General relativity was for too long the ugly duckling of science. In the 50s and 60s the dominant impression was of the difficulty of the equations, solvable only by arcane techniques inapplicable elsewhere; of the scarcety of significant experimental tests; of the prohibitive cost of computational solutions, compounded by a lack of rigorous approximation techniques; and of the isolation of the subject from the physics of the other fundamental forces. This led to a situation where, even in the 70s, much theoretical work was becoming increasing irrelevant to physics. Exact solutions proliferated but (with the exception of cosmology) attempts at physical interpretation were few and unconvincing. Mathematical investigations in the wake of the singularity theorems became increasingly sophisticated, but few were applied to actual physical models. In the 70s and 80s, however, all this changed, with the growth of experimental relativity, the trend to geometrical methods in high energy physics, and the inception of numerical relativity. The workshop reported in this book marks the complete clearing of this last hurdle, as reliable and practical computational techniques are established.

It brought together numerical and classical relativists, and showed that the cultural gap between them was closing fast. Dramatically increased standards of reliability and accuracy had been set, and were being achieved in many cases, so that numerical work can no longer be seen merely as providing a rough indication for the ‘proper’ work of analysis.

Abstract. The difficulties in solving the characteristic initial value problem in general relativity when matter fields are included are discussed. A scheme is proposed with new dependent variables, coordinates and tetrad which should alleviate some of the problems.

THE CHARACTERISTIC INITIAL VALUE PROBLEM

Almost all calculations in numerical relativity are based on the Arnowitt-Deser-Misner (1962) formalism in which spacetime is foliated by spacelike hypersurfaces. The Einstein equations are decomposed into elliptic constraint equations which are intrinsic to the slices, and hyperbolic evolution equations which govern the evolution from slice to slice. In principle the constraint equations have only to be solved on the initial slice, but even this requires considerable effort and a large computer.

Stewart and collaborators, Friedrich and Stewart (1982), Corkill and Stewart (1983), developed an alternative approach based on a fundamentally different principle. In this spacetime was foliated by null hypersurfaces, and originally two families of null hypersurfaces were used. A helpful analogy is to consider the Schwarzschild solution described by double null coordinates. Examination of that solution will reveal that such coordinates have many theoretical advantages. One can explore the spacetime through the regular event horizon right up to the singularity at r = 0. Further one can proceed out along surfaces u = const, to future null infinity, the area inhabited by distant observers. Indeed one can bring infinity in to a finite point by the technique of conformal compactification, due originally to Penrose.

Abstract. Criteria are presented for choosing a matter model in analytical or numerical investigations of the Einstein equations. Two types of matter, the perfect fluid and the collisionless gas, are treated in some detail. It is discussed how the former has a tendency to develop singularities which have little to do with gravitation (matter-generated singularities) whereas the latter does not seem to suffer from this problem. The question of how the concept of a matter-generated singularity could be defined rigorously is considered briefly.

INTRODUCTION

In any investigation of the Einstein equations it is necessary to make some assumptions about the energy-momentum tensor. One possibility is simply to require that some energy conditions be satisfied. (In that case it might be more appropriate to say that the object of study is the ‘Einstein inequalities’.) Despite the fact that this is sufficient to obtain important results including the singularity theorems and the positive mass theorem, it is very likely that there are significant results concerning the qualitative behaviour of solutions of the Einstein equations which require more specific assumptions. In any case the choice of a definite matter model is indispensable for numerical calculations and for analytic work based on the use of a well-posed initial value problem. The particular kind of matter chosen will of course depend on the problem being studied.

Abstract. We review the classical and modern work on the stability of rotating fluid configurations with particular interest in astrophysical scenarios likely to produce gravitational radiation. We describe a hybrid method for numerically generating axisymmetric equilibrium models in rapid differential rotation based on the self consistent field approach. We include a description of the 3-D hydrodynamics code that we have developed to model the production of gravitational radiation, and present the results of a 3-D test case simulating the growth of the dynamical bar mode instability in a rapidly rotating polytrope.

