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In summary: I was looking at the conditions involving G and R. But you are right, in terms of T there are only 3, but that still leads to over determination except for the idea that differential conditions are... weak?

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It might help if you gave some more context about why you are asking the question.

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PAllen said:All metrics with Weyl curvature and no Ricci curvature have vanishing SET

Yes; but I would expect these to be described as "vacuum" solutions, not "zero pressure" solutions.

PAllen said:there would be an infinite number of configurations of pressureless dust

These are only pressureless in one coordinate chart; in other coordinate charts they are not. Or, for a more physical description, they are only pressureless to comoving observers; they are not pressureless to non-comoving observers. This is the kind of thing I was referring to in my previous post.

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But there is an invariant definition of a pressureless dust solution. See, for example, the criterion of contractions of the Einstein tensor given here:PeterDonis said:Yes; but I would expect these to be described as "vacuum" solutions, not "zero pressure" solutions.

These are only pressureless in one coordinate chart; in other coordinate charts they are not. Or, for a more physical description, they are only pressureless to comoving observers; they are not pressureless to non-comoving observers. This is the kind of thing I was referring to in my previous post.

https://en.m.wikipedia.org/wiki/Dust_solution#Dust_model

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PAllen said:there is an invariant definition of a pressureless dust solution

I'm not disputing that the solution has an invariant definition. I'm just pointing out, for the OP's benefit, that "pressureless" only correctly describes that solution with respect to comoving observers. That's because the OP did not ask specifically about "pressureless dust" solutions defined as you say; he asked about "pressure zero" solutions, and he probably does not realize the limitations of that description.

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Even more trivially, it is true forPAllen said:

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This case is not so obvious by that type of criteria, which is why I made a physical argument. One can sort of argue the SET is over constrained for a pressureless dust solution. You start with arbitrary symmetric tensor fields with 10 functional degrees of freedom. First, by coordinate invariance, they form equivalence classes leaving only 6. Then, vanishing divergence is 4 more conditions. But then 4 conditions need to be satisfied for a pressureless dust solution. Of course, vanishing divergence are differential conditions, which are weaker.Orodruin said:Even more trivially, it is true foranydifferential equation for which a sufficient number of boundary conditions have not been specified.

I would be interested if you can add anything in this area.

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I know that, but I am a little bothered by the counting argument I just gave. Is the flaw just that differential conditions are very weak?martinbn said:

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Hm, not sure. Are these independent? What are the 4 conditions for dust?

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See the earlier link I gave to Wikipedia.martinbn said:Hm, not sure. Are these independent? What are the 4 conditions for dust?

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I guess what I am confused about is what 4 conditions need to be satisfied? The SET is ##T_{\mu\nu}=\rho u_\mu u_\nu##, why does this impose any restriction on the metric other than the EFE?PAllen said:See the earlier link I gave to Wikipedia.

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I’m reasoning directly from the SET as a symmetric tensor field. 4 conditions are given on T itself.martinbn said:I guess what I am confused about is what 4 conditions need to be satisfied? The SET is ##T_{\mu\nu}=\rho u_\mu u_\nu##, why does this impose any restriction on the metric other than the EFE?

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Oh, I see. But why do you say 4, the condition for no pressure should be 3 conditions, no?

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I was looking at the conditions involvingmartinbn said:Oh, I see. But why do you say 4, the condition for no pressure should be 3 conditions, no?

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When you say, that the differential conditions are weaker, do you mean that they can have many solutions. One differential equation for one unknown has infinitely many solutions, unless more information is specified, say boundary conditions.

The Einstein Field Equations are a set of equations in general relativity that describe the relationship between the curvature of spacetime and the distribution of matter and energy within it. When the pressure of the matter and energy is zero, the equations simplify to a form known as the Einstein Field Equations with zero pressure.

There are three possible solutions for the Einstein Field Equations with zero pressure, known as the Friedmann-Lemaitre-Robertson-Walker (FLRW) solutions. These solutions describe a homogeneous and isotropic universe, with different curvatures: closed, open, or flat.

The number of solutions for the Einstein Field Equations with zero pressure is important because it determines the possible shapes and evolutions of our universe. Each solution represents a different curvature and expansion rate, which can have significant implications for the fate of the universe.

Currently, observations suggest that our universe has a flat curvature and is expanding at an accelerating rate. This is consistent with the FLRW solution with zero pressure and a cosmological constant, also known as the Lambda-CDM model. However, further research and observations are necessary to confirm this model.

While the Einstein Field Equations with zero pressure have been successful in describing the large-scale structure of the universe, they do have limitations. For example, they do not take into account the effects of dark matter and dark energy, which are believed to make up a significant portion of the universe. Therefore, modifications to the equations may be necessary to fully understand the behavior of the universe.

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