Initial Data for Binary Systems

Gregory B. Cook, Wake Forest University

The Center for Gravitational Wave Physics at the Pennsylvania State University held its first Focus Session on March 29-30, 2002. The topic was ``Initial Data for Binary Systems''. Organized by Pablo Laguna and myself, the meeting hosted eight invited speakers and thirty participants. The primary goal of the workshop was to foster discussion on current open questions and possible future directions related to the construction of astrophysically relevant initial data representing binary systems containing black holes and neutron stars. This is a very difficult standing problem for the LIGO/LISA source-modeling community. There are currently a number of different approaches that can be used to construct compact-binary initial data. Unfortunately, the only thing we are sure of is that, in one way or another, all of these current methods fall short of the goal of representing astrophysical systems with sufficient fidelity. With these initial data serving as the starting point for full numerical simulations of the plunge and coalescence of compact binaries, it is clear that, at the very least, the quality of the initial data needs to be well understood.

Why is it so hard to construct realistic initial data for compact binary systems? A combination of factors come into play. The initial value problem of general relativity is a constrained hyperbolic system. At any given instant of time, the gravitational fields must satisfy four constraints that can be posed as a set of elliptic boundary-value equations. Solving these equations represents the primary computational difficulty in constructing initial data, and this aspect of the problem has received considerable attention. However, solving the constraints only fixes one-third of the degrees-of-freedom of the gravitational fields. The remaining degrees-of-freedom are divided evenly between the gauge freedom of the theory and the freely specifiable initial dynamical content of the gravitational fields. Historically, the choices for these latter degrees-of-freedom have been based largely on what would simplify the problem of solving the constraints, not on what would produce the most realistic data. There are of course additional problems. The exact definition of astrophysically realistic initial data is not fully understood. Furthermore, given only the initial data for a gravitational system, it is impossible (except for special cases) to determine its full physical content. This requires evolving the data.

The schedule for the Focus Session was designed to foster active discussion. There were four sessions over two days. Each session was limited to two half-hour invited talks, with each talk followed by an hour of discussion. Participants were encouraged to prepare one or two transparencies and to ask the session chairs for time to present these at an appropriate time during the discussion sessions. The first day offered talks by Peter Diener, Philippe Grandclément, Richard Matzner and myself.

Philippe Grandclément and I each presented new approaches for constructing binary black hole initial data. These approaches are very similar and try to extend to black holes an approach that has been very fruitful for the case of neutron stars. They incorporate an approximate helical Killing vector, or a notion of quasi-equilibrium, into the process of constructing the data. The most important feature of these approaches is that they employ a much different method for fixing the extrinsic curvature. For black holes, essentially all approaches for constructing initial data have used an analytic solution of the momentum constraints (the Bowen-York solution) to fix the extrinsic curvature. Recent attempts to improve black hole data have used superpositions of the extrinsic curvature of a boosted Kerr black hole as a foundation for the extrinsic curvature. A difficulty with either of these approaches is that they incorporate an unknown contribution to the freely specifiable dynamical content of the data. Loosely speaking, some amount of unphysical junk is built into the data. In the new approaches discussed, the only assumption built into the extrinsic curvature is that it should, at least instantaneously, lead to a stationary geometry. (Technically only the conformal three-geometry is instantaneously stationary.) This new approach for specifying the initial data for black-hole binaries in quasi-circular orbits seems to yield results that are in better agreement with post-Newtonian results.

Richard Matzner presented a talk on using a superposition of boosted Kerr geometries as background data for solving the constraints. He also discussed many issues related to the physics of close binaries: the meaning of the innermost stable circular orbit (ISCO), the effects of spin and frame dragging, and tidal forces. This use of superposed boosted Kerr geometries as background data for solving the constraint equations is among the first attempts to employ more realistic choices for the freely specifiable initial dynamical content of the initial data.

Another theme of the meeting was to explore the possibility of incorporating, into initial data, information from post-Newtonian solutions for binaries in circular orbits. Peter Diener presented what may be the first attempt to use this information in fixing the background fields for both the metric and extrinsic curvature. This approach relies on using post-Newtonian solutions in the ``ADM transverse-traceless'' gauge. It seems that a major difficulty with the idea of incorporating post-Newtonian information is that different parts of the post-Newtonian solution are only well defined in certain regions (near-zone, wave-zone, far-zone).

These issues were explored again on the second day of the Focus Session when talks were given by Bala Iyer, Thomas Baumgarte, Koji Uryu, and Olivier Sarbach. Bala Iyer gave an overview of how waveforms for inspiralling compact binaries are computed in post-Newtonian formalisms. This led to an extensive discussion about what information from the various post-Newtonian calculations could be incorporated into a background metric for use in solving the constraints. The consensus was that much of the wave information cannot be taken directly from current post-Newtonian calculations. One suggestion, however, was to use some kind of numerical post-Minkowski approach to obtain wave information that could be incorporated into a background metric.

Thomas Baumgarte and Koji Uryu each presented approaches that used sequences of individual initial-data sets to model the inspiral of a neutron-star binary system. While their approaches were rather different, both made the assumption of a quasi-adiabatic inspiral and computed the gravitational waveforms being produced. In Thomas Baumgarte's approach, each initial-data configuration was evolved in a full dynamical code, with the restriction that the neutron-star matter sources were ``frozen'' in the corotating frame. After several orbits, the configuration reached a steady state and the gravitational waves being emitted were extracted. Koji Uryu's approach differed in that it solved for a perturbation of the initial metric, evolved a linearized system, and expanded the perturbation in spherical harmonics. Although they followed different approaches, they both computed the gravitational-wave luminosity and then estimated the radiation reaction timescale in order to produce waveforms. These approaches seem to be promising avenues for computing the gravitational waveforms down to a point near the ISCO, and for providing corrections to the background metric which can be used to improve initial-data computations.

In the final talk, Olivier Sarbach presented results for a rather new approach for constructing black hole initial data. This approach differs from others in that it constructs initial data that is strictly of the Kerr-Schild type. While it isn't clear if this restriction will allow for constructing astrophysically realistic binary initial data, it is well suited for constructing data that can be used in ``close-limit'' perturbative evolutions. Such computations are especially useful for comparison with full nonlinear evolution codes.

I want to close by offering my thanks to the directors and staff of the Center for Gravitational Wave Physics for their support in running this Focus Session. They did a great job. Especially enjoyable was a wonderful banquet, hosted by Abbay Ashtekar, held Friday evening at a local Indian restaurant. A great time was had by all.

A full listing of the talks, along with copies of the speakers' slides, can be found at

Jorge Pullin 2002-09-23