Many numerical codes now under development to solve Einstein's equations of general relativity in 3+1 dimensional spacetimes employ the standard ADM form of the field equations. This form involves evolution equations for the raw spatial metric and extrinsic curvature tensors. Following Shibata and Nakamura, we modify these equations by factoring out the conformal factor and introducing three "connection functions". The evolution equations can then be reduced to wave equations for the conformal metric components, which are coupled to evolution equations for the connection functions. We evolve small amplitude gravitational waves and make a direct comparison of the numerical performance of the modified equations with the standard ADM equations. We find that the modified form exhibits much improved stability.

We construct relativistic equilibrium models of differentially rotating neutron stars and show that they can support significantly more mass than their nonrotating or uniformly rotating counterparts. We dynamically evolve such "hypermassive" models in full general relativity and show that there do exist configurations that are dynamically stable against radial collapse and bar formation. Our results suggest that the remnant of binary neutron star coalescence may be temporarily stabilized by differential rotation, leading to delayed collapse and a delayed gravitational wave burst.

We perform hydrodynamical simulations of neutron-star mergers for a large sample of temperature-dependent, nuclear equations of state, and determine the threshold mass above which the merger remnant promptly collapses to form a black hole. We find that, depending on the equation of state, the threshold mass is larger than the maximum mass of a nonrotating star in isolation by between 30 and 70 per cent. Our simulations also show that the ratio between the threshold mass and maximum mass is tightly correlated with the compactness of the nonrotating maximum-mass configuration. We speculate on how this relation can be used to derive constraints on neutron-star properties from future observations.

Black hole-neutron star (BHNS) binary mergers are candidate engines for generating both shorthard gamma-ray bursts (SGRBs) and detectable gravitational waves. Using our most recent conformal thin-sandwich BHNS initial data and our fully general relativistic hydrodynamics code, which is now AMR-capable, we are able to efficiently and accurately simulate these binaries from large separations through inspiral, merger, and ringdown. We evolve the metric using the BSSN formulation with the standard moving puncture gauge conditions and handle the hydrodynamics with a high-resolution shock-capturing scheme. We explore the effects of BH spin (aligned and anti-aligned with the orbital angular momentum) by evolving three sets of initial data with BH:NS mass ratio q = 3: the data sets are nearly identical, except the BH spin is varied between a/MBH = −0.5 (anti-aligned), 0.0, and 0.75. The number of orbits before merger increases with a/MBH, as expected. We also study the nonspinning BH case in more detail, varying q between 1, 3, and 5. We calculate gravitational waveforms for the cases we simulate and compare them to binary black-hole waveforms. Only a small disk (< 0.01M⊙) forms for the anti-aligned spin case (a/MBH = −0.5) and for the most extreme mass ratio case (q = 5). By contrast, a massive (M disk ≈ 0.2M⊙), hot disk forms in the rapidly spinning (a/MBH = 0.75) aligned BH case. Such a disk could drive a SGRB, possibly by, e.g., producing a copious flux of neutrino-antineutino pairs.

Numerical relativity is the most promising tool for theoretically modeling the inspiral and coalescence of neutron star and black hole binaries, which, in turn, are among the most promising sources of gravitational radiation for future detection by gravitational wave observatories. In this article we review numerical relativity approaches to modeling compact binaries. Starting with a brief introduction to the 3+1 decomposition of Einstein's equations, we discuss important components of numerical relativity, including the initial data problem, reformulations of Einstein's equations, coordinate conditions, and strategies for locating and handling black holes on numerical grids. We focus on those approaches which currently seem most relevant for the compact binary problem. We then outline how these methods are used to model binary neutron stars and black holes, and review the current status of inspiral and coalescence simulations.

