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.

The workshop concluded with a panel discussion, chaired by Chris Clarke, with John Miller, Silvano Bonazzola and Matt Choptuik comprising the panel. The session was recorded by (what turned out to be) a rather inadequate tape recorder. I have tried on the one hand to make sense in the transcription of the passages which were unclear, and on the other to edit out some of the more repetitious moments, but I must apologise if I have ended up misreporting any of the participants. (Ed.)

Chris Clarke:

I would like the panel to start off by saying what they think are the main highlights of the meeting, the things which have struck them most about the results which have been presented here and also, if possible, what they think are the main problems of where we're going.

John Miller:

A particular impression which I have got from listening to the contributions in this meeting is that the situation now is really very different from what it was only a few years ago in similar sorts of meeting. At that time there was really one sort of method that everybody used. There were experiments with other sorts of method but there was a very strong brand leader. This is still true, but the brand leader has much stronger competition now and I have been very interested to hear during this meeting about how the various competing methods are coming along.

Abstract. Penrose has described a method for computing a solution for the characteristic initial value problem for the spin-2 equation for the Weyl spinor. This method uses the spinorial properties in an essential way. From the symmetrized derivatives of the Weyl spinor which are known from the null datum on a cone one can compute all the derivatives by using the field equation and thus one is able to write down a power series expansion for a solution of the equation. A recursive algorithm for computing the higher terms in the power series is presented and the possibility of its implementation on a computer is discussed.

INTRODUCTION

Due to the nonlinear nature of general relativity it is very difficult to obtain exact solutions of the field equations that are in addition of at least some physical significance. Prominent examples are the Schwarzschild, Kerr and Friedmann solutions. Given a concrete physical problem it is more often than not rather hopeless to try to solve the equations using analytical techniques only. Therefore, in recent years, attention has turned towards the methods of numerical relativity where one can hope to obtain answers to concrete questions in a reasonable amount of time given enough powerful machines. However, it is still a formidable task to obtain a reliable code. There is first of all the inherent complexity of the field equations themselves when written out in full without the imposition of symmetries or other simplifying assumptions.

Abstract. Initial value problems involving hyperboloidal hypersurfaces are pointed out. Characteristic properties of hyperboloidal initial data and rigorous results concerning the construction of smooth hyperboloidal initial data are discussed.

INTRODUCTION

In this article I shall discuss some properties of “hyperboloidal hypersurfaces”. These occur naturally in a number of interesting initial value problems. I became first interested in them in the context of abstract existence proofs for solutions of Einstein's field equations which fall off in null directions in such a way that they admit the construction of a smooth conformal boundary at null infinity (Friedrich (1983)). But it appears to me that hyperboloidal hypersurfaces should also be of interest, in particular if questions concerning gravitational radiation are concerned, in various numerical studies.

Let us consider solutions to Einstein's field equations with vanishing cosmological constant and possibly massive sources of spatially compact support and long range fields like Maxwell fields. We call a space-like hypersurface in such a space-time “hyperboloidal” if it extends to infinity in such a way that it ends on null infinity. We assume that the hypersurface remains space-like in the limit when it “touches null infinity”. The standard examples of such hypersurfaces are the space-like unit hyperbolas in Minkowski space, which motivate the name hyperboloidal. In the standard picture of Minkowski space it is seen that these hypersurfaces are asymptotic to certain null cones.

Abstract. Gravitational radiation from the first phase of the gravitational collapse of a stellar core, i.e. the dynamical phase which precedes the formation of a shock and a bounce, is studied by means of a 3-D pseudo-spectral self-gravitating hydro code. It is shown that the efficiency of this process is very low (of the order of a few percent) and insensitive to the equation of state and to whether the initial configuration is axisymmetric, with an initial quadrupole of rotational or tidal origin, or fully asymmetric. An attempt to treat shock waves in asymmetric situations is described and preliminary results obtained from stellar core bounce are presented.

INTRODUCTION

The gravitational collapse of a stellar core is one of the sources of gravitational radiation which is likely to be detected by the next generation of interferometric gravitational wave detectors (e.g. VIRGO and LIGO projects). However, we need an accurate prediction of the gravitational wave form for a wide range of collapse models in order to interpret the results of the gravitational wave observations.

During the last decade, various attempts have been made to predict the efficiency of this process and to predict wave forms. Most of these papers are based on numerical simulations. Some of them take account of the microphysics and some of them were performed in the framework of General Relativity. However, most of these preliminary works assumed axisymmetry (a complete review of this field can be found in Finn (1989)).

Abstract. We have constructed sequences of equilibrium numerical models for selfgravitating thin discs around rotating black holes. The multigrid method has been used for solving numerically the stationary and axisymmetric Einstein equations describing the problem.

