where Q is a scalar unperturbed quantity, is the angular frequency of the mode, as measured by a distant inertial observer and f (r) represents the radial dependence of the perturbation. Normal modes of nonrotating stars are degenerate in m, and it suffices to study the axisymmetric (m =0) case.
The perturbation of the metric, , can be expressed in terms of spherical, vector and tensor harmonics. These are either of ``polar'' or ``axial'' parity. Here, parity is defined as 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
For given (l, m), a solution exists for any value of the eigenfrequency , and it consists of ingoing- and outgoing-wave parts. Outgoing 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):
Are slowly damped modes analogous to the Newtonian fluid pulsations:
|Rotating Stars in Relativity
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