Quasi-normal modes of oscillation

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

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,

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 $δ𝜀$ 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 (49

) 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):

Polar fluid modes are slowly damped modes analogous to the Newtonian fluid pulsations:
- f (undamental)-modes: surface modes due to the interface between the star and its surroundings ( $f0 ∼ 2 kHz$ , $τ0 < 1 s$ ),
- p(ressure)-modes: nearly radial ( $f > 4 kHz 0$ , $τ > 1 s 0$ ),
- g(ravity)-modes: nearly tangential, only exist in stars that are non-isentropic or that have a composition gradient or first order phase transition ( $f0 < 500 Hz$ , $τ0 > 5 s$ ).
Axial and hybrid fluid modes:
- inertial modes: degenerate at zero frequency in nonrotating stars. In a rotating star, some inertial modes are generically unstable to the CFS instability; they have frequencies from zero to kHz and growth times inversely proportional to a high power of the star’s angular velocity. Hybrid inertial modes have both axial and polar parts even in the limit of no rotation.
- r(otation)-modes: a special case of inertial modes that reduce to the classical axial r-modes in the Newtonian limit. Generically unstable to the CFS instability with growth times as short as a few seconds at high rotation rates.
Polar and axial spacetime modes:
- w(ave)-modes: Analogous to the quasi-normal modes of a black hole (very weak coupling to the fluid). High frequency, strongly damped modes ( $f0 > 6 kHz$ , $τ0 ∼ 0.1 ms$ ).

For a more detailed description of various types of oscillation modes, see [177 , 176 , 225 , 56 , 180 ].