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On the small redshift limit of steady states of the spherically symmetric Einstein–Vlasov system and their stability

Published online by Cambridge University Press:  05 October 2015

MAHIR HADŽIĆ  and
GERHARD REIN
MAHIR HADŽIĆ
Affiliation:
King's College London, Department of Mathematics Strand, London, WC2R 2LS. e-mail: mahir.hadzic@kcl.ac.uk
GERHARD REIN
Affiliation:
Mathematisches Institut der Universität Bayreuth D-95440 Bayreuth, Germany. e-mail: gerhard.rein@uni-bayreuth.de
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Abstract

Families of steady states of the spherically symmetric Einstein–Vlasov system are constructed, which are parametrised by the central redshift. It is shown that as the central redshift tends to zero, the states in such a family are well approximated by a steady state of the Vlasov–Poisson system, i.e., a Newtonian limit is established where the speed of light is kept constant as it should be and the limiting behavior is analysed in terms of a parameter which is tied to the physical properties of the individual solutions. This result is then used to investigate the stability properties of the relativistic steady states with small redshift parameter in the spirit of recent work by the same authors, i.e., the second variation of the ADM mass about such a steady state is shown to be positive definite on a suitable class of states.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 3 , November 2015 , pp. 529 - 546
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

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