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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...
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.Ibix said:not only is the black hole singularity present, but it extends beyond the FLRW region
Yes.Ibix said:saying the singularity spanning the causal future of the event horizon is tautological, I think
What don't you understand about it?Ibix said:it's a thing I don't completely understand.
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##.PeterDonis said:What don't you understand about it?
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.Ibix said:I find it interesting that the collapsing matter doesn't seem to explain the singularity.
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.Ibix said:The singularity and the event horizon remain properties primarily of the Schwarzschild vacuum spacetime.
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.Ibix said: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##.
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.Ibix said:I think I'm just observing in more detail that this isn't a satisfactory model of gravitational collapse.
The Oppenheimer-Snyder Model is a theoretical model in general relativity that describes the gravitational collapse of a spherically symmetric star. It was proposed by J. Robert Oppenheimer and H. Snyder in 1939.
The model explains that when a massive star runs out of nuclear fuel, it can no longer produce enough energy to counteract the inward pull of its own gravity. This leads to the collapse of the star under its own weight, resulting in a black hole.
The model assumes that the star is spherically symmetric, that it has no rotation, and that it is made up of a perfect fluid. It also assumes that the collapse happens slowly enough for the star to reach thermal equilibrium at each stage of the collapse.
The Oppenheimer-Snyder Model was one of the first models to predict the formation of black holes. It also provided a framework for understanding the physics of gravitational collapse and helped to advance our understanding of general relativity.
Yes, the Oppenheimer-Snyder Model has some limitations. It does not take into account the effects of rotation, magnetic fields, and other factors that may affect the collapse of a star. It also does not fully account for the quantum effects that may occur during a collapse.