Reza Mansouri (1977): uniformly collapsing perfect fluid ball with pressure has no solution in general relativity

We wrote a blog entry about this paper on July 13, 2019.

Let us again analyze Reza Mansouri's 1977 proof that a uniformly collapsing perfect fluid ball has no solution in general relativity, if the pressure p is not zero, and p is only dependent on the density ρ of the fluid.

The assumption of a spatially uniform collapse is not realistic. There is no reason why the collapse should be uniform. Imagine, for example, a uniform ball with a uniform pressure. A surface layer will expand outward relative to the interior. If the metric is comoving with matter, the radial metric close to the surface expands, while the tangential metric stays roughly unchanged. It is not uniform.

Matching the internal solution of the ball to the external Schwarzschild metric

https://www.researchgate.net/publication/259392290_On_the_Non-existence_of_Fluid_Sphere_in_General_Relativity_Obeying_an_Equation_of_State

We have not yet checked the correctness of Mansouri's calculations. He uses a formalism developed by McVittie (1967) and Taub (1968).

Mansouri shows in Section 4 that if he tries to match the solution inside the ball to the Schwarzschild metric outside the ball, he cannot get the border conditions satisfied.

He concludes that there is no uniform collapse with the pressure p(ρ) where general relativity would be satisfied.

Question. Can we define ρ and p(ρ) in such a way that the collapse is at least almost uniform for some time, and condition (66) in the picture above fails?

Solving the collapse problem in a computer simulation

Denotations in Mansouri's paper are complicated. Let us attack the problem from a different point of view: using a numerical simulation of the collapse process. We assume that we have stored the initial state into a computer. Let us calculate a short timestep Δt forward, using the metric and the pressure. After that, calculate an approximate new metric and the pressure.

Is there any reason why the numerical simulation would fail? Maybe we are not able to calculate a new metric because there exists no metric which matches the new configuration? Or the calculation may blow up: some parameter runs away to infinity.

These questions are general problems of the existence of a solution. There is no proof of the existence.

The calculation looks innocuous. It would be somewhat surprising if the calculation does not converge.

Conclusions

Reza Mansouri proved that a uniform collapse is not possible in general relativity if there is a pressure p(ρ). It may be that the pressure simply makes a uniform collapse impossible. This does not prove that there is anything wrong with general relativity.

The calculations are complicated and we will not try to develop them further to study nonuniform collapses.

A simple computer simulation argument suggests that general relativity can handle a nonuniform collapse.