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 Ringström (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.