The methods of proofs in the cases described in Sections 3.3 and 3.4, where global existence has been shown, are all tailored to treat special classes of initial data and they will likely not apply in more general situations. In this section we discuss some attempts to treat general initial data. These results are all conditional in the sense that assumptions are made on the solutions, and not only on the initial data.

The first study on global existence for general initial data is [146], which is carried out in Schwarzschild coordinates. The authors introduce the following variables in the momentum space adapted to spherical symmetry,

where is the square of the angular momentum and is the radial component of the momenta. A consequence of spherical symmetry is that angular momentum is conserved along the characteristics. In these variables the Vlasov equation for becomes where

The main result in [146] shows that as long as there is no matter in the ball

In [156] Rendall shows global existence outside the center in maximal-isotropic coordinates. The bound on is again obtained by estimating each term in the characteristic equation. In this case there are no point-wise terms in contrast to the case with Schwarzschild coordinates. However, the terms are, in analogy with the Schwarzschild case, strongly singular at the center.

A recent work [12] gives an alternative and simplified proof of the result in [146]. In particular, the method avoids the point-wise terms by using the fact that the characteristic system can be written in a form such that Green’s formula in the plane can be applied. This results in a combination of terms involving second-order derivatives, which can be substituted for by one of the Einstein equations. This method was first introduced in [7] but the set-up is different in [12] and the application of Green’s formula becomes very natural. In addition, the bound of is improved compared to (49) and reads

The method in [12] also applies to the case of maximal-isotropic coordinates studied in [156]. There is an improvement concerning the regularity of the terms that need to be estimated to obtain global existence in the general case. A consequence of [12] is accordingly that the quite different proofs in [146] and in [156] are put on the same footing. We point out that the method can also be applied to the case of maximal-areal coordinates.

The results discussed above concern time gauges, which are expected to be singularity avoiding so that the issue of global existence makes sense. An interpretation of these results is that “first singularities” (where the notion of “first” is tied to the causal structure), in the non-trapped region, must emanate from the center and that this case has also been shown in double null-coordinates by Dafermos and Rendall in [64]. The main motivation for studying the system in these coordinates has its origin from the method of proof of the cosmic-censorship conjecture for the Einstein–scalar field system by Christodoulou [60]. An essential part of his method is based on the understanding of the formation of trapped surfaces [58]. In [62] it is shown that a single trapped surface or marginally-trapped surface in the maximal development implies that weak cosmic censorship holds The theorem holds true for any spherically-symmetric matter spacetime if the matter model is such that “first” singularities necessarily emanate from the center. The results in [146] and in [156] are not sufficient for concluding that the hypothesis of the matter needed in the theorem in [62] is satisfied, since they concern a portion of the maximal development covered by particular coordinates. Therefore, Dafermos and Rendall [64] choose double-null coordinates, which cover the maximal development, and they show that the mentioned hypothesis is satisfied for Vlasov matter.

Living Rev. Relativity 14, (2011), 4
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