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*Astron. Astrophys. 354, 1115-1122 (2000)*
## Liouville's equation in post Newtonian approximation
### II. The post Newtonian modes
**
Y. Sobouti**^{ 1,2,3} and
V. Rezania^{ 1}
^{1} Institute for Advanced Studies in Basic Sciences, P.O. Box 45195-159, Zanjan 45195, Iran
^{2} Physics Department, Shiraz University, Shiraz 71454, Iran
^{3} Center for Theoretical Physics and Mathematics, AEOI, P.O. Box 11345-8486,Tehran, Iran (sobouti@iasbs.ac.ir; rezania@iasbs.ac.ir)
*Received 23 April 1998 / Accepted 3 December 1999*
**Abstract**
We use the post-Newtonian (*pn*) order of Liouville's equation
to study the normal modes of oscillation of a spherically symmetric
relativistic system. Perturbations that are neutral in Newtonian
approximation develop into a new sequence of normal modes. In the
first *pn* order; a) their frequency is an order *q* smaller
than the classical frequencies, where *q* is a *pn*
expansion parameter; b) they are not damped, for there is no
gravitational wave radiation in this order; c) they are not coupled
with the classical modes in *q* order; d) because of the
spherical symmetry of the underlying equilibrium configuration, they
are designated by a pair of angular momentum eigennumbers,
(), of a pair of phase space angular
momentum operators (). The
eigenfrequencies are, however, *m*-independent. Hydrodynamics of
these modes is also investigated; a) they generate oscillating
macroscopic toroidal motions that are neutral in the classical case;
and b) they give rise to an oscillatory
component of the metric tensor that
otherwise is zero in the unperturbed system. The conventional
classical modes, which in their hydrodynamic behaviour emerge as
*p* and *g* modes are, of course, perturbed to order
*q*. These, however, have not been of concern in this paper.
**Key words:** methods:
numerical
stars: general
stars: oscillations
This article contains no SIMBAD objects.
### Contents
© European Southern Observatory (ESO) 2000
Online publication: February 25, 2000
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