A critical issue related to the discussions of the previous sections is the excitation of the QNMs. The truth is that, although the QNMs are predicted by our perturbation equations, it is not always clear which ones will be excited and under what initial conditions. As we have already mentioned in the introduction there is an excellent agreement between results obtained from perturbation theory and full nonlinear evolutions of Einstein equations for head-on collisions of two black holes. Still there is a degree of arbitrariness in the definition of initial data for other types of stellar or black hole perturbations. This due to the arbitrariness in specifying the gravitational wave content in the initial data.
The construction of acceptable initial data for the evolution of perturbation equations is not a trivial task. In order to specify astrophysically relevant initial data one should first solve the fully nonlinear 3-dimensional initial value problem for (say) a newly formed neutron star that settles down after core collapse or two colliding black holes or neutron stars. Afterwards, starting from the Cauchy data on the initial hypersurface, one can evolve forward in time with the linear equations of perturbation theory instead of the full nonlinear equations. Then most of the long-time evolution problems of numerical relativity (throat stretching when black holes form, numerical instabilities or effects due to the approximate outer boundary conditions) are avoided. Additionally, the interpretations of the computed fields in terms of radiation is immediate . This scheme has been used by Price and Pullin  and Abrahams and Cook  with great success for head-on colliding black holes. The success was based on the fact that the bulk of the radiation is generated only in the very strong-field interactions around the time of horizon formation and the radiation generation in the early dynamics can be practically ignored (see also the discussion in ). The extension of this scheme to other cases, like neutron star collisions or supernovae collapse, is not trivial. But if one can define even numerical data on the initial hypersurface then the perturbation method will be probably enough or at least a very good test for the reliability of fully numerical evolutions. For a recent attempt towards applying the above techniques in colliding neutron stars see [5, 20].
Before going into details we would like to point out an important issue, namely the effect of the potential barrier on the QNMs of black holes. That is, for any set of initial data that one can impose, the QNMs will critically depend on the shape of the potential barrier, and this is the reason that the close limit approximation of the two black-hole collision used by Price and Pullin  was so successful, because whatever initial data you provide inside the region (the peak of the potential barrier is around ) the barrier will “filter” them and an outside observer will observe only the QNM ringing (see for example recent studies by Allen, Camarda and Seidel ). This point of view is complementary to the discussion earlier in this section, since roughly speaking even before the creation of the final black hole the common potential barrier has been created and anything that was to be radiated had to be “filtered” by this common barrier.
© Max Planck Society and the author(s)