# Collapse into a black hole in general relativity

I am looking for a paper that would study a mathematical model for the collapse of a star into a black hole (no QM please, just plain old GR). I know about the "dust bowl" model of Oppenheimer and Snyder, but I would like something slightly more realistic that takes pressure into account. For instance, there are the TOV (Tolman - Oppenheimer - Volkoff) equations, where density can increase with pressure. To ask a more specific question, is it known that for a star that's massive enough, the TOV model will lead to the formation of a black hole? Is it known that it does not lead to the formation of a black hole ? Or is that still an open question? Assuming spherical symmetry is fine if that helps.

Edit: Answers have been slow to come in, so I'll add a couple of words on what I would view as an acceptable answer. For models taking pressure into account, there's probably no exact solution (we do have the interior Schwarzschild solution, but it is static: no collapse here). However, it is often possible to study with rigorous math the evolution of a system even in the absence of explicit solutions. This is essentially what mathematicians working on differential equations, PDE, and dynamical systems in general do for a living, right?

In the absence of a rigorous mathematical analysis, a pointer to a paper presenting numerical simulations would be the next best thing.

• To my knowledge any star that exceeds some critical $M/R$ ratio becomes a black hole. Its formation starts with diverging pressure at the center by finite energy density. In case a known solution (metric) the critical ratio is defined by $g_{00}(0,M/R)=0$.
– JanG
Commented Oct 22, 2022 at 14:43
• Jan Gogolin: Yes, I think that is the conventional wisdom in the field. But can you point out to a paper or a book where this is worked out for a specific model, apart from the dust bowl model ? Commented Oct 23, 2022 at 12:31
• Pascal: Unfortunately not. I am looking for it, too. Pressure, understood as a mean hydrostatic stress, plays crucial role by forming of initial (central) event horizon. That can be seen on example of Schwarzschild's interior (static perfect fluid) solution.
– JanG
Commented Oct 23, 2022 at 14:58