We are convinced of a genuine need for a monograph describing the many facets and new developments in numerical relativistic hydrodynamics. Such calculations are crucial to several areas of current research in the physics of stellar collapse, supernovae, and black hole formation, as well as the merging of the final orbits of coalescing binary neutron stars. Both problems are only now entering the level of sophistication where three-dimensional relativistic hydrodynamics simulations are both possible and necessary. In the former problem such calculations are crucial to understand the explosion mechanism. In the latter problem, a great deal of interest in such calculations has recently been inspired by the development of next-generation gravity wave detectors to search for such events, and as a possible explanation of the physics underlying observed astrophysical γ-ray bursts.

The field of numerical relativistic hydrodynamics has developed over the past 30 years, but there has not been written a technical text explaining the many techniques relevant to this discipline, many of which are much different than standard general relativity textbook approaches. This book will present such a review of techniques for numerical general relativistic hydrodynamics developed by one of the pioneers of this field over the past three decades.

We begin by developing the equations and differencing schemes for special relativistic hydrodynamics as an introduction to the metric formulation of the problems. Here, the basic numerical techniques and a number of test problems and applications will be discussed.

Following this, the formalism for matter flows in the curved spacetime of general relativity will be presented in the usual (3+1) formalism.

Progress in computing full general relativistic hydrodynamics in three spatial dimensions has been slow. The problem is not in the hydrodynamics, but in the solution of the field equations. The equation for extrinsic curvature, Kij, is particularly unstable. For example, at the writing of this book no strong field fully relativistic calculation has computed more than two orbits of a neutron star binary without becoming unstable. Currently, there seems to be some promise, however, in a modified version of the ADM equations based upon a conformal decomposition as originally proposed by Shibata and Nakamura and later reinvented by Baumgarte and Shapiro. At the end of this chapter we briefly summarize this method for completeness. First, however, we summarize a useful alternative which the authors have developed for solving strong field systems which avoids the nonlinearities of the full Einstein equations by reducing the problem to an implementation of constraint equations.

The conformally flat approximation

For most gravitating systems studied so far (e.g.), only a relatively small amount of energy is emitted by gravitational waves. Even for the merger of two black holes it is expected that only a few tenths of a percent of the rest mass will be radiated away in gravitation. For the case of two neutron stars we would not expect any more radiation to be emitted during the last few orbits than for a two black hole merger, i.e. during the inspiral, the radiated energy per orbit is a minuscule fraction of the energy in orbital motion. Furthermore, an explicit treatment of the radiation reaction is exceedingly difficult.

As discussed in Chapter 3, the treatment of the spherical collapse of a massive star to produce a supernova is sufficiently complex to warrant a separate chapter. It is believed that a Type II supernova arises from the delicate balance between energy deposited by escaping neutrinos from the core and the gravitational energy of collapsing outer layers. Thus, the relativistic energy and mass transport must be considered with high numerical accuracy to obtain a believable simulation. The model discussed herein includes the experience of about 30 years of development and should be of some guidance to those who wish to understand this fascinating phenomenon.

Collapse supernovae

A brief review of the scenario is as follows. Massive stars (i.e. 10 M/M⊙ 30) evolve until the iron core exceeds ∼1–1.3 M⊙. At this point there can be no more nuclear energy generation in the core. Neutrino emission, electron capture, and photodisintegration cool the inner ∼1 M⊙ and remove pressure support from the core. The central density then rises. When the central density ρc approaches ρc 109 g cm−3, neutrino emission is so large that collapse becomes supersonic, i.e vmax > cs, where cs is the speed of sound. As the core collapses, the inner ∼0.7 M⊙ collapses homologously (e.g.). Once the core density exceeds nuclear density, ρc > 2.5×1014 g cm−3, the pressure rises rapidly and collapse is halted. Matter continues to fall inward, however, so an outward moving shock wave is formed. This is referred to as the core bounce.

Most cosmology calculations utilize a homogeneous background spacetime in either a Friedmann–Robertson–Walker (FRW) metric or an equivalent extension. Calculations of local physical variables are treated as a perturbation on the background homogeneous expansion.

A natural question, however, is whether strong gravity waves in the early universe can cause a significant departure from this FRW treatment. There are also other cosmological questions which may require significant deviations from a FRW-plus-perturbations approach, such as the formation of primordial black holes, or the development of an inflating spacetime from inhomogeneous initial conditions. All such problems require the ability to solve for cosmological evolution in a metric more general than that of a simple perturbed FRW. Here we describe some attempts to model such cosmologies numerically.

Planar cosmology

In a linear numerical cosmology program was developed to study how strong waves might affect the physics of the early universe. In particular, the paradigms for inflation, nucleosynthesis, and microwave anisotropy have been analyzed in this context, along with the question of whether strong gravity waves steepen in the early universe. That is, the nonlinear nature of general relativity could cause colliding waves to produce an even stronger superposition wave. In spherical symmetry such superpositions can even form black holes.

The simplest cosmology to study is that of planar symmetry. A system with planar symmetry can have gravity waves. In one-dimensional simulations, physical quantities are restrained to vary in one direction, say z. All fluid quantities are then functions of z and t only.

Relativistic numerical hydrodynamics is currently a field of intense interest. On the one hand, the development of next-generation laser interferometric and cryogenic gravity wave detectors is opening a new window of astronomy, one which will peer into a world of multidimensional rapidly varying matter and gravity fields such as occur in and around neutron stars, black holes, supernovae, compact binary systems, dense clusters, collapsing stars, the early universe, etc. At the same time, X-ray and γ-ray observatories are providing (or will soon provide) a wealth of data on the evolution of matter in and around X-ray and γ-ray emitting compact objects such as accreting black holes and neutron stars. Such systems can only be realistically analyzed by a detailed numerical study of the spacetime and matter fields.

A quantitative understanding of these systems as well as a host of other astrophysical phenomena such as stellar collapse leading to supernovae, the evolution of massive stars, and the origin of γ-ray bursts, the origin and evolution of relativistic jets, all require multidimensional complex relativistic numerical simulations in three spatial dimensions. Since analytic and post-Newtonian methods are only applicable for systems of special symmetry and/or relatively weak fields, numerical relativistic hydrodynamics is the only viable method to model such highly dynamical asymmetrical strong field systems.