Quasicircular binary data

3.4 Quasicircular binary data

One of the primary driving forces behind the development of black-hole initial data has been the two-body problem of general relativity: the inspiral and coalescence of a pair of black holes. This problem is of fundamental importance. Not only is the relativistic two-body problem the most fundamental dynamical problem of general relativity, it is also considered one of the most likely candidates for observation with the upcoming generation of gravitational wave laser interferometers. Because of the circularizing effects of gravitational radiation damping, we expect the orbits of most tight binary systems to have small eccentricities. It is therefore desirable to have a method that can discern which data sets, within the very large parameter space of binary black-hole initial-data sets, correspond to black-hole binaries in a nearly circular (quasicircular) orbit.

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 [87 ]. 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 $EADM$ is the total ADM energy of the system measured at infinity, and $M1$ and $M2$ are the masses of the individual black holes. Quasicircular orbital configurations are obtained by minimizing the effective potential (defined as the nondimensional binding energy $E ∕μ b$ (where $μ ≡ M M ∕(M + M ) 1 2 1 2$ ) as a function of separation, while keeping the ratio of the masses of the black holes $M1 ∕M2$ , the spins of the black holes $2 S1∕M 1$ and $2 S2∕M 2$ , and the total angular momentum $J∕(M1M2 )$ constant.

This approach is limited primarily by the ambiguity in defining the individual masses of the black holes, $M 1$ and $M 2$ . 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. [87 ]). 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 [87 ], 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 [88 ].

To date, the results of the effective-potential method have not matched well to the best result from post-Newtonian approximations [47 ]¹¹ . 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.