Comment Re:And what about dark matter? (Score 1) 109
Well, I wasn't a particular fan of that comment either...
1) The Friedman equation, which is the "Hamiltonian constraint" and which can be interpreted as an energy constraint equation
2) The matter continuity equations, which arise directly from conservation of equation of matter
(Technically, and apologies for the ugly LaTeX notation here, matter is described in relativity by "stress-energy" or "energy-momentum" tensors, which bundle together the classical concepts of energy, momentum flux, energy flow, and momentum flux density. The gradient of the more familiar Momentum flux density produces momentum conservation, while the time derivative and gradient of the energy give, well, energy conservation otherwise known as matter continuity. In terms of the stress-energy tensors, if energy is conserved then the covariant gradient of these is conserved -- written T^{\mu \nu}_{; \nu} = 0 where the \mu and \nu describe the four spacetime coordinates, and the repeated \nu means we sum across them. In Euclidean spacetime this immediately reduces to normal energy and momentum conservation. That semi-colon means "covariant derivative", meaning it has contributions coming from interactions with the spacetime geometry; if instead you insist on writing the conservation equation as a partial derivative (as in familiar classical mechanics and fluid dynamics) then you get T^{\mu \nu}_{, \nu} = [gravitational terms]^\mu and if you were unaware of the form of a covariant derivative this would look like a violation of energy and momentum conservation rather than what it is -- an interaction of the stress-energy of matter with the geometry of spacetime.
(The energy density, with respect to some chosen time coordinate which can be described as the vector n^\mu normal to a purely spatial three-dimensional surface, is found by the contraction \rho = n^\mu n^\nu T_{\mu \nu} which basically just means the scalar product of the stress-energy tensor with the time coordinate, or phrased differently, the stress-energy tensor projected along the time coordinate. This energy is conserved. Similarly, if we take the full Einstein equations they can be written as G^\mu_\nu = 8\pi G T^\mu_\nu where the G^\mu_\nu is the "Einstein tensor" that basically characterises the local gravitational field, and the G is newton's constant which infuriatingly has the same god damned symbol. If we project these equations along the time coordinate then we get out an energy on the right-hand side, and something that therefore can be interpreted as embodying the "gravitational energy" on the left-hand side. The result of this is the Hamiltonian constraint, and in the specific case of a cosmological spacetime the result is the Friedmann equation. Both these equations therefore express energy conservation -- one for the gravitational field, and the other for the matter.)
Said a bit less technically, the second equation is simply what you expect -- for normal matter we except the density changes with scale factor as rho = rho_0 / a^3. That is, the density decays as the volume of the universe increases. For photons we would expect instead rho=rho_0 / a^4, where we've got the normal volume dilution (1/a^3) and, since a photon's energy is proportional to length and the expansion of the universe will stretch that length, we should have an extra decay in the energy of 1/a. So rho = rho_0 / a^4. To a good approximation the same can be said for neutrinos (although to be exact we should make these massive, albeit extremely light.) Put these together and you have rho = rho_matter / a^3 + rho_photons / a^4. This is nothing but what happens when space expands.
That's well and good but it doesn't answer your issue with energy conservation, because it seems to violate it. That's where the other equation comes in, because these equations are only *half* the story. In GR (and other related, geometric theories of gravity) gravity also has an associated energy. It's hard to describe where the Friedmann equation comes from in this picture without the full theory, but it's obvious that if the energy density of photons is being diluted there must be some impact on the geometry that underpins what we call "gravity". In cosmology there is basically one quantity that determines that geometry -- the same scale factor, a. So the energy equation should determine how a evolves in time. Given that we can characterise it with the Hubble rate, which is the change in a with time, divided by a. This doesn't have the units of energy which (since we've set the speed of sound equal to 1, meaning we measure seconds and centimetres with the same stick) has units of 1/time^2. So let's square the Hubble rate to get something in the right units to be an energy. That gives us a crude estimate of the gravitational "energy conservation" as H^2 = ?. First guess for the right hand side has to be to slap in something proportional to the matter energy. This then gives us H^2 = \kappa \rho. This isn't a particularly convincing way to justify the Friedmann equation (which is what that is), but it's better than the usual approach which is to assume an expanding sphere of pure dust and which misses the point of gravitational energy completely.
The point of all of that, both technical and ineptly non-technical, is that while if you just look at the matter sector alone and forget that you have to take the geometry into account, you conclude that energy is not being conserved. However, remembering that in the theory cosmology is based on we have to include interactions with gravity (which can also be described as the "energy" of the gravtiational field though I'd advise caution in taking such interpretations too strongly) everything suddenly falls into place -- the apparent loss of energy from matter is balanced by the gravitational field.
Jesus. Sorry for the length of that one, you didn't deserve that. Oh well, hopefully it interested you -- ultimately I'd agree with the statement "physics has problems too" because if it didn't there'd be one hell of a lot of unemployed physicists mooching about looking unhappy.