3.5 On global existence for general initial data

As was mentioned at the end of Section 3.1, the issue of global existence for general initial data is only relevant in certain time gauges since there are initial data leading to singular spacetimes. However, it is reasonable to believe that global existence for general data may hold in a polar time gauge or a maximal time gauge, cf. [116Jump To The Next Citation Point], and it is often conjectured in the literature that these time slicings are singularity avoiding. However, there is no proof of this statement for any matter model and it would be very satisfying to provide an answer to this conjecture for the Einstein–Vlasov system. A proof of global existence in these time coordinates would also be of great importance due to its relation to the weak cosmic censorship conjecture, cf. [61Jump To The Next Citation Point, 62Jump To The Next Citation Point, 65Jump To The Next Citation Point].

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 [146Jump To The Next Citation Point], which is carried out in Schwarzschild coordinates. The authors introduce the following variables in the momentum space adapted to spherical symmetry,

x ⋅ v L := |x |2|v|2 − (x ⋅ v)2,w =----, (47 ) r
where L is the square of the angular momentum and w 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 f = f (t,r,w, L) becomes
w ( L ) ∂tf + eμ− λ--∂rf − λtw + eμ−λμrE − eμ−λ ---- ∂wf = 0, (48 ) E r3E
where
āˆ˜ -------------- E = E (r,w, L) = 1 + w2 + Lāˆ•r2.

The main result in [146Jump To The Next Citation Point] shows that as long as there is no matter in the ball

{x ∈ ā„3 : |x| ≤ šœ–},
the estimate
Q (t) ≤ elogQ(0)eC(šœ–)t, (49 )
holds. Here C (šœ–) is a constant, which depends on šœ–. Thus, in view of the continuation criterion this can be viewed as a global existence result outside the center of symmetry for initial data with compact support. This result rules out shell-crossing singularities, which are present when, e.g., dust is used as a matter model. The bound of Q is obtained by estimating each term individually in the characteristic equation associated with the Vlasov equation (48View Equation) for the radial momentum. This involves a particular difficulty. The Einstein equations imply that
m- 2λ 2λ μr = r2e + 4πrpe
where
∫ r m (t,r) = 4π η2ρ (t,η)dη, (50 ) 0
is the quasi local mass. Thus, using (38View Equation) the characteristic equation consists of the two terms T = 4 πreμ+λ(jw + pE ) 1, and T = eμ+λm- 2 r2, together with a term, which is independent of the matter quantities. There is a distinct difference between the terms T1 and T2 due to the fact that m can be regarded as an average, since it is given as a space integral of the energy density ρ, whereas j and p are point-wise terms. The method in [146Jump To The Next Citation Point] makes use of a cancellation property of the radial momenta in T1 so that outside the center this term is manageable but in general it seems very unpleasant to have to treat point-wise terms of this kind.

In [156Jump To The Next Citation Point] Rendall shows global existence outside the center in maximal-isotropic coordinates. The bound on Q (t) 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 [12Jump To The Next Citation Point] gives an alternative and simplified proof of the result in [146Jump To The Next Citation Point]. 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 [12Jump To The Next Citation Point] and the application of Green’s formula becomes very natural. In addition, the bound of Q is improved compared to (49View Equation) and reads

Q (t) ≤ Q(0)eC(1+t)āˆ•šœ–t.
This bound is sufficient to conclude that global existence outside the center also holds for non-compact initial data. In addition to the global existence result outside the centre, it is shown in [12Jump To The Next Citation Point] that as long as 3m (t,r) ≤ r and j ≤ 0, singularities cannot form. Note that in Schwarzschild coordinates 2m (t,r) ≤ r always, and that there are closed null geodesics if 3m = r in the Schwarzschild static spacetime.

The method in [12Jump To The Next Citation Point] also applies to the case of maximal-isotropic coordinates studied in [156Jump To The Next Citation Point]. 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 [12Jump To The Next Citation Point] is accordingly that the quite different proofs in [146Jump To The Next Citation Point] and in [156Jump To The Next Citation Point] 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 [64Jump To The Next Citation Point]. 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 [60Jump To The Next Citation Point]. An essential part of his method is based on the understanding of the formation of trapped surfaces [58Jump To The Next Citation Point]. In [62Jump To The Next Citation Point] 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 [62Jump To The Next Citation Point] is satisfied, since they concern a portion of the maximal development covered by particular coordinates. Therefore, Dafermos and Rendall [64Jump To The Next Citation Point] choose double-null coordinates, which cover the maximal development, and they show that the mentioned hypothesis is satisfied for Vlasov matter.


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