3.2 Effect of Rotation on 3 Oscillations and Stability3 Oscillations and Stability

3.1 Quasi-Normal Modes of Oscillation

The spacetime of a nonrotating star is static and spherically symmetric. A general linear perturbation can be written as a sum of quasi-normal modes that are characterized by the indices (l, m) of the spherical harmonic tex2html_wrap_inline2203 and have angular and time-dependence of the form


where Q is a scalar unperturbed quantity, tex2html_wrap_inline2207 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, tex2html_wrap_inline2213, 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 tex2html_wrap_inline2215 . A polar perturbation has parity tex2html_wrap_inline2217, while an axial perturbation has parity tex2html_wrap_inline2219 . 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 tex2html_wrap_inline2207, 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 (tex2html_wrap_inline2225 and tex2html_wrap_inline2227 are typical frequencies and damping times of the most important modes in the nonrotating limit):

  1. Polar fluid modes

    Are slowly damped modes analogous to the Newtonian fluid pulsations:

  2. Axial fluid modes
  3. Polar and axial spacetime modes
For a more detailed description of various modes see [86, 87, 88, 89, 90].

3.2 Effect of Rotation on 3 Oscillations and Stability3 Oscillations and Stability

image Rotating Stars in Relativity
Nikolaos Stergioulas
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de