{d\rho}{\rho} = -4 \frac{da}{a}
\ln\left(\rho\right) = -4 \ln\left(a\right) + C
\rho(a) = \rho(0) a^{-4}
Since the volume is increasing as a^3, but the energy density is falling as a^{-4}, this represents an energy loss per unit volume, taken directly from the conservation of stress-energy and making use of the fact that p = \rho/3 for photons.
Note that you can follow this process in the exact same way for any form of matter where p = w\rho with w = constant. For w = 0, energy is conserved in a comoving volume. For w < 0, energy grows with expansion. For w > 0, energy drops with expansion.
