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Stability criterion for selfsimilar solutions with perfect fluids in general relativity

Tomohiro Harada
(
- Waseda U.
)

Sep, 2001

18 pages

Published in:

Class.Quant.Grav. 18 (2001) 4549-4568

e-Print:

gr-qc/0109042 [gr-qc]

DOI:

10.1088/0264-9381/18/21/311

Report number:

WU-AP-137-01

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Abstract: (arXiv)

A stability criterion is derived for self-similar solutions with perfect fluids which obey the equation of state

P=k\rho

in general relativity. A wide class of self-similar solutions turn out to be unstable against the so-called kink mode. The criterion is directly related to the classification of sonic points. The criterion gives a sufficient condition for instability of the solution. For a transonic point in collapse, all primary-direction nodal-point solutions are unstable, while all secondary-direction nodal-point solutions and saddle-point ones are stable against the kink mode. The situation is reversed in expansion. Applications are the following: the expanding flat Friedmann solution for

1/3 \le k < 1

and the collapsing one for

0< k \le 1/3

are unstable; the static self-similar solution is unstable; nonanalytic self-similar collapse solutions are unstable; the Larson-Penston (attractor) solution is stable for this mode for 0

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