INTRODUCTION

Overview

The study of the effects of rotation on equilibrium fluid bodies was begun by Newton in Book III of the Principia where he investigated the consequences of rotation on the figure of the earth, and concluded that the result would be a flattening at the poles to give the earth a slightly oblate shape. Much of the classical work accomplished since that time has been concerned with equilibrium configurations for fluids with uniform density and/or rigid rotation. Recent advances in computing technology, however, have allowed more detailed investigations involving differential rotation, various equations of state, and dynamical evolution of self gravitating systems. This work has fostered a variety of astrophysical applications, notably in the study of the formation of single and binary stars from collapsing gas clouds, and of the structure of compact objects, such as white dwarfs and neutron stars.

Abstract. The characteristic initial value problem is reviewed and a number of possible schemes for implementing it are discussed. Particular attention is given to choosing variables and choosing a minimal set of equations in the Newman–Penrose formalism. A particular scheme which is based on null cones and involves giving the free gravitational data in terms of Ψ0 is presented. The question of regularity at the vertex is briefly discussed and asymptotic expansions for the spin coefficients near the vertex are given.

INTRODUCTION

In studying problems in which gravitational radiation plays an important rôle, a description of the geometry which is adapted to the wavefronts of the radiation is obviously useful. Thus both the Bondi formalism and Newman–Penrose formalism have proved very helpful in understanding gravitational radiation at null infinity J+. From the point of view of an initial value problem this suggests that rather than specifying data on a spacelike surface one should specify data on a null surface and look instead at the characteristic initial value problem (CIVP).

There are a number of technical advantages that one gets from looking at the CIVP. The first of these is that the variables one uses are precisely those one needs to calculate the physically important quantities such as the amount of gravitational radiation, the Bondi momentum and so on. The second advantage is that the elliptic constraints which play such an important rôle in the spacelike case are effectively eliminated and one can freely specify the appropriate null data.

Abstract. We use the multiquadric approximation scheme for the solution of a three-dimensional elliptic partial differential equation occurring in 3 + 1 numerical relativity. This equation describes two-black-hole initial data, which will be a starting point for time-evolution computations of interacting black holes and gravitational wave production.

INTRODUCTION

Adopting the Arnowitt-Deser-Misner (ADM) 3 + 1 description of general relativity (1962) has, over the years, proved to be a fruitful approach for numerical relativity calculations. Using this description spacetime is constructed as a foliation of spacelike hyper surfaces. This split into space plus time leads to a constrained system of equations so that initial data must be specified on a spatial hypersurface and evolved into the future. The specification of initial data necessarily involves the solution of elliptic partial differential equations; these being the Hamiltonian and momentum constraints. When combined with York's conformal approach (1979) the system of elliptic equations is well-posed for solution by numerical techniques.

Until a few years ago the standard approach adopted by numerical relativists for the construction of initial data consisted of finite-differencing the constraint equations and applying iterative techniques, such as simultaneous-over-relaxation, to the resulting matrix of algebraic equations. More recently direct matrix solvers such as conjugate gradient and its variations have been employed (Evans 1986; Oohara and Nakamura 1989; Laguna et al 1991). A sophisticated multilevel iterative scheme developed by Brandt (1977) has also been used in numerical relativity calculations, mainly by Choptuik (1982, 1986), Lanza (1986, 1987) and Cook (1990, 1991).

Abstract. We study the effect of a moving grid on the stability of the finite difference approximations to the wave equation. We introduce two techniques, which we call “causal reconnection” and “time-symmetric ADI” that together provide efficient, accurate and stable integration schemes for all grid velocities in any number of dimensions.

INTRODUCTION

In the numerical study of wave phenomena it is often necessary to use a reference frame that is moving with respect to the medium in which the waves propagate. In this paper, by studying the simple wave equation, we show that the consistent application to such a problem of two fundamental physical principles — causality and time-reversal-invariance — produces remarkably stable, efficient and accurate integration methods.

