3.8 Numerical studies on critical collapse

In [147Jump To The Next Citation Point] a numerical study on critical collapse for the Einstein–Vlasov system was initiated. A numerical scheme originally used for the Vlasov–Poisson system was modified to the spherically-symmetric Einstein–Vlasov system. It has been shown by Rein and Rodewis [148] that the numerical scheme has desirable convergence properties. (In the Vlasov–Poisson case, convergence was proven in [167], see also [77]).

The speculation discussed above that there may be no naked singularities formed for any regular initial data is in part based on the fact that the naked singularities that occur in scalar field collapse appear to be associated with the existence of type II critical collapse, while Vlasov matter is of type I. The primary goal in [147Jump To The Next Citation Point] was indeed to decide whether Vlasov matter is type I or type II.

These different types of matter are defined as follows. Given small initial data, no black holes form and matter will disperse. For large data, black holes will form and consequently there is a transition regime separating dispersion of matter and formation of black holes. If we introduce a parameter A on the initial data such that for small A dispersion occurs and for large A a black hole is formed, we get a critical value Ac separating these regions. If we take A > Ac and denote by mB (A ) the mass of the black hole, then if mB (A) → 0 as A → Ac we have type II matter, whereas for type I matter this limit is positive and there is a mass gap. For more information on critical collapse we refer to the review paper by Gundlach [89].

The conclusion of [147Jump To The Next Citation Point] is that Vlasov matter is of type I. There are two other independent numerical simulations on critical collapse for Vlasov matter [128, 21Jump To The Next Citation Point]. In these simulations, maximal-areal coordinates are used rather than Schwarzschild coordinates as in [147Jump To The Next Citation Point]. The conclusion of these studies agrees with the one in [147].


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