Stephen R. Green, Robert M. Wald
(Submitted on 25 Jan 2016)
"We provide a simple discussion of our results on the backreaction effects of density inhomogeneities in cosmology, without mentioning one-parameter families or weak limits. Emphasis is placed on the manner in which "averaging" is done and the fact that one is solving Einstein's equation. The key assumptions and results that we rigorously derived within our original mathematical framework are thereby explained in a heuristic way."
Let us analyze from a mathematician's point of view what the authors have proved.
https://arxiv.org/abs/1505.07800
We will show that Thomas Buchert, George F.R. Ellis, Syksy Räsänen, et al. (2015) are at least partially right in their criticism of the papers and claims of Green and Wald.
https://arxiv.org/abs/1505.07800
We will show that Thomas Buchert, George F.R. Ellis, Syksy Räsänen, et al. (2015) are at least partially right in their criticism of the papers and claims of Green and Wald.
The assumptions
The authors start from an assumption that the real universe fulfills the Einstein equations
G_ab g_ab + Λ g_ab = 8π T_ab,
where
g_ab = g^(0)_ab + γ_ab,
and g^(0)_ab is a standard FLRW metric and γ_ab is "small".
T_ab is the stress-energy tensor of the real universe where we live.
Let us analyze the assumptions. The authors assume that:
1. There is an exact solution g_ab for general relativity with the stress-energy tensor T_ab.
2. The solution is "near" a standard FLRW metric g^(0)_ab.
Assumption 1 is something people have tried to prove for 104 years, but have failed.
Assumption 2 is not self-evident either. It might well be that general relativity has a solution, but it is not "near" a standard FLRW metric.
Assumption 1 is something people have tried to prove for 104 years, but have failed.
Assumption 2 is not self-evident either. It might well be that general relativity has a solution, but it is not "near" a standard FLRW metric.
The theorem
The authors define a stress-energy tensor T^(0)_ab where the mass content of the universe is spread evenly and T^0_ab is the stress-energy tensor for some FLRW metric.
They proceed to show that
G_ab g^(0)_ab + Λ g^(0)_ab - 8π T^(0)_ab
= T_diff.
is "small" in a limiting case λ -> 0, where λ is a parameter that specifies a whole family of metrics g_ab(λ).
If T_diff = 0, then g^(0)_ab is an exact solution for the averaged stress-energy tensor T^(0)_ab.
What if T_diff is not zero? Since T^(0)_ab is the stress-energy tensor for some FLRW metric, there exists an exact FLRW metric solution for it. Can we prove that this solution is "close to" g^(0)_ab?
Here we would need a way to compare metrics and how "close" they are to each other. In the case of standard FLRW metrics, we might define that the metrics are close if they give almost the same predictions of the future of the universe for an astronomer. Did the authors prove this?
They proceed to show that
G_ab g^(0)_ab + Λ g^(0)_ab - 8π T^(0)_ab
= T_diff.
is "small" in a limiting case λ -> 0, where λ is a parameter that specifies a whole family of metrics g_ab(λ).
If T_diff = 0, then g^(0)_ab is an exact solution for the averaged stress-energy tensor T^(0)_ab.
What if T_diff is not zero? Since T^(0)_ab is the stress-energy tensor for some FLRW metric, there exists an exact FLRW metric solution for it. Can we prove that this solution is "close to" g^(0)_ab?
Here we would need a way to compare metrics and how "close" they are to each other. In the case of standard FLRW metrics, we might define that the metrics are close if they give almost the same predictions of the future of the universe for an astronomer. Did the authors prove this?
Why the authors did not prove that the backreaction is negligible?
The reason is the assumptions 1 and 2 above.
1. They did not prove that a solution exists at all.
2. They did not prove that if a solution exists, it is "close to" a standard FLRW solution.
Since they did not prove that the assumptions are true, they did not prove that any of the conclusions are true.
Assumption 2 says that the solution is close to an FLRW metric, and the theorem says, among other things, that the solution is close to an FLRW metric. The theorem is, to some extent, circular reasoning
Our remarks above highlight the fact that all current cosmological models may be broken in the sense that they might not approximate any solution of general relativity. It is not the models' fault. The problem is that existence of physically realistic solutions of general relativity is an open problem.
Assumption 2 says that the solution is close to an FLRW metric, and the theorem says, among other things, that the solution is close to an FLRW metric. The theorem is, to some extent, circular reasoning
Our remarks above highlight the fact that all current cosmological models may be broken in the sense that they might not approximate any solution of general relativity. It is not the models' fault. The problem is that existence of physically realistic solutions of general relativity is an open problem.
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