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 [2]. This scheme has been used by Price and Pullin [170] and Abrahams and Cook [1] 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 [3]). 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 [6
,
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 [170] was so successful, because whatever initial data you provide
inside the
r
< 3
M
region (the peak of the potential barrier is around 3
M) 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 [7]). 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.
![]() |
Quasi-Normal Modes of Stars and Black Holes
Kostas D. Kokkotas and Bernd G. Schmidt http://www.livingreviews.org/lrr-1999-2 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |