Currently, only one approach has been developed for locating
quasicircular orbits in a parameter space of binary black-hole
initial data [39]. It is based on the fact that minimizing the energy of a binary
system while keeping the orbital angular momentum fixed will
yield a circular orbit in Newtonian gravity. The idea does not
hold strictly for general relativistic binaries since they emit
gravitational radiation and cannot be in equilibrium. However,
for orbits outside the innermost stable circular orbit, the
gravitational radiation reaction time scale is much longer than
the orbital period. Thus it is a good approximation to treat such
binaries as an equilibrium system. Called an ``effective
potential method'', this approach was used originally to find the
quasicircular orbits and innermost stable circular orbit (ISCO)
for equal-sized nonspinning black holes [39]. In this work, the initial data for binary black holes were
computed using the conformal-imaging approach outlined in
Section
3.2.1
. The approach was also applied to binary black-hole data
computed using the puncture method [10], where similar results were found. Configurations containing a
pair of equal-sized black holes with spin also have been
examined [86
]. In this case, the spins of the black holes are equal in
magnitude, but are aligned either parallel to, or anti-parallel
to, the direction of the orbital angular momentum.
The approach defines an ``effective potential'' based on the binding energy of the binary. The binding energy is defined as
where
is the total ADM energy of the system measured at infinity, and
and
are the masses of the individual black holes. Quasicircular
orbital configurations are obtained by minimizing the effective
potential (defined as the nondimensional binding energy
(where
) as a function of separation, while keeping the ratio of the
masses of the black holes
, the spins of the black holes
and
, and the total angular momentum
constant.
This approach is limited primarily by the ambiguity in
defining the individual masses of the black holes,
and
. There is no rigorous definition for the mass of an individual
hole in a binary configuration and some approximation must be
made here. There is also no rigorous definition for the
individual spins of holes in a binary configuration. The problem
of defining the individual masses becomes particularly pronounced
when the holes are very close together (see Ref. [86
]). The limiting choice of conformal flatness for the 3-geometry
also has proven to be problematic. The effects of this choice
have been clearly seen in the case of quasicircular orbits of
spinning black holes [86], but it is also believed to be a serious problem for any binary
configuration because binary configurations are not conformally
flat at the second post-Newtonian order [87].
To date, the results of the effective-potential method have
not matched well to the best result from post-Newtonian
approximations [47]
. It will be interesting to see if the results from
post-Newtonian approximations and numerical initial-data sets
converge, especially when the approximation of conformal flatness
is eliminated.
![]() |
Initial Data for Numerical Relativity
Gregory B. Cook http://www.livingreviews.org/lrr-2000-5 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |