Binary neutron stars have always been of great interest to relativists and astrophysicists. Binary neutron stars are known to exist. Approximately a half-dozen have been identified to date in our own galaxy, and, for some of these, general relativistic effects in the binary orbit have been measured to high precision. The discovery of the first binary pulsar, PRS 1913 + 16, by Hulse and Taylor (1975), led to the observational confirmation of Einstein's quadrupole formula for gravitational wave emission in the slow-motion, weak-field regime of general relativity. The inspiral and coalescence of binary neutron stars is one of the most promising scenarios for the generation of gravitational waves detectable by laser interferometers. With the construction of the first of these interferometers completed, and planned upgrades already scheduled, it is of growing urgency that theorists be able to predict the gravitational waveform emitted during the merger of the two stars. The low-frequency inspiral waveform is emitted early on, before tidal distortions of the stars become important, and it can be calculated fairly accurately by performing high-order post-Newtonian expansions of the equations of motion for two point masses. The high-frequency coalescence waveform is emitted at the end, during the epoch of tidal distortion, disruption and merger, and it requires the combined machinery of relativistic hydrodynamics (or MHD) and numerical relativity. These tools are necessary to determine not only the waveform in the strong-field regime but also the final fate of the merged remnant. One of the key issues is determining whether a merged remnant collapses to a black hole immediately after coalescence (“prompt collapse”) or instead forms a transient, dynamically stable, differentially rotating, hypermassive star that only later undergoes collapse due to dissipative secular effects (“delayed collapse”). These different outcomes will leave distinguishing imprints on the late-epoch gravitational waveform.

In this chapter we wish to construct initial data for quasiequilibrium binary black holes. That is, we seek solutions corresponding to two black holes in stable, nearly circular orbit about each other. In contrast to Newtonian theory, a stellar binary in general relativity can never be in strict equilibrium, with the companions moving in exactly circular orbits at constant separation for all time. Instead, gravitational radiation emission inevitably leads to loss of orbital energy and angular momentum, causing the orbit to decay. The resulting trajectory then traces out an inspiral rather than a perfect circle. For sufficiently large separations, the binary motion is nearly Newtonian, hence the orbit is nearly circular, decaying very little during one orbital period. If isolated from outside perturbations (e.g., gravitational encounters with other stars), it is expected that astrophysical binaries composed of compact stars (i.e., compact binaries) will ultimately evolve to a quasiequilibrium state following their formation at large separation. The reason is that gravitational radiation loss drives orbital circularization as well as decay, as we will discuss in the next section. Only when the orbits become very close and highly relativistic, just prior to radial plunge and binary merger, do the deviations from circular motion become large.

The construction of quasiequilibrium binary initial data poses a number of conceptual challenges. Getting started, however, is fairly straightforward. To find solutions we shall follow the approaches outlined in Chapter 3.

What is numerical relativity?

General relativity – Einstein's theory of relativistic gravitation – is the cornerstone of modern cosmology, the physics of neutron stars and black holes, the generation of gravitational radiation, and countless other cosmic phenomena in which strong-field gravitation plays a dominant role. Yet the theory remains largely untested, except in the weak-field, slow-velocity regime. Moreover, solutions to Einstein's equations, except for a few idealized cases characterized by high degrees of symmetry, have not been obtained as yet for many of the important dynamical scenarios thought to occur in nature. With the advent of supercomputers, it is now possible to tackle these complicated equations numerically and explore these scenarios in detail. That is the main goal of numerical relativity, the art and science of developing computer algorithms to solve Einstein's equations for astrophysically realistic, high-velocity, strong-field systems.

Numerical relativity has become one of the most powerful probes of relativistic spacetimes. It is the tool that allows us to recreate cataclysmic cosmic phenomena that are otherwise inaccessible in the conventional laboratory – like gravitational collapse to black holes and neutron stars, the inspiral and coalescence of binary black holes and neutron stars, and the generation and propagation of gravitational waves, to name a few. Numerical relativity picks up where post-Newtonian theory and general relativistic perturbation theory leave off. It enables us to follow the full nonlinear growth of relativistic instabilities and determine the final fate of unstable systems.

Binary black hole–neutron stars have received significantly less attention than binary black holes or binary neutron stars. No black hole–neutron star binary has been identified to date. However, stellar population synthesis models suggest that such systems represent a significant fraction of all compact binary mergers ultimately visible in gravitational waves by the LIGO detector. In addition, the study of black hole–neutron star mergers is important in light of the localizations of short-hard gamma-ray bursts. These GRB sources are found in galactic regions of low star-formation devoid of supernovae associations, ruling out massive stars as progenitors: massive stars have very short lifetimes and would need to be replenished more rapidly than is possible in low star-formation regions to account for these bursts. A more plausible progenitor for a short-hard GRB is a compact binary containing a neutron star, i.e., either a binary neutron star or binary black holeneutron star. The short-hard burst time scales and energetics are consistent with GRB models based on the coalescence of such compact binaries, and the evolution time scale of over 1 Gyr between formation and merger is consistent with the low star-formation rate.

Black hole-neutron star binaries can merge in two distinct ways. The neutron star may either be tidally disrupted by the black hole companion before being consumed, or it may be swallowed by the black hole more or less intact.

Spherically symmetric spacetimes, which we discussed in Chapter 8, do not admit gravitational radiation. Once we relax this symmetry restriction, as we shall do in the following chapters, we will encounter spacetimes that do contain gravitational radiation. In fact, simulating promising sources of gravitational radiation and predicting their gravitational wave signals are among the most important goals of numerical relativity. These goals are especially urgent in light of the new generation of gravitational wave laser interferometers which are now operational. A book on numerical relativity therefore would not be complete without a discussion of gravitational waves.

In this chapter we review several topics related to gravitational waves. We start in Section 9.1 with a discussion of linearized waves propagating in nearly Minkowski spacetimes and the role that these waves play even in the case of nonlinear sources of gravitational radiation. In Section 9.2 we survey plausible sources of gravitational waves, highlighting those that seem most promising from the perspective of gravitational wave detection. We briefly describe some of the existing and planned gravitational wave detectors in Section 9.3. Finally, in Section 9.4 we make contact with numerical relativity, and review different strategies that have been employed to extract gravitational radiation data from numerical relativity simulations.

Linearized waves

Most of this book deals with strong-field solutions of Einstein's equations, including black holes, neutron stars, and binaries containing these objects. As long as these solutions are dynamical and nonspherical, they will emit gravitational radiation.