Cosmological models: is there any solution for the Einstein equations?

FLRW models typically assume a perfectly uniform mass density ρ and pressure p throughout an expanding, spatially finite universe.

https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric

A realistic universe must contain variations in the mass density, though. Let us investigate if the Einstein equations have a solution at all for a realistic universe.

If the universe contains differences in the mass density ρ, and those differences are not made "static" by a pressure, then a rogue metric variation spoils any possible solution of the Einstein equations.

A static uniform sphere M in an FLRW universe

https://articles.adsabs.harvard.edu//full/1956MNRAS.116..678G/0000678.000.html

C. Gilbert (1956) presents a solution. Gilbert has to assume that the universe had a uniform mass density ρ > 0, and that the sphere M condensed from that mass, making a hole in the mass distribution. Then the gravity pull at the edge of the hole is equal from all directions. An infinitesimal rogue metric variation cannot move matter to a lower gravity potential? That is correct. This explains why Gilbert was only able to find the solution in the case where M is exactly the mass of the hole, condensed.

If we assume that a condensation process did take place, then the sphere M was born in a dynamic process, and we can use a rogue variation of the metric to show that there is no solution of the Einstein equations for that process.

Very rogue metric variations break also the FLRW models: we have to ban such variations

Recall from May 26, 2024 that a very rogue variation δg makes a particle to go backward in the time coordinate (of the old coordinates). In the Oppenheimer-Snyder collapse, we used a very rogue metric variation to show that their solution is erroneous. We wrote that particles can "legally" travel to a smaller time coordinate (smaller in the old coordinate), because clocks tick at different rates there, and the clock time is used as the time coordinate.

In an FLRW model, a very rogue variation genuinely takes a particle back in time. It is not an artefact of coordinates.

A very rogue variation can make a particle to have three copies of itself at a single time coordinate. The variation keeps the spacetime geometry constant: therefore, the action integral S of the Ricci scalar R is unchanged. But the integral of LM is changed since a single particle has several copies of itself at the same time of the old coordinates.

What is the problem here? The FLRW metric is a solution of the Einstein field equations. How can we have a very rogue metric variation δg which changes S?

The explanation is that the variation of LM becomes very complicated if we allow travel back in time. The stress-energy tensor is defined as the variation of the integral of LM, when we vary the metric g. The stress-energy tensor is no longer the simple

        T  =

              ρ     0     0     0

              0     0     0     0

              0     0     0     0

              0     0     0     0,

if we allow very rogue variations.

Very rogue variations must in this case be banned in general relativity.

Conclusions

The FLRW model with a uniform density ρ can be called a "dynamic" system, and it does have a solution for the Einstein equations. We can even embed into the FLRW universe static stars which reside is a spherical hole in the otherwise uniform mass distribution. The mass of the star must be equal to the mass missing from the hole. Otherwise, a rogue variation would spoil this solution of the Einstein equations.