3.1 Quasi-normal modes of oscillation

A general linear perturbation of the energy density in a static and spherically symmetric relativistic star can be written as a sum of quasi-normal modes that are characterized by the indices (l,m ) of the spherical harmonic functions Ylm and have angular and time dependence of the form
δ𝜀 ∼ f(r)P m(cos 𝜃)ei(mĪ•+ωit), (48 ) l
where δ indicates the Eulerian perturbation of a quantity, ωi is the angular frequency of the mode as measured by a distant inertial observer, f(r) represents the radial dependence of the perturbation, and P m (cos 𝜃) l 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 δua can be expressed in terms of vector harmonics, while the metric perturbation δgab 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 π. A polar perturbation has parity l (− 1), while an axial perturbation has parity l+1 (− 1). 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,

δ(Gab − 8πT ab) = 0, (49 )
and the perturbation of the conservation of the stress-energy tensor,
δ(∇aT ab) = 0, (50 )
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 δ𝜀 and into equations for the three spatial components of the perturbation in the 4-velocity δua. As linear perturbations have a gauge freedom, at most six components of the perturbed field equations (49View Equation) need to be considered.

For a given pair (l,m ), a solution exists for any value of the frequency ωi, 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 (f0 and τ0 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 and hybrid fluid modes:
  3. Polar and axial spacetime modes:

For a more detailed description of various types of oscillation modes, see [17717622556180].

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