The goal of numerical relativity is to study spacetimes that cannot be studied by analytic means. The focus is therefore primarily on dynamical systems. Numerical relativity has been applied in many areas: cosmological models, critical phenomena, perturbed black holes and neutron stars, and the coalescence of black holes and neutron stars, for example. In any of these cases, Einstein’s equations can be formulated in several ways that allow us to evolve the dynamics. While Cauchy methods have received a majority of the attention, characteristic and Reggi calculus based methods have also been used. All of these methods begin with a snapshot of the gravitational fields on some hypersurface, the initial data, and evolve these data to neighboring hypersurfaces.
The focus of this review is on the initial data needed for Cauchy evolutions of Einstein’s equations. These initial data cannot be freely specified in their entirety. Rather they are subject to certain constraints which must be satisfied. Because of the nonlinearity of Einstein’s equations, there is no unique way of choosing which pieces of the initial data can be freely specified and which are constrained. In Section 2, I will look at the various formalisms that exist for expressing the initial-value equations. I will use the 3 + 1 (or ADM) decomposition of Einstein’s equations throughout and I begin the review with a brief introduction of this in Section 2.1. In Section 2.2, Section 2.3, and Section 2.4, I will explore the most important and widely used decompositions that have been developed.
In the remainder of the review, I will focus on initial data for black holes and neutron stars. Section 3 deals with black-hole initial data, which has received considerable attention over the years. The evolution of black-hole spacetimes is of particular importance because it allows for the study of pure geometrodynamics. The spacetime is either pure vacuum or any matter can be hidden behind an event horizon. Thus, black-hole spacetimes allow us to study the dynamics of gravity alone. In Section 3.1, I begin with a review of some of the classic analytic solutions of the initial-data equations. Section 3.2 covers current methods for constructing general multi-black-hole initial data sets and also explores some of the limitations of these data sets. Section 3.3 deals with a class of black-hole initial data that has received little attention until recently and which I feel may play an important role in future work on constructing black-hole initial data. Finally, Section 3.4 examines the issue of constructing black-hole initial data for quasiequilibrium binaries.
This last topic is of extreme importance. Several laser interferometer gravitational wave detectors will become operational in the near future (cf. Ref.  for a Living Review on this topic) and the coalescence of black-hole binaries is considered to be one of the strongest candidates for detection by the earliest generation of detectors. The chances of detecting these events and then unraveling the information contained in the gravitational wave signals will be greatly increased if we have accurate numerical simulations of black-hole binary coalescence. While post-Newtonian techniques can be used to simulate the inspiral of a compact binary system, the final plunge and coalescence must be simulated numerically. Astrophysically realistic initial data will be needed before these simulations can provide reliable results.
The final plunge and coalescence of neutron-star binaries must also be simulated numerically. Section 4 examines the construction of neutron-star initial data. The major difference between black-hole and neutron-star initial data is the need to deal with the neutron star’s matter. In this section, I will deal with the neutron star matter in general, without considering any particular equation of state. In Section 4.1, I look at the issue of hydrostatic equilibrium of matter. Section 4.2 takes a brief look at constructing equilibrium initial data for isolated neutron stars. Finally, Section 4.3 examines the issues involved in constructing quasiequilibrium neutron-star binary initial data. In particular, we look at recent advances in constructing irrotational fluid models.
This article is far from a complete review of all the work that has been done on initial data for numerical relativity. I offer my apologies to all whose work I have not included. In particular, I have not covered any of the issues associated with existence and uniqueness of solutions to the constraint equations, nor dealt extensively with the issue of asymptotic falloff rates. I welcome correspondence on topics that you feel should be covered in this review, references that you think should be included, and the inevitable typographical errors.
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