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We express the Einstein-Vlasov system in spherical symmetry in terms of adimensionless momentum variable $z$ (radial over angular momentum). Thisregularises the limit of massless particles, and in that limit allows us toobtain a reduced system in independent variables $(t,r,z)$ only. Similarly, inthis limit the Vlasov density function $f$ for static solutions depends on asingle variable $Q$ (energy over angular momentum). This reduction allows us toshow that any given static metric which has vanishing Ricci scalar, is vacuumat the centre and for $r>3M$ and obeys certain energy conditions uniquelydetermines a consistent $f=\bar k(Q)$ (in closed form). Vice versa, any $\bark(Q)$ within a certain class uniquely determines a static metric (as thesolution of a system of two first-order quasilinear ODEs). Hence the space ofstatic spherically symmetric solutions of Einstein-Vlasov is locally a space offunctions of one variable. For a simple 2-parameter family of functions $\bark(Q)$, we construct the corresponding static spherically symmetric solutions,finding that their compactness is in the interval $0.7\lesssim \rmmax_r(2M/r)\le 8/9$. This class of static solutions includes one that agreeswith the approximately universal type-I critical solution recently found byAkbarian and Choptuik (AC) in numerical time evolutions. We speculate on whatsingles it out as the critical solution found by fine-tuning generic data tothe collapse threshold, given that AC also found that {\em all} staticsolutions are one-parameter unstable and sit on the threshold of collapse.

physical review. d/physical review. d.

The Einstein-Vlasov system in spherical symmetry: reduction of the  equations of motion and classification of single-shell static solutions, in  the limit of massless particles