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 regularizes 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 that has vanishing Ricci scalar, is vacuum at the center and for r>3M and obeys certain energy conditions uniquely determines a consistent f=¯k(Q) (in closed form). Vice versa, any ¯k(Q) within a certain class uniquely determines a static metric (as the solution of a system of two first-order quasilinear ordinary differential equations). Hence the space of static spherically symmetric solutions of the Einstein-Vlasov system is locally a space of functions of one variable. For a simple two-parameter family of functions ¯k(Q), we construct the corresponding static spherically symmetric solutions, finding that their compactness is in the interval 0.7?maxr(2M/r)?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 all static solutions are one-parameter unstable and sit on the threshold of collapse.

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ORCID for Carsten Gundlach: orcid.org/0000-0001-9585-5375

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