INTRODUCTION

We solved numerically Einstein's equations for equilibrium configurations made by self–gravitating thin discs around rapidly rotating black holes. These configurations may play an important role in modelling active galactic nuclei (AGN) since the self–gravity may induce the so–called “runaway” instability which may be connected with X–ray variability observed in AGNs (Abramowicz et al., 1980, see however Wilson, 1984). Also, such configurations are seen formed in numerical simulations of general relativistic collapse to a black hole (Nakamura, 1981, Stark and Piran, 1986, Nakamura et al., 1987) for some intial conditions.

We consider here only the case in which the disc is thin; for pressure dominated discs (thick discs) work is in progress (see Nishida, Eriguchi and Lanza, 1992). Self–gravitating discs and rings have been considered in the past by Bardeen and Wagoner (1971) (BW) without central body; Will (1974, 1975) has studied weakly self–gravitating rings around slowly rotating black holes.

In order to solve such a highly non–linear problem we employed the multigrid method (MG) as a strategy to solve the finite difference equations which derive from the discretization of Einstein's equations.

Abstract. This paper is concerned with the axisymmetric characteristic initial value problem (CIVP). Tests on the accuracy and evolution stability of the code are described. The results compare reasonably well with expectations from numerical analysis. It is shown explicitly how to compactify CIVP coordinates so that a finite grid extends to future null infinity. We also investigate the feasibility of interfacing Cauchy algorithms in a central region with CIVP algorithms in the external vacuum.

INTRODUCTION

The construction of a new generation of gravitational wave detectors has important implications for numerical relativity. LIGO (Laser Interferometry Gravitational Observatory) is likely to detect gravitational waves from various astrophysical events within the next few years. Numerical relativity will be the main tool for interpreting such data, and will need to be able to calculate waveforms at infinity as accurately as possible. Much work on numerical relativity has been based on the standard 3 + 1 Cauchy problem, where data is specified on a spacelike hypersurface and then evolved to the future. An alternative approach is the characteristic initial value problem (CIVP) based on a 2 + 2 decomposition of space-time. It would seem that the CIVP is more appropriate in vacuum, but that it loses this advantage in the presence of matter, whose characteristics do not coincide with those of the gravitational field. Another consideration is the present state of development of numerical codes.

This volume derives from a workshop entitled “Approaches to Numerical Relativity” which was held in the week 16-20th December, 1991, in the Faculty of Mathematical Studies at Southampton University, England. It was held principally because it was thought that the time was opportune to begin a dialogue between theorists in classical general relativity and practitioners in numerical relativity. Numerical relativity - the numerical solution of Einstein's equations by computer - is a young field, being possibly only some fifteen years old, and yet it has already established an impressive track record, despite the relatively small number of people working in the field. Part of this dialogue involved bringing participants up to date with the most recent advances. To this end, international experts in the field were invited to attend and give presentations, including Joan Centrella, Matt Choptiuk, John Miller, Ken-Ichi Oohara, Paul Shellard and Jeff Winicour. In addition, a significant number of European scientists, both theoreticians and practitioners in numerical relativity, were invited, the majority of whom attended. In the event, there were some 35 participants, most of whom gave presentations. This volume is largely comprised of the written versions of these presentations (their length being roughly proportional to the time requested by the authors for their presentations).

In an attempt to highlight the distinctive nature of the workshop, I have divided the contributions into Part A, Theoretical Approaches and Part B, Practical Approaches.

Abstract. We investigate the stability of charged boson stars in the framework of general relativity. The constituents of these stars are scalar bosons which interact not only via their charge and mass but also via a short–range Higgs potential U. Our stability analysis is based on catastrophe theory which is capable of providing more information than perturbation theory. In fact, it predicts novel oscillation and collapse regimes for a certain range of the particle number.

INTRODUCTION

In the early universe, spin-zero particles, such as the scalar Higgs particles, may have played an important rôle [1]. At that early time it is conceivable that clouds of particles created stars which are kept together by their own gravitational field, the so–called boson stars [2]. These stars could make up a considerable fraction of the hypothetical dark matter.

The boson star consists of many particles and may have a very large mass comparable or larger than that of a neutron star. The latter depends upon the form of a self–interaction between the bosons [3]. Generally speaking, the boson star is in many ways analogous to the neutron star [4,5]. Both stars consist of one matter component. Recently, Higgs particles interacting with gauge field have been studied [6]. If we attribute charge to the bosons, they will interact also via electromagnetic forces. Because of the repulsive nature of this interaction there exists a critical total charge of these scalar particles beyond which the star becomes unstable [6].