Our principal motivation for studying these techniques is the development of algorithms for the numerical simulation of moving, interacting black-holes. If we imagine a black hole moving “through” a finite difference grid then some requirements become clear. As the hole moves, grid points ahead of it will fall inside the horizon, while others will emerge on the other side. This requires grids that shift faster than light. Moreover, in situations when the dynamical time scale is large, one would like to be free of the Courant stability condition on time-steps, i.e. one wants to use implicit methods. Full implicit schemes require the inversion of huge sparse matrices. Alternating Direction Implicit (ADI) schemes reduce the computational burden by turning the integration into a succession of one-dimensional implicit integrations.

Abstract. Much of physics concerns temporal dynamics, which describes a spatial world (or Cauchy surface) evolving in time. In Relativity, the causal structure suggests that null dynamics is more relevant. This article sketches Lagrangian and Hamiltonian formalisms for dual-null dynamics, which describes the evolution of initial data prescribed on two intersecting null surfaces. The application to the Einstein gravitational field yields variables with recognisable geometrical meaning, initial data which divide naturally into gravitational and coordinate parts, and evolution equations which are covariant on the intersection surface and free of constraints.

INTRODUCTION

The ADM or “3+1” formalism [1,2] is a natural approach to the Cauchy problem in General Relativity, and has been used widely both analytically and numerically. By comparison, null (or characteristic) evolution problems are more appropriate to the study of problems involving radiation, whether gravitational or otherwise, since radiation propagates in null directions. Null surfaces also have a central place in the causal structure of General Relativity which spatial surfaces do not.

A distinction should be drawn between the null-cone problem discussed elsewhere in this volume, in which the initial surface is a null cone, and the dual-null problem, in which there are two intersecting null initial surfaces. The latter problem was originally described by Sachs [3], with existence and uniqueness proofs being given by Müller zum Hagen and Seifert [4], Friedrich [5] and Rendall [6], and a general “2+2” formalism being developed by d'Inverno, Smallwood and Stachel [7–9].

Abstract. We have extended some high-resolution shock-capturing methods, designed recently to solve nonlinear hyperbolic systems of conservation laws, to the general-relativistic hydrodynamic system of equations and applied them to the study of the gravitational collapse of spherically symmetric configurations.

INTRODUCTION

Several topics are of current interest among astrophysicists working in the field of stellar collapse: (i) The equation of state for both subnuclear and supranuclear densities. Alongside the theoretical problems concerned here, there is also the technical problem of making both approaches consistent with each other, as well as sufficiently fast to compute in stellar collapse calculations (see Lattimer and Swesty, 1992). (ii) The coupling between neutrinos and matter in connection with the feasibility of the so-called delayed mechanism. (iii) The correct modelling of shocks in order to conserve total energy along the propagation of the shock formed in the collapse after bounce. In the last years, a part of our research has been addressed to this point.

In a previous paper (Martí et al., 1990, in the next MIM90) we have focussed on the shock formation and propagation such as it appears in the standard scenario of the prompt mechanism. In MIM90 we have undertaken Newtonian stellar collapse calculations with two codes: (i) A standard finite-difference scheme which uses an artificial viscosity technique. (ii) A Godunov-type method which uses a linearized Riemann solver. The initial model and the equation of state was kept fixed in order to be able to compare both methods directly.

Abstract. An outline is given of a scheme being used for making computations of the growth of single hadronic bubbles during the cosmological quark-hadron transition. The code uses a standard Lagrangian finite-difference scheme for flow within the bulk of each phase together with continuous tracking of the phase interface across the grid by means of a characteristic method with iterative solution of junction conditions.

INTRODUCTION

In view of the subject of this meeting, our emphasis here will be on the computational aspects of our study of the cosmological quark-hadron transition (Miller & Pantano 1989, 1990; Pantano 1989). However, as a preliminary, it is good to recall some fundamental points of the physics lying behind the calculations.

