Today Helmut Friedrich told me about an article by Pankaj Joshi in the February 2009 issue of Scientific American on the subject of cosmic censorship. I consider this article one-sided and consequently misleading and I find it sad that it appeared in such a prominent place. Here I present an alternative view.

General relativity predicts the occurrence of spacetime singularities where the gravitational force and/or matter density become unboundedly large, implying that known physical laws break down. If a singularity of this type can influence distant regions – this is referred to as a naked singularity – then the predictability, and hence the viability, of general relativity is undermined. Thus there is a strong motivation for ruling out this type of scenario. The cosmic censorship hypothesis of Roger Penrose suggests that the dynamics of the theory contrive to hide the effects of singularities from faraway observers – this is the cosmic censor. In fact there are two variants of this, called weak and strong cosmic censorship. In the first case the singularities are hidden because they are covered by the event horizon of a black hole while in the second they are made invisible to any observer outside the singularity.

The basic equations of general relativity are the Einstein equations which describe the gravitational field coupled with matter equations which describe the sources of the gravitational field. There are mathematical proofs of the existence of singularities in general relativity under rather general physically reasonable conditions. These stem from the singularity theorems of Penrose and Hawking which appeared starting in the mid 1960’s. It is not a priori clear how to define a singularity in this context but a mathematically clear definition has emerged which is useful and widely accepted. It is formulated in terms of geodesic incompleteness. The mathematics entering the proofs of these theorems is differential geometry and ordinary differential equations. The input is less than the full Einstein-matter equations with a definite choice of matter model – rather only certain inequalities on the energy-momentum tensor, the right hand side of the Einstein equations, are required. These inequalities are the so-called energy conditions.

In contrast there is no proof or disproof of cosmic censorship and even finding a precise formulation is difficult. There a couple of concrete obstacles involved:

1. There are very special classes of solutions known, for instance some with high symmetry, which do contain naked singularities. To avoid these a genericity assumption is required.

2. There are certain matter models with pathological properties which can lead to singularities even in the absence of gravity. The most notorious of these is dust, a fluid without pressure. A formulation of cosmic censorship which has a chance of being true must include some restriction on the matter model.

I like to hope that cosmic censorship is true. I strongly believe that it is impossible to prove it using differential geometry and ordinary differential equations alone. A proof will require taking the Einstein-matter equations seriously as a system of partial differential equations. It so happens that proving results about solutions of partial differential equations is much harder than proving analogous results about ordinary differential equations. Thus it is not surprising that progress in proving cosmic censorship has been limited, even taking the optimistic view that it is true. Another difficulty is that relevant parts of PDE theory simply do not belong to the usual mathematical repertoire of people working in general relativity.

Now I come back to the article of Joshi. The author appears rather fond of naked singularities and has mentioned a lot of work tending to support their existence. My aim here is not to give a detailed criticism of the work he cites. Instead I will describe important work providing support for cosmic censorship which is completely ignored in the article. As already mentioned, there is no proof of cosmic censorship. The character of the results I will mention here is that they concern model problems (with some assumptions on symmetry and the choice of matter model) where the mechanisms which are invoked to make cosmic censorship work can be seen in action. The first mechanism is genericity. I give two examples. The first is the spherically symmetric scalar field, as studied by Christodoulou (Ann. Math. 149, 183 (1999)). He showed that in this model naked singularities do occur but that they are eliminated by a genericity assumption. The second is the case of vacuum Gowdy spacetimes where a conceptually similar result was obtained by RingstrÃ¶m (to appear in Ann. Math., see http://www.math.kth.se/~hansr). The second mechanism is the influence of the choice of matter model. Gerhard Rein, Jack Schaeffer and I showed that a large class of naked singularities (the non-central ones) in solutions of the Einstein-dust equations can be removed by replacing dust by collisionless matter. There are many other papers tending to support cosmic censorship. Here I have just mentioned some which illustrate important aspects well. For a valuable account of the development of ideas about cosmic censorship I recommend the Prologue to the recent book of Christodoulou “The formation of black holes in general relativity”. A preprint version is available as gr-qc/0805.3880.

The jury is still out on cosmic censorship. I think that the appropriate expert witnesses who might be called to open the way to a just verdict are the specialists in partial differential equations.

April 26, 2009 at 2:26 am |

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!

April 26, 2009 at 3:28 pm |

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.

September 15, 2010 at 5:16 am |

[...] 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. [...]