Oppenheimer-Snyder Model: Overview of Gravitational Collapse

[PeterDonis]

Most people who have spent any time at all studying GR are familiar with the Schwarzschild solution. (A series of Insights articles discusses the key properties of that solution.) Much of that familiarity probably derives from the fact that the Schwarzschild solution describes a black hole. However, for some reason, much less attention is paid to the simplest solution that actually describes the collapse of a massive object to a black hole. This solution was discovered by J. Robert Oppenheimer and Hartland Snyder in 1939 and is called the Oppenheimer-Snyder model. In this article, we will briefly sketch how this model is constructed, and examine its key properties.
Our starting point is an idealized massive object which is perfectly spherically symmetric, with constant density in its interior, of finite extent, and surrounded by vacuum. Of course, such an object is highly unrealistic. But it makes the math tractable, in the sense that we can actually find closed-form solutions for...

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[Drakkith]

Nice article, Peter!

 

[Ibix]

The Oppenheimer-Snyder model eliminates the white hole and its singularity, the wormhole, and the other exterior that are all present in maximally extended Schwarzschild spacetime. But I find it fascinating that not only is the black hole singularity present, but it extends beyond the FLRW region. On a Penrose diagram, the finite FLRW region collapses to a point in the upper left corner, but the singularity is a line that extends horizontally from there. So the singularity completely spans the causal future of the stellar matter as it crosses its own event horizon, and not just its actual future (in the sense of being more than "where/when the stellar matter completely collapses").

I'm not sure if that's a completely trivial observation (at least saying the singularity spanning the causal future of the event horizon is tautological, I think), but it's a thing I don't completely understand.

 

[PeterDonis]

not only is the black hole singularity present, but it extends beyond the FLRW region

Yes; the FLRW region actually ends up as just a single point at one "end" of the spacelike singularity at $r = 0$. (That's the upper left corner of the Penrose diagram you describe.) The other "end" is at future timelike infinity, and all of the singularity in between is in the vacuum region.

saying the singularity spanning the causal future of the event horizon is tautological, I think

Yes.

it's a thing I don't completely understand.

What don't you understand about it?

 

[Ibix]

What don't you understand about it?

I find it interesting that the collapsing matter doesn't seem to explain the singularity. The singularity is a spacelike line all but one point of which is not in the present/future of the matter distribution when it becomes singular. (I should probably state that in terms of a limit as we approach the singularity, but I don't think my laziness is problematic here.) So all the collapsing matter does is eliminate the obviously crazy aspects of maximally extended Schwarzschild spacetime, and consequently removes all the symmetry evident in a Kruskal diagram. The singularity and the event horizon remain properties primarily of the Schwarzschild vacuum spacetime. And it would still be there (I think) if we put in by hand some mechanism that stopped the matter collapsing at some $r<R_S$.

I suspect I'm not saying anything particularly deep here. In fact, I think I'm just observing in more detail that this isn't a satisfactory model of gravitational collapse. I'm just surprised, not that it goes singular, but at the way it goes singular. I probably need to look closer at the maths.

 

[PeterDonis]

I find it interesting that the collapsing matter doesn't seem to explain the singularity.

I'm not sure what you mean. The collapsing matter satisfies the premises of the classic singularity theorems, just as any closed matter-dominated FRW spacetime does.

The singularity and the event horizon remain properties primarily of the Schwarzschild vacuum spacetime.

In the sense that "most of" the singularity lies in the vacuum region and not the collapsing matter, yes, this is true. But that doesn't change the fact that both regions are valid solutions of the EFE and both end in a singularity. The two regions, including the way they end in singularities, still have to match at the boundary between them.

And it would still be there (I think) if we put in by hand some mechanism that stopped the matter collapsing at some $r<R_S$.

There is no such mechanism possible. That's one of the implications of the singularity theorems: any surface with $r < R_S$ is a trapped surface, and the presence of a trapped surface, given that the other requirements for the singularity theorems are satisfied, is a sufficient condition for the collapsing matter to end in a singularity. The singularity theorems make no assumptions whatever about the details of the collapse; you can add whatever pressure, shock waves, etc. you want, and that won't change the fact that, once you have a trapped surface, a singularity is inevitable.

What would change that outcome is for the equation of state of the "matter" to change in a way that violated the energy conditions. There are models in the literature that do this, for example the models that were discussed in a recent thread on the Bardeen black hole and models derived from it. In those models, the deep interior has a de Sitter-like geometry, which is one possibility for what quantum fields might do under conditions of strong enough spacetime curvature. But in these models, there is not only no singularity, but no event horizon and no black hole at all.

In short, there are no conditions that I am aware of under which the collapsing matter would fail to produce a singularity, but there would still be an event horizon and a black hole, with a singularity in the vacuum region.

I think I'm just observing in more detail that this isn't a satisfactory model of gravitational collapse.

Again, I'm not sure what you mean. Of course the presence of singularities in any GR model is taken by most physicists to mean that GR breaks down at sufficiently large spacetime curvatures. But that doesn't seem to be what you're referring to here.