6.2 Relativistic stars

For the time independent case the system of Equations (51, 52, 53) inside the star reduces to a 4th order system of ODEs [79]. One can then even treat it as two coupled time independent wave equations⁴. The first equation will correspond to the fluid and the second equation will correspond to spacetime perturbations. In this way one can easily work in the Cowling approximation (ignore the spacetime perturbations) if the aim is the calculation of the QNM frequencies of the fluid modes (, , , …) or the Inverse Cowling Approximation [22] (ignore the fluid perturbations) if the interest is in -modes. The integration procedure inside the star is similar to those used for Newtonian stars and involves numerical integration of the equations from the center towards the surface in such a way that the perturbation functions are regular at the center of the star and the Lagrangian variation of the pressure is zero on the surface (for more details refer to [141, 130]). The integrations inside the star should provide the values of the perturbation functions on the surface of the star where one has to match them with the perturbations of the spacetime described by Zerilli’s equation (21, 23, 24).

In principle the integrations of the wave equation outside the star can be treated as in the case of the black holes. Leaver’s method of continued fraction has been used in  [119, 138], Andersson’s technique of integration on the complex plane was used in [21] while a simple but effective WKB approach was used by Kokkotas and Schutz [130, 211].

Finally, there are a number of additional approaches used in the past which improved our understanding of stellar oscillations in GR. In the following paragraphs they will be discussed briefly.

• Resonance Approach. This method was developed by Thorne [199], the basic assumption being that there are no incoming or outgoing waves at infinity, but instead standing waves. Then by searching for resonances one can identify the QNM frequencies. The damping times can be estimated from the half-width of each resonance. This is a simple method and can be used for the calculations of the fluid QNMs. In a similar fashion Chandrasekhar and Ferrari [59] have calculated the QNM frequencies from the poles of the ratio of the amplitudes of the ingoing and outgoing waves.
• Direct Numerical Integration. This method was used by Lindblom and Detweiler [141] for the calculation of the frequencies and damping times of the -modes for various stellar models for thirteen different equations of state. In this case, after integration of the perturbation equations inside the star, one gets initial data for the integration of the Zerilli equation outside. The numerical integration is extended up to “infinity” (i.e. at a distance where the solutions of Zerilli equations become approximately simple sinusoidal ingoing and outgoing waves), where this solution is matched with the asymptotic solutions of the Zerilli equations which describe ingoing and outgoing waves. The QNM frequencies are the ones for which the amplitude of the incoming waves is zero. This method is more accurate than the previous one at least in the calculation of the damping times as has been verified in [19], but still is not appropriate for the calculation of the -modes.
• Variational Principle Approach. Detweiler and Ipser [78] derived a variational principle for non-radial pulsational modes. Associated with that variational principle is a conservation law for the pulsational energy in the star. The time rate of change of that pulsational energy, as given by the variational principle, is equal to minus the power carried off by gravitational waves. This method was used widely for calculating the  [74] and -modes [87] and in studies of stability [78, 75].
• WKB. This is a very simple method but quite accurate, and contrary to the previous three methods it can be used for the calculation of the QNM frequencies of the -modes (this was the first method used for the derivation of these modes). In practice one substitutes the numerical solutions of the Zerilli equation with their approximate WKB solutions and identifies the QNM frequencies as the values of the frequency for which the amplitude of the incoming waves is zero [130, 211].
• WKB-Numerical. This is a combination of direct integration of the Zerilli equation and the WKB method. The trick is that instead of integrating outwards, one integrates inwards (using initial data at infinity for the incoming wave solution); this procedure is numerically more stable than the outward integration. On the stellar surface one needs of course not the solution for incoming waves but the one for outgoing ones. But through WKB one can derive approximately the value of the outgoing wave solution from the value of the incoming wave solution. In this way errors are introduced, nevertheless the results are quite accurate [130].