According to present ideas, hadrons are composed of quarks which move freely within a hadron but are strongly constrained from leaving. A phenomenological description of this is provided by the MIT bag model (Chodos et al 1974) where the region occupied by the quarks is associated with a false vacuum state characterized by a uniform vacuum energy density B and an associated negative pressure – B. If normal hadronic matter were compressed to high enough density, the individual hadrons would overlap and the quarks would become free to move within the entire interior region, giving rise to a quark-gluon plasma. Heavy-ion collision experiments at CERN and Brookhaven are aiming to create transient plasma in the course of collisions and to look for signatures of its decay.

Abstract. This article contains some proposals for the construction of an algorithm for the evolution of initial data in general relativity which will apply to generic initial values. One of the main issues is to allow a dynamic refinement of the discretisation which will be local and vary according to local values of the initial data. I outline some of the main problems which will have to be addressed in any implementation of the general scheme. There are also some suggestions for a construction of a smooth solution of the Einstein equations which is near to the discrete evolution.

INTRODUCTION

At the present time, computer codes for general relativity are written specifically for particular problems such as stellar collapse or coalescing binary systems. In the longer run relativists are interested in using the computer as a mathematical tool to investigate the properties of solutions which seem inaccessible by analytic means, or to formulate hypotheses which may then be attacked analytically. This requires the construction of an algorithm which applies to generic initial data and which also has a sufficiently solid framework which allows analytic investigation of the error of the approximation.

The approach I would like to suggest is based on triangulations. One of the problems of numerical relativity is that the degree of discretisation that is required to approximate given data well is dependent on that data. However one cannot predict — in advance — how this will evolve as the data evolves with time.

Abstract. The results of a detailed numerical investigation of the strong-field, dynamical behaviour of a collapsing massless scalar field coupled to the gravitational field in spherical symmetry are summarized. A variety of non-linear phenomena suggestive of a type of universality in the model have been discovered using a finite difference approach combined with an adaptive mesh algorithm based on work by Berger & Oliger. A derivation of the equations of motion for the system is sketched, the adaptive algorithm is described, and representative examples of the strong-field behaviour are displayed.

INTRODUCTION

The problem of the collapse of a massless scalar field coupled to the Einstein gravitational field in spherical symmetry has been studied in considerable detail, both analytically (Christodoulou 1986a, 1986b, 1987a, 1987b), and through numerical work (Choptuik 1986, 1989, 1991, Goldwirth & Piran 1987, Goldwirth et al 1989, Gómez & Winicour 1989, 1992, Gómez et al 1992). In many ways, the system provides an ideal model for addressing a variety of basic issues in numerical relativity. The scalar field provides the model with a radiative degree of freedom which is necessarily absent from any “dynamics” of the Einstein (or Maxwell) field in spherical symmetry. At the same time, by suitable choice of initial data, the self-gravitation of the scalar field can be made arbitrarily strong, so that processes such as curvature scattering of radiation and black-hole formation can be studied.

Abstract. It is argued that having a good conceptual understanding of relativistic effects is very important when undertaking large computations in numerical relativity. The radius of gyration (the square root of the ratio of the specific angular momentum to the angular velocity) is proposed as a useful quantity for the analysis of effects which are related to rotation.

INTRODUCTION

This paper is concerned with the effects of rotation in general relativity and parts of it draw heavily on joint work carried out together with M. A. Abramowicz of NORDITA and Z. Stuchlik of the Silesian University of Opava (see Abramowicz et al. 1992).

Among the areas of particular interest for current work in numerical relativity are the following:

1. (i) collisions of neutron stars or black holes;

2. (ii) realistic three-dimensional relativistic gravitational collapse;

3. (iii) non-axisymmetric behaviour of compact objects;

4. (iv) accretion onto compact objects;

5. (v) processes in the early universe;

6. (vi) behaviour of gravitational waves;

7. (vii) formation of singularities.

In much of this, the calculations are intrinsically three-dimensional and rotation plays a crucial role. The question arises: how well do we understand the effects of rotation in general relativity? One could argue that this is unimportant; there is a system of equations to be solved for given initial conditions and boundary conditions and the process is a mechanical one which leads to predictions which could then be tested experimentally, at least in principle.