Numerical Techniques

6 Numerical Techniques

For each candidate source of gravitational waves, gravitational wave astronomy needs answers to two questions. Firstly, how much energy will be carried away by the emitted gravitational waves? Secondly, which are the “preferred” frequencies at which a $10 M ⊙$ black hole or a neutron star will oscillate? The answer to the first question is that the energy will depend on the degree of asymmetry that the process generates, and it will depend critically on the initial data. In the previous section we have tried to provide some guesstimates for the energy emitted during the oscillation phase of black holes and neutron stars. The answer to the second question is related to the numerical solution of the perturbation equations. The numerical schemes developed for this purpose will be described in this section.

Let us describe why the numerical calculation of quasi-normal mode frequencies is delicate. Consider again the case treated in Section 2 of the wave equation with a potential with compact support. We try to find a complex number $s$ with negative real part such that the solution which is $−sx e$ for large positive $x$ , is $esx$ for large negative $x$ . Note that these solutions grow exponentially with $|x|$ and therefore one has to be very careful to make sure that there is no exponentially decaying part in the solution. The situation becomes even more complicated if we do not know $f±$ explicitly because one can not characterize the correct solution by some growth property. This is for example the case for the Schwarzschild solution.

6.1 Black holes
  6.1.1 Evolving the time dependent wave equation
  6.1.2 Integration of the time independent wave equation
  6.1.3 WKB methods
  6.1.4 The method of continued fractions
6.2 Relativistic stars