6.2 Relativistic stars

Although the perturbation equations in the exterior of a star are similar to those of the black-hole and the techniques described earlier can be applied here as well, special attention must be given to the interior of the star where the perturbation equations are more complicated.

For the time independent case the system of Equations (51View Equation52View Equation53View Equation) 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 (f, p, g, …) or the Inverse Cowling Approximation [22] (ignore the fluid perturbations) if the interest is in w-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 [141Jump To The Next Citation Point130Jump To The Next Citation Point]). 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 (21View Equation23View Equation24View Equation).

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  [119138], Andersson’s technique of integration on the complex r plane was used in [21] while a simple but effective WKB approach was used by Kokkotas and Schutz [130Jump To The Next Citation Point211Jump To The Next Citation Point].

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.


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