Black hole-neutron star (BHNS) binaries are expected to be among the leading sources of gravitational waves observable by ground-based detectors, and may be the progenitors of short-hard gamma ray bursts (SGRBs) as well. We discuss our new fully general relativistic calculations of merging BHNS binaries, which use high-accuracy, low-eccentricity, conformal thin-sandwich configurations as initial data. Our evolutions are performed using the moving puncture method and include a fully relativistic, high-resolution shock-capturing hydrodynamics treatment. Focusing on systems in which the neutron star is irrotational and the black hole is nonspinning with a 3:1 mass ratio, we investigate the inspiral, merger, and disk formation in the system. We find that the vast majority of material is promptly accreted and no more than 3% of the neutron star's rest mass is ejected into a tenuous, gravitationally bound disk. We find similar results for mass ratios of 2:1 and 1:1, even when we reduce the NS compaction in the 2:1 mass ratio case. These ambient disks reach temperatures suitable for triggering SGRBs, but their masses may be too small to produce the required total energy output. We measure gravitational waveforms and compute the effective strain in frequency space, finding measurable differences between our waveforms and those produced by binary black hole mergers within the advanced LIGO band. These differences appear at frequencies corresponding to the emission that occurs when the NS is tidally disrupted and accreted by the black hole. The resulting information about the radius of the neutron star may be used to constrain the neutron star equation of state.

Aimed at students and researchers entering the field, this pedagogical introduction to numerical relativity will also interest scientists seeking a broad survey of its challenges and achievements. Assuming only a basic knowledge of classical general relativity, the book develops the mathematical formalism from first principles, and then highlights some of the pioneering simulations involving black holes and neutron stars, gravitational collapse and gravitational waves. The book contains 300 exercises to help readers master new material as it is presented. Numerous illustrations, many in color, assist in visualizing new geometric concepts and highlighting the results of computer simulations. Summary boxes encapsulate some of the most important results for quick reference. Applications covered include calculations of coalescing binary black holes and binary neutron stars, rotating stars, colliding star clusters, gravitational and magnetorotational collapse, critical phenomena, the generation of gravitational waves, and other topics of current physical and astrophysical significance.

We perform fully relativistic calculations of binary neutron stars in quasiequilibrium circular orbits. We integrate Einstein's equations together with the relativistic equation of hydrostatic equilibrium to solve the initial-value problem for equal-mass binaries of arbitrary separation. We construct sequences of constant rest mass and identify the innermost stable circular orbit and its angular velocity. We find that the quasiequilibrium maximum allowed mass of a neutron star in a close binary is slightly larger than in isolation. [S0031-9007(97) PACS numbers: 04.25. Dm, 04.30.Db, 04.40.Dg, 97.60.Jd The two-body problem is one of the outstanding, unsolved problems in classical general relativity. And yet, neutron star binary systems are known to exist, even within our own galaxy [1]. For some of these systems (including PSR B1913+16 and B1534+12) general relativistic orbital effects have been measured to high precision [2]. Binary neutron stars are among the most promising sources for gravitational wave detectors now under construction, like LIGO, VIRGO, and GEO. This has triggered an intense theoretical effort to predict the gravitational wave form emitted during the inspiral and coalescence of the two stars.Much of the work on binary neutron stars has been performed within the framework of Newtonian hydrodynamics [3]. Many investigators have also studied the problem in post-Newtonian (PN) theory. As long as the PN stars are well separated, they can be approximated by point sources [4], but for close binaries, hydrodynamical effects must also be taken into account [5][6][7][8][9].Fully general relativistic treatments of the problem are complicated by the nonlinearity of Einstein's equations and the requirement of very large computational resources. Numerical simulations are currently only in their infancy [6]. Recently, Wilson and Mathews [10] reported preliminary results obtained with a relativistic numerical evolution code. Their dynamical calculations suggest that the neutron stars may collapse to black holes prior to merger. They also find that, typically, binaries have too large a total angular momentum to form a Kerr black hole immediately upon merger (see also [11]). Their results are in disagreement with predictions of Newtonian [12] and PN calculations [7], which show that tidal fields stabilize neutron stars against radial collapse.In this Letter we report the first calculations in full relativity of quasiequilibrium, equal mass, neutron star binaries in synchronized circular orbits. We numerically integrate a subset of the Einstein equations, coupled to the equations of relativistic hydrodynamics, to solve the initial value problem for binaries. Such quasiequilibrium models provide initial data for future dynamical evolution calculations. We construct quasiequilibrium sequences of constant rest mass configurations at varying separation. These sequences mimic evolutionary sequences in which the stars undergo slow inspiral on nearly circular orbits due to the emission of gravitational waves. We id...