The University of Southampton
University of Southampton Institutional Repository

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
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.
1550-7998
1-16
Gundlach, Carsten
586f1eb5-3185-4b2b-8656-c29c436040fc
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), 1-16. (doi:10.1103/PhysRevD.94.124046).

Record type: Article

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
Download (2MB)

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

Catalogue record

Date deposited: 31 Jan 2017 13:20
Last modified: 20 Jul 2018 00:34

Export record

Altmetrics

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of https://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×