Formation of black holes in vacuum, part 2

I have just returned from a conference at the Mathematical Sciences Research Institute (MSRI) in Berkeley with the title ‘Hot topics: black holes in relativity‘. The central theme of this conference was the work of Demetrios Christodoulou on the formation of black holes in vacuum which I discussed in a previous post

On the first day of the conference I gave a talk on the characteristic initial value problem in general relativity. This was based on a paper I wrote more than twenty years ago (Proc. R. Soc. Lond. A427, 221 – I find it difficult to believe that it has been so long). The result of this paper is used in Christodoulou’s work and this was the main justification for the talk. In the ordinary initial value problem (Cauchy problem) for a hyperbolic system, or for the Einstein equations, initial data are prescribed on a spacelike hypersurface. The idea of the characteristic initial value problem is to instead prescribe data on a characteristic hypersurface. In fact it is necessary to use a singular characteristic hypersurface (such as a cone) or a pair of smooth hypersurfaces which intersect transversely. The result of Christodoulou is formulated in terms of the first of these possibilities, with data prescribed on a light cone. However he assumes that these data coincide with flat space data near the vertex of the cone, which allows the problem to be reduced to the second, easier possibility and it is the latter which I treated in my paper. In the result of that paper, which applies to smooth initial data, existence and uniqueness results for the characteristic initial value problem are deduced from the corresponding results for the Cauchy problem. In the ordinary Cauchy problem for the Einstein equations it is necessary to solve the constraint equations, which means solving an elliptic problem. In the characteristic case the constraints reduce to a hierarchical system of ordinary differential equations, which can be a big advantage.

During the conference Christodoulou gave five talks about his theorem and its proof and I found these very enlightening. I feel I have a much better understanding of the basics of this work now that I did before. One aspect of the result is that the data used are in one sense small (close to flat space data) and in another sense large. If they were small in a sufficiently strong sense then this should lead to a global existence result which in particular rules out the formation of black holes due to the theorem of Christodoulou and Klainerman on the stability of Minkowski space. On the other hand the interpretation of the result (formation of a trapped surface starting from a weak-field situation) requires that the data be small in some sense. Combining these two requirements (smallness and largeness in different senses) is a key feature of the theorem. It is also the case that the data are in some sense close to being spherically symmetric but in another sense far from spherical symmetry. Intuitively, it is necessary to have data which represent a sufficiently strong pulse of gravitational radiation. Spherical symmetry rules out gravitational radiation and this might be extrapolated to say that being close to spherical symmetry means restricting to a small amount of radiation.

In the proof of the theorem the solution is parametrized in the following way. The initial hypersurface is a null cone C_0. It can be foliated by surfaces which are of constant affine distance from the vertex. Through each of these there is a null hypersurface transverse to C_0 which is taken to be a level hypersurface of a function \underline{u}. This function agrees with the affine distance (suitably normalized) on C_0. A function u is defined to be constant on the null cones of the points on a timelike curve passing through the vertex of the cone. Things are always set up so that these null hypersurfaces have no caustics. The two functions define a foliation by spheres by means of the intersections of their level hypersurfaces. This foliation is in a sense the analogue of that by symmetry orbits in a spherically symmetric problem. The fact that the problem is almost spherically symmetric is witnessed by the fact that the Gaussian curvature of these spheres is almost constant. Note that the gradients of the functions u and \underline{u} do not commute as vector fields in general. Thus they are not tangent to surfaces and this is an important difference from spherical symmetry.

The initial data is such that a suitable energy density on the cone changes suddenly from being zero to being sufficiently large. This is the basis of the short pulse method, which is the central new technique in the proof of the theorem. What is this energy density? It is the norm squared of the trace-free part of the second fundamental form of the spheres in the direction along the cone.

When Christodoulou had completed his last lecture someone in the audience asked, ‘What’s next?’ In reply he announced that this had been his last project in general relativity, which came as quite a shock to the audience. The word ‘announced’ is perhaps not appropriate since it sounds too formal – he just said it spontaneously. This is sad news for the field of mathematical relativity but perhaps it is less sad in a wider context. After all, Christodoulou has a number of fascinating projects he is working on in other areas. At the same time the theorem I have been talking about here will probably be a beginning rather than an end. At the conference Igor Rodnianski gave a talk on work he has been doing with Sergiu Klainerman aimed at generalizing this result while understanding it more deeply. I look forward to seeing where that will lead.

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4 Responses to “Formation of black holes in vacuum, part 2”

  1. Uwe Brauer Says:


    I am also shocked that Christodoulou is “leaving GR”. So what
    are these

    fascinating projects he is working on in other areas.

    Uwe Brauer

  2. hydrobates Says:

    Of course I do not have a complete overview of Christodoulou’s projects but I do know of two which are fascinating to say the least. In previous posts I have mentioned his work on the formation of shock waves in (special) relativistic fluids. In the book on fluids he already published he followed the solutions up to and including the shock surface, with no symmetry assumptions. This was set up so that the whole analysis could be done in the context of irrotational and isentropic fluids. At the shock surface both entropy and rotation will be produced, even if there was none beforehand. Christodoulou intends to analyse the evolution beyond the shock hypersurface. This analysis will not be global in time but should control the evolution in some well-defined region which would allow processes like the production of vorticity to be analysed. This is the first project. The second concerns nonlinear electrodynamics. This means replacing the Lagrangian |E|^2+|B|^2 of ordinary electrodynamics by some other function of E and B. The corresponding Euler-Lagrange equations define a hyperbolic system which will in general develop shocks. The results of an analysis of these equations could in principle lead to testable physical predictions, although this is beyond the range of present experimental techniques. Both of these priojects are part of a more general attempt to better understand quasilinear hyperbolic equations whose characteristics are more complicated than the single quadratic cone of the vacuum Einstein equations or irrotational fluids.

  3. Bootstrap arguments « Hydrobates Says:

    [...] and Klainerman and the theorem on formation of trapped surfaces by Christodoulou discussed in previous posts. The name comes from a story where someone pulls himself up by his bootstraps, leather attachments [...]

  4. hydrobates Says:

    A preprint on the work of Klainerman and Rodnianski mentioned in this post has just appeared on ArXiv (

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