The characteristic initial value problem in general relativity in a double null foliation

Question

In a Paper by Rendall, it is shown that the characteristic initial value problem for the Einstein equations is well-posed. In fact, if the data are specified in some coordinates, then one can extend these coordinates to be harmonic in the future development. In work of Christodoulou on the formation of trapped surfaces, Christodoulou seems to use a version of this result to set up his initial data [Edit: I really should have said "to start his argument by obtaining a solution to the Einstein equations locally from his initial data"]. However, the coordinates he gets for his solution are what he calls "canonical coordinates." In short, if he chooses coordinates $v,\theta$ on one null hypersurface and $u,\theta$ on the other, then it seems to me like $u$,$v$ should extend to optical functions in the interior, and that the gradient of $u$ should be $\frac{-1}{2}\frac{\partial}{\partial v}$ on the initial hypersurface.

My question is this: (1) is my interpretation accurate? and if so, (2) what's the idea of the proof? The theorem of Rendall is argued by using harmonic coordinates and reduction to the Cauchy problem (which has long been solved). If one tries a similar approach to the initial data of Christodoulou, one ends up trying to solve the null Bianchi equations, which are not of hyperbolic character and don't appear to have a general theory, so even obtaining a local existence theorem seems like very hard work.

Answer 1

1. Christodoulou does not use Rendall's result to "set up his initial data". What happened in the end of Chapter 2 is that he appealed to Rendall's result to show that, given the initial data he has prescribed on the two initial intersecting cones, there is a local solution (in harmonic gauge). Given that a solution does exist, his arguments (the various estimates he proves in the middle part of the paper) guarantees the existence of a double null foliation along which the required boundedness of various fluxes hold. (In other words, once you have a solution, you can start proving a priori estimates, even in the situation when the coordinates relative to which you are trying to prove estimates are dynamic; this is the standard bootstrapping procedure in quasilinear PDEs.)

2. To get a solution that exists for long enough time, it is not sufficient to appeal to simply Rendall's result. So Christodoulou proves an extension theorem in Chapter 16: the proof spans page 559 - 580 on the arXiv version. The local extension theorem works with space-like initial data in harmonic coordinates. The reason that you can use space-like initial data is that, remembering that we are only interested in a region near the incoming cone, and within the incoming cone by domain of dependence arguments we know the metric is flat, as soon as the initial region of existence (which is guaranteed by Rendall's result) is found, one can draw a space-like hypersurface, which will, in principle, lie in the past of the first trapped surface that is to be shown.

   (This is where the initial assumption of having a flat causal diamond is very useful. In principle you can also replace the Minkowski causal diamond with a causal diamond from any suitable known regular solution [for example a diamond from any of the Christodoulou-Klainerman small data solutions]. In terms of what Rendall did, the idea is that for Rendall's proof to work he needs to "fill in" the region to the past of his cone with "stuff" which he then subtracts as an inhomogeneity when solving the space-like initial value problem. In Christodoulou's case, he started in a situation where the "stuff" in the past of the cone is already known [in fact Minkowski], so he does not need to go back and fill it in.)

   The extension theorem is a bit long to prove in part because he needs to translate the estimates he has proven in the double null foliation to estimates that would be useful to control the evolution in harmonic coordinates.

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Let me add some (badly drawn) pictures to help clarify.

Initially we have a Minkowski diamond (green) with a short pulse outside prescribed on an outgoing line cone (red).

Rendall's existence theorem guarantees that the blue region exists.

The next picture shows the induction argument to construct regions to the future of what is guaranteed by Rendall's existence theorem. Note that what is used is a space-like initial data (blue curve in the picture below) from which local existence in a lens-like domain (area between the blue and red curves) is found. Part of this area is known (the Minkowski part under the green line), but part of it is new (the shaded red part).

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See J. Luk, On the Local Existence for the Characteristic Initial Value Problem in General Relativity for a treatment of related questions.