Liouville's equation in post Newtonian approximation
II. The post Newtonian modes
Y. Sobouti 1,2,3 and
V. Rezania 1
Received 23 April 1998 / Accepted 3 December 1999
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
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© European Southern Observatory (ESO) 2000
Online publication: February 25, 2000