Comments on: In defence of cosmic censorship

By: Self-similar solutions of the Einstein-Vlasov system « Hydrobates

Self-similar solutions of the Einstein-Vlasov system « Hydrobates — Wed, 15 Sep 2010 05:16:51 +0000

[...] Self-similar solutions of the Einstein-Vlasov system By hydrobates The Einstein-Vlasov system describes the evolution of a collisionless gas in general relativity. The unknown in the Vlasov equation is a function , the number density of particles with position and velocity at time . A regular solution is one for which the function is smooth (or at least ). These equations can be used to model gravitational collapse in general relativity, i.e. the process by which a concentration of matter contracts due to its own weight. I concentrate here on the case that the configuration is spherically symmetric since it is already difficult enough to analyse. It has been known for a long time that a solution of this system corresponding to a sufficiently small concentration of matter does not collapse. The matter spreads out at late times, with the matter density and the gravitational field tending to zero. More recently it has been proved that there is a class of data for which a black hole is formed. In particular singularities occur in these equations. It is of interest to know whether singularities can occur which are not contained in black holes. This is the question of cosmic censorship. [...]

By: hydrobates

hydrobates — Sun, 26 Apr 2009 15:28:38 +0000

Hi,

Thanks for your comment. Let me just first remark that the geometric flows you listed are nonlinear. This is what makes them difficult to study and interesting. The Penrose inequality arose from considerations related to cosmic censorship. It was noticed that, on a heuristic level, if the Penrose inequality failed then cosmic censorship would fail. So in other words, on the same level, if cosmic censorship is true then the Penrose inequality is true. The (Riemannian) Penrose inequality was then proved by Huisken/Ilmanen and Bray using quite different ideas. In any case, there was a flow of information from general relativity to geometric flows. So far there has not been such a strong flow in the opposite direction (apart from the proof of the Penrose inequality itself) but it is reasonable to hope that such a thing could happen. Strong cosmic censorship means controlling the evolution of geometry determined by the Einstein equations. Thus analogies to geometric flows like Ricci flow are to be expected. The biggest difference is that the Einstein equations are hyperbolic and the usual geometric flow equations parabolic. Another possible connection is to use geometric flows to define good coordinates or gauge conditions for the Einstein evolution. The best example I know of an interaction of this kind between hyperbolic and parabolic equations is the paper
0411345 of Tao on the ArXiv.

Concerning the recommendation you asked for, the PDEs which are most interesting from the point of view of general relativity are quasilinear hyperbolic equations. A key model problem is that of wave maps. (The analogue of that in the context of geometric flows is the harmonic map heat flow.) I hope this is of some use to you.

By: Geometrick

Geometrick — Sun, 26 Apr 2009 02:26:04 +0000

Hi Mr. Rendall,

I remember reading that SciAm article a few months ago. I really like your blog post about Cosmic Censorship.

(1) What are your views on using geometric flows to cast some light on CCC? I know you mentioned PDEs and of course geometric flows are just geometric PDEs. Of course the work by Bray using IMCF to prove Riemannian Penrose inequality is a very good result. But what about in the broader view of CCC itself?

(2) What fields in PDEs in particular would you recommend someone pursuing? I ask because I’m currently a math grad student, but I have some heavy tendencies towards General Relativity. I believe most of the geometric flows that are employed are linear: Ricci, Yamabe, Mean Curvature and Inverse Mean Curvature.

Thanks, love your blog!