### 3.8 Numerical studies on critical collapse

In [147] 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 [147] 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 on the initial
data such that for small dispersion occurs and for large a black hole is formed, we get a critical
value separating these regions. If we take and denote by the mass of the black
hole, then if as 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 [147] is that Vlasov matter is of type I. There are two other independent numerical
simulations on critical collapse for Vlasov matter [128, 21]. In these simulations, maximal-areal coordinates
are used rather than Schwarzschild coordinates as in [147]. The conclusion of these studies agrees with the
one in [147].