[Seems I don't get PF's alert system. I saw a quote the same day, but there wasn't any response when I looked. Now there is a response...]
PeterDonis said:
My first link was intended as an entropy reference. As can be seen from their figs 6 & 7, whether one include the cosmological event horizon the total entropy is effectively constant and so the density goes down with the expansion.
But I was thinking of comparing it to a black hole as in the OP, which I think has maximum entropy density.
PeterDonis said:
However, the claim that there is 0 energy density seems obviously false, not only because the universe obviously contains galaxies, but because current cosmological models have nonzero density of all main types of stress-energy (ordinary matter, dark matter, and dark energy).
The reference I was thinking of when I wrote "there are indicators (using the same GR Lagrangian that are used for quantizing it to get gravitons, another non-classical extension)" is Faraoni's and Cooperstocks's "ON THE TOTAL ENERGY OF OPEN FRIEDMANN-ROBERTSON-WALKER UNIVERSES", The Astrophysical Journal, 587:483–486, 2003 April 20 ;
http://iopscience.iop.org/0004-637X/587/2/483/pdf/56020.web.pdf ]
"The idea that the universe has zero total energy when one includes the contribution from the gravitational field is reconsidered. A Hamiltonian is proposed as an energy for the exact equations of FRW cosmology: it is then shown that this energy is constant. Thus, open and critically open FRW universes have the energy of their asymptotic state at infinite dilution, which is Minkowski space with zero energy. It is then shown that de Sitter space, the inflationary attractor, also has zero energy, and the argument is generalized to Bianchi models converging to this attractor."
I think that is effectively the gravitational Lagrangian (applied to a FRW universe) that is used to get gravitons at low energies and large scales, but of course I haven't checked. [
https://golem.ph.utexas.edu/~distler/blog/archives/000639.html ]
That energy is balanced and conserved in cosmology may not be such an astonishing idea. I've seen this attempt (which I lack the mathematical chops to verify - no tensors, only the most basic differential geometry), but I don't know if it has been published outside of arxiv:
"These are questions that surfaced relatively recently. As I mentioned in my history post, the original dispute over energy conservation in general relativity began between Klein, Hilbert and Einstein in about 1916. It was finally settled by about 1957 after the work of Landau, Lifshitz, Bondi, Wheeler and others who sided with Einstein. After that it was mostly discussed only among science historians and philosophers. However, the discovery of cosmic microwave background and then dark energy have brought the discussion back, with some physicists once again doubting that the law of energy conservation can be correct.
Energy in the real universe has contributions from all physical fields and radiation including gravity and dark energy. It is constantly changing from one form to another, it also flows from one place to another. It can travel in the form of radiation such as light or gravitational waves. Even the energy loss of binary pulsars in the form of gravitational waves has been observed indirectly and it agrees with experiment. None of these processes is trivial and energy is conserved in all cases. But what about energy on a truly [sic!] universal scale, how does that work?
On scales larger than the biggest galactic clusters, the universe has been observed to be very close to homogeneous and isotropic. ...
In a previous post I gave the equation for the Noether current in terms of the fields and an auxiliary vector field that specifies the time translation diffeomorphisms. The Noether current has a term called the Komar superpotential but for the standard cosmology this is zero. The remaining terms in the zero component of the current density come from the matter fields and the spacetime curvature and are given by
J^0 = \rho + \frac{\gamma}{a} + \frac{\Lambda c^2}{\kappa} - 3 \frac{\dot{a}^2}{\kappa a^2}
The first term is the mass-energy from cold matter, (including dark matter) at density \rho . The second term is the energy density from radiation. The third term is dark matter energy density and the last term is the energy in the gravitational field. Notice that the gravitational energy is negative. By the field equations we know that the value of the energy will be zero. This equation is in fact one of the Freidmann equations that is used in standard cosmology.
If you prefer to think of total energy in an expanding region of spacetime rather than energy density, you should multiply each term of the equation by a volume factor a^3.
It should now be clear how energy manages to be conserved in cosmology on large scales even with a cosmological constant. The dark energy in an expanding region increases with the volume of the region that contains it, but at the same time the expansion of space accelerates exponentially so that the negative contribution from the gravitational field also increases in magnitude rapidly. The total value of energy in an expanding region remains zero, and therefore constant. This is not a trivial result because it is equivalent to the Friedmann equation that captures the dynamics of the expanding universe.
So there you have it; the cosmological energy conservation equation that everybody has been asking about is just this
E = M c^2 + \frac{\Gamma}{a} + \frac{\Lambda c^2}{\kappa} a^3 – \frac{3}{\kappa}\dot{a}^2 a = 0
It is not very complicated or mysterious, and it’s not trivial because it describes gravtational dynamics on the scale of the observable universe."
[
http://blog.vixra.org/2010/08/17/energy-is-conserved-in-cosmology/ and its references. Among other things the arxiv paper claims that there is a slicing that covers black holes, so that aspect of GR is good too, assuming it is correctly done.]
