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

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

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.

1-16

Gundlach, Carsten

586f1eb5-3185-4b2b-8656-c29c436040fc

27 December 2016

Gundlach, Carsten

586f1eb5-3185-4b2b-8656-c29c436040fc

Gundlach, Carsten
(2016)
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.
*Physical Review D*, 94 (124046), .
(doi:10.1103/PhysRevD.94.124046).

## 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.

Text

** 1610.08908.pdf
- Accepted Manuscript**
## More information

Accepted/In Press date: 7 December 2016

e-pub ahead of print date: 27 December 2016

Published date: 27 December 2016

Organisations:
Applied Mathematics

## Identifiers

Local EPrints ID: 405142

URI: https://eprints.soton.ac.uk/id/eprint/405142

ISSN: 1550-7998

PURE UUID: bd40938b-8f76-4245-a6d3-5cf4f85119ff

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Date deposited: 31 Jan 2017 13:20

Last modified: 20 Jul 2018 00:34

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