Formation of black holes in vacuum

A black hole can be described by an explicit solution of Einstein’s equations, the Schwarzschild solution, which was discovered shortly after Einstein formulated general relativity. Karl Schwarzschild wrote his paper on the subject while serving on the Russian front in 1915. He died in May of the following year, possibly as a result of the autoimmune disease pemphigus vulgaris. The physical interpretation of the solution remained obscure for decades and the name ‘black hole’ arose in the 1960’s and is generally attributed to John Wheeler.

Physically the Schwarzschild solution represents an eternal black hole. Not only will it always be there, it has always been there. The maximal extension of the solution contains a white hole region which is of doubtful physical relevance. The situation of real physical interest is that where a black hole is formed by the collapse of matter. In fact, since the energy carried by the gravitational field itself gravitates in general relativity, it is possible to imagine that a black hole can be formed from an initial state with small gravitational fields without the intervention of matter. That this scenario can take place within general relativity has recently been shown in a milestone work of Demetrios Christodoulou which took the research community by surprise. The long introduction to that paper provides a beautiful introduction to the history of the mathematical study of black holes.

The Schwarzschild solution is spherically symmetric – it is a perfectly spherical configuration. As a consequence it would be very convenient if the process of formation of a black hole could also be studied in the context of spherical symmetry. Unfortunately this cannot be done in the vacuum case. The obstruction is Birkhoff’s theorem, which shows that a spherically symmetric solution of Einstein’s vacuum equations is not dynamical. Hence if it is desired to profit from the simplification of the mathematics provided by spherical symmetry matter must be included. This has been done in the past and results on the formation of black holes have been obtained by several authors including Christodoulou, Dafermos, Andréasson, Kunze and Rein.

The new work of Christodoulou quoted above concerns vacuum solutions of the Einstein equations and correspondingly has to manage without symmetry. In fact the mathematical result whose proof occupies almost all of this long text is that a trapped surface can be formed from weak initial data by the evolution defined by the Einstein equations in vacuum. The concept of a trapped surface was introduced by Roger Penrose in 1965 as a criterion for the occurrence of singularities in solutions of the Einstein equations (Penrose singularity theorem). Intuitively the presence of a singularity serves to diagnose the formation of a black hole.

So what exactly happens in the solutions whose existence is demonstrated by Christodoulou? The initial configuration is an ingoing pulse of gravitational waves of high density but short duration. Initially the condition for the presence of a trapped surface is arbitrarily far from being satisfied. As the pulse moves inwards in the course of the evolution the energy is focussed until a trapped surface is formed. A key point of the proof is to show that no spacetime singularity is formed before the trapped surface has time to develop. The techniques used are major developments of those used by Christodoulou and Klainerman in their proof of the nonlinear stability of flat space under the evolution defined by the Einstein vacuum equations.

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