This has gotta be the quickest I've ever read a paper.

After about 2 weeks (8 working days to be exact), I finished "reading" (to be more precise, I skimmed the paper without checking all of the calculations, but the fundamental ideas all are clear by looking at the form of the calculations) the 594 page paper of Demetrios Christodoulou on The Formation of Black Holes in General Relativity. With Arick Shao, Phillip Whitman, and David Parlongue, I have been holding discussion sessions (twice so far, and one more tomorrow) about the result. So I think I now have a pretty good understanding of what goes into this proof.

First, a brief description and motivation of the result: the starting point is the 1965 singularity theorem of Sir Roger Penrose. Penrose proved that assuming the space-time admits (i) a non-compact Cauchy surface (ii) its energy-momentum content satisfies the dominant energy condition and (iii) there exists a closed trapped surface then the space-time must be geodesically incomplete (e.g. admits a singularity). The proof of the theorem is by contradiction: basically if we assume that the space-time is geodesically complete, the future of the trapped surface defines a compact region in space-time from which via the geodesics we have a one-to-one map with the Cauchy hypersurface: but it is not possible to have a one-to-one map of a compact region with a non-compact region.

A crucial realization is that the Cauchy hypersurface need not be strictly space-like: any achronal Cauchy surface will do. So it is possible to reformulate the singularity theorem with a complete null cone in place of the non-compact Cauchy surface.

The question the Christodoulou seeked to answer is this: is the development of a black hole dynamic? Can we start from a space-time that does not have a black hole and then form one? For the space-time with matter, he answered the question in the affirmative for the scalar field problem: there exists some threshold for the energy density such that above that limit it is possible for a black hole to develop from a previously regular space-time. (Contrast with the white hole case: it is known that a white hole cannot sponaneously appear with reasonable matter fields, and also that once you have a black hole you can't uncreate it [under classical general relativity of course; Stephen Hawking has put forth evaporation arguments for black holes to dissipate due to quantum fluctuations].) The question that Christodoulou attempts to tackle in this paper is that, whether a black hole can form from the focusing of gravitational waves in the absence of matter!

One of the most classic predictions of general relativity is that gravitation can propagate as waves: some think of them as ripples in space-time continuum. This prediction is what the gravitational interferometers like LIGO seek to verify. Since the equations of general relativity reduces to a nonlinear wave equation, it seems reasonable to expect that these gravitational waves should interact (i.e. unlike waves on a string or electromagnetic waves in vacua, both of which allows the principle of superposition and that two waves will pass through each other unchanged); given that gravity is an attractive force, it also seems reasonable to ask whether this attraction will be strong enough to trap the waves in a finite region of space, forcing it to interact with itself and providing a feed-back that eventually collapses as a black hole!

And this is what Christodoulou has shown in his work.

Mathematically, the paper was not difficult. All of his insights are demonstrated in the very beginning as he wrote down the initial ansatz. The rest is just verifications.

The basic ingredients of the proof are the following:

• The characteristic initial value problem As was shown by Alan Rendall, there is a local existence and uniqueness theorem for the initial value problem in general relativity when the data is prescribed on two transversely intersecting null hypersurfaces. Christodoulou, in his work, takes it to be the intersection of an incoming null cone with an outgoing null cone, with the data on the incoming null cone being trivial/Minkowski and the data prescribed on an interval of length δ on the outgoing cone starting from the sphere of intersection. The region of space-time we are interested in is the δ neighborhood of the incoming null cone.
  • Rendall's result states that there is a small neighborhood of the sphere of intersection where the initial value problem is solved.
  • Prescription of initial data on a null cone does not suffer from a big problem of the normal Cauchy problem--that of the constraint equation. Einstein's equation is over-determined. When prescribing initial data, the initial data must satisfy some compatibility equation so that the equation even has an iota of chance of be solvable. For the normal problem, the compatibility equation is a nonlinear elliptic partial differential equation that many geometric analysts spend their entire life studying; for the characteristic initial value problem, the compatibility equation becomes some ordinary differential equations along the generators of the null cones. In other words, and being more precise, we have complete freedom in prescribing a conformal metric for the initial data, and all the other initial data--the null second fundamental form and the conformal factor in the metric--can be determined by solving ordinary differential equations on the cone. This means that by passing from the Lie group to Lie algebra (via the expoential map), the space of initial data for the equation is a vector space and we circumvent the problem of trying to figure out what exactly is meant by "small data" and have an explicit way of making the data small.
  • Using the explicit way of making the data small, and using the fact that the vacuum Einstein equation is conformally invariant, Christodoulou chose the scaling

    ψ(r,θ) ← δ^½ ψ(r/δ,θ)

    and thus enforcing a particular homogeneity for the solution.
• Choquet-Bruhat's local existence theorem We can glue onto the underside of our picture (the region beneath the incoming null cone) a Minkowskian region. Then Rendall's theorem shows that up to time a little bit more than the time of the sphere of intersection, there is a local solution. Using Choquet-Bruhat's theorem, we can extend this a bit more in time.
• Bootstrapping/Method of Continuity We look for the last time before some fixed time τ such that certain quantities--the size of the space-time curvature in particular--remains controllable. So we look for a priori estimates on the sizes of those quantities and try to bootstrap. It turns out that Einstein's equations can be written as first order propagation equations for the most part, and these type of estimates can be deduced by simply integrating from the boundary where the inital data is prescribed and using Gromwell's inequality. (Well, to get better control on the derivatives, one realizes that the propagation equations can be turned into propagation equations of Hodge systems, and elliptic regularity gives improved control.)

The above techniques are enough to produce a satisfactory local existence theorem for the initial data up to time τ if we assume the scale δ is small enough. To produce the trapped surface, however,

• One integrates for trχ, the trace of the out-going null second fundamental form relative to the double null foliation, a quantity that measures the derivative of the volume of the sphere of intersection, from the incoming hypersurface. One shows that this quantity, by the choice of ansatz, is independent of δ, when we measure it at the outer edge of the solution on the time-slice for τ (which is a fixed number chosen at the start). So that by choosing the initial data ψ big enough, we can force the value of trχ to become negative. And thus get a trapped surface.

In some sense, the idea is that the problem depends on two different norms that scales differently (the rescaling ansatz defines a scaling). One of the norms, if made small, can be used to guarantee local existence. The other of the norms, if made big, can be used to guarantee appearance of a trapped surface. By chosing an initial data and chosing an appropriate scale (basically two degress of freedom) we can make the first norm as small as we want while keeping the second norm as big as we want (using up our two degrees of freedom). This is vaguely reminiscent of problems in Schroedinger type equations where we have two conserved quantities in different scales, and thus have different local behaviour.