where
indicates the Eulerian perturbation of a quantity,
is the angular frequency of the mode as measured by a distant
inertial observer,
f
(r) represents the radial dependence of the perturbation, and
are the associated Legendre polynomials. Normal modes of
nonrotating stars are degenerate in
m
and it suffices to study the axisymmetric (m
=0) case.
The Eulerian perturbation in the fluid 4-velocity
can be expressed in terms of vector harmonics, while the metric
perturbation
can be expressed in terms of spherical, vector, and tensor
harmonics. These are either of ``polar'' or ``axial'' parity.
Here, parity is defined to be the change in sign under a
combination of reflection in the equatorial plane and rotation by
. A polar perturbation has parity
, while an axial perturbation has parity
. Because of the spherical background, the polar and axial
perturbations of a nonrotating star are completely decoupled.
A normal mode solution satisfies the perturbed gravitational field equations,
and the perturbation of the conservation of the stress-energy tensor,
with suitable boundary conditions at the center of the star
and at infinity. The latter equation is decomposed into an
equation for the perturbation in the energy density
and into equations for the three spatial components of the
perturbation in the 4-velocity
. As linear perturbations have a gauge freedom, at most six
components of the perturbed field equations (49
) need to be considered.
For a given pair (l,
m), a solution exists for any value of the frequency
, consisting of a mixture of ingoing and outgoing wave parts.
Outgoing quasi-normal modes are defined by the discrete set of
eigenfrequencies for which there are no incoming waves at
infinity. These are the modes that will be excited in various
astrophysical situations.
The main modes of pulsation that are known to exist in
relativistic stars have been classified as follows (
and
are typical frequencies and damping times of the most important
modes in the nonrotating limit):
For a more detailed description of various types of oscillation modes, see [179, 178, 225, 56, 177].
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Rotating Stars in Relativity
Nikolaos Stergioulas http://www.livingreviews.org/lrr-2003-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei.mpg.de |