Lee Samuel Finn, Northwestern
Since late 1993, a wide collaboration of relativists have been engaged in an effort to solve numerically for the final inspiral and coalescence of a binary black hole system. A quantitative understanding of black hole binary coalescence is needed to complete our solution of the relativistic Kepler problem, whose beginnings (in a nearly Newtonian binary undergoing slow, adiabatic inspiral) and endings (in a quiescent, single Kerr black hole) are already understood separately. The gravitational radiation arising from this final stage of binary inspiral/coalescence may also be detectable in the interferometric detectors now under construction; thus, the waveforms predicted by these calculations may play an important role in the associated data analysis.
To connect the initial and final states of the relativistic Kepler problem, or to use the predicted waveforms to learn something of the character of an observed coalescing binary, it is necessary that the initial data for the numerical calculation be firmly related to a binary system involving two distinct black holes of definite mass and spin in an orbit of certain energy and orbital angular momentum. Herein lies two problems:
To highlight the urgency of these problems, the Binary Black Hole Grand Challenge Alliance sponsored a one-day meeting at Caltech on 27 July 1996. This meeting, hosted by the Caltech Relativity Group, brought together, in person or by teleconference from Cardiff and Potsdam, many of the experts in the fields of post-Newtonian binary evolution calculations and numerical relativity for a discussion of these problems and possible approaches to their solution.
The meeting began with an overview by Richard Matzner, principal-investigator of the Binary Black Hole Grand Challenge Team, on the project status, followed by a presentation by Takashi Nakamura on the on-going efforts in Japan to approach the same problem. Discussion then turned, with presentations by Nakamura, Sasaki and Wiseman, and by Seidel and Matzner, to the second question described above: what is the minimum separation for which existing post-Newtonian methods can give reliable results for a symmetric black hole binary, and what is the maximum separation at which the numerical calculations can begin if they are to carry the evolution reliably through coalescence to the final state of a single, perturbed black hole?
Several proposals were discussed for bridging the gap between the ending point of the reliable perturbative techniques used for the adiabatic inspiral and the fully numerical techniques being pursued for the coalescence. Two of these proposals convey the range of options discussed. Steve Detweiler described very promising work, just nearing completion, on a post-Minkowskii approximation scheme for iteratively constructing spacetimes (not spacetime slices) that satisfy the full field equations to fixed order in G. On the other hand, Kip Thorne suggested that an adiabatic approximation to the field equations be sought that would allow the numerical solution to be carried out from larger separations. One element of this approximation, which would deal with the ``dynamics'' associated with the motion of the black holes through the coordinate grid used in the numerical calculations (but not the dynamics associated with the physical propagation of radiation), is the use of a coordinate system that co-rotates with the binary. Such a coordinate system introduces a light-cylinder, where the character of the coordinates change (some of the coordinates becoming light-like as one crosses the cylinder), and there was considerable discussion over the difficulties of handling this transition region, posing boundary conditions, and identifying the other components of the adiabatic approximation.
The discussion then turned briefly to the problem of identifying the numerical initial data for the coalescence calculations with a binary evolved from large separations. Here, again, discussion covered the full range of possibilities. Larry Kidder discussed a method under investigation with Sam Finn where the multipolar decomposition of the spatial metric and extrinsic curvature on a near-zone two-sphere surrounding the binary in the numerical initial data slice is compared to an identical decomposition of a similar slice through, e.g., a post-Newtonian spacetime. In the restricted context of binary black hole initial data and a point-mass binary post-Newtonian spacetime, intuition suggests that agreement of the moments with suggests that the evolution of the numerical initial data represents an approximate continuation of the binary system evolved by post-Newtonian (or other) means from large separation, and that this approximation should become better as increases. The principle concern, voiced by Kip Thorne, is the identification of a prescription that identifies unambiguously equivalent two-spheres and multipole moments in the numerical initial data slice and the post-Newtonian spacetime. On the other hand, Lee Lindblom suggested that if the evolution scheme used for the early, adiabatic inspiral could be made sufficiently accurate ( i.e., satisfy the constraints with sufficiently small residuals) at small separation, that a slice through the resulting spacetime could be used directly for as initial data for the fully numerical evolution, thus eliminating the ``seam'' that Kidder and Finn were attempting to sew.
Finally, Richard Price described on-going work with Andrew Abrahams, Jorge Pullin and other collaborators on ``naive'' application of perturbation theory to black hole coalescence. Following-up on earlier work by Abrahams and Cook, Abrahams, Price, Pullin and collaborators use either the Zerilli equation for Schwarzschild perturbations or the Teukolsky equation for perturbations of Kerr to evolve the highly perturbed single black holes that exist immediately following the formation of a single event horizon in a binary black hole coalescence. Doing so, they have found a remarkable and unexpected agreement with the radiated energy of the fully numerical simulation. This work suggests that, for at least some purposes, the validity of black hole linear perturbation theory may extend far into the regime traditionally considered a large perturbation.