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

Abstract

We express the Einstein-Vlasov system in spherical symmetry in terms of a dimensionless momentum variable $z$ (radial over angular momentum). This regularises the limit of massless particles, and in that limit allows us to obtain a reduced system in independent variables $(t,r,z)$ only. Similarly, in this limit the Vlasov density function $f$ for static solutions depends on a single variable $Q$ (energy over angular momentum). This reduction allows us to show that any given static metric which has vanishing Ricci scalar, is vacuum at the centre and for $r>3M$ and obeys certain energy conditions uniquely determines a consistent $f=\bar{k}(Q)$ (in closed form). Vice versa, any $\bar{k}(Q)$ within a certain class uniquely determines a static metric (as the solution of a system of two first-order quasilinear ODEs). Hence the space of static spherically symmetric solutions of Einstein-Vlasov is locally a space of functions of one variable. For a simple 2-parameter family of functions $\bar{k}(Q)$, we construct the corresponding static spherically symmetric solutions, finding that their compactness is in the interval $0.7\lesssim \mathrm{ma}{\mathrm{x}}_{\mathrm{r}}(2M/r)\le 8/9$. This class of static solutions includes one that agrees with the approximately universal type-I critical solution recently found by Akbarian and Choptuik (AC) in numerical time evolutions. We speculate on what singles it out as the critical solution found by fine-tuning generic data to the collapse threshold, given that AC also found that {\em all} static solutions are one-parameter unstable and sit on the threshold of collapse.

Version accepted for publication in PRD

References in corpus (3)

• A numerical investigation of the stability of steady states and critical phenomena for the spherically symmetric Einstein-Vlasov system
• On the steady states of the spherically symmetric Einstein-Vlasov system
• Critical collapse in the spherically-symmetric Einstein-Vlasov model

Cited by in corpus (4)

• Accretion of the Vlasov gas on Reissner-Nordström black holes
• The Einstein-Vlasov system in spherical symmetry II: spherical perturbations of static solutions
• Gravitational collapse in the vicinity of the extremal black hole critical point
• Existence of steady states of the massless Einstein-Vlasov system surrounding a Schwarzschild black hole