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].

Abstract. We review the present status of the null cone approach to numerical evolution being developed by the Pittsburgh group. We describe the simplicity of the underlying algorithm as it applies to the global description of general relativistic spacetimes. We also demonstrate its effectiveness in revealing asymptotic physical properties of black hole formation in the gravitational collapse of a scalar field.

INTRODUCTION

We report here on a powerful new approach for relating gravitational radiation to its matter sources based upon the null cone initial value problem (NCIVP), which has been developed at the University of Pittsburgh. We are grateful to the many graduate students and colleagues who have made important contributions: Joel Welling (Pittsburgh Supercomputing Center), Richard Isaacson (National Science Foundation), Paul Reilly, William Fette (Pennsylvania State University at McKeesport) and Philipos Papadopoulous.

As will be detailed, the NCIVP has several major advantages for numerical implementation, (i) There are no constraint equations. This eliminates need for the time consuming iterative methods needed to solve the elliptic constraint equations of the canonical formalism, (ii) No second time derivatives appear so that the number of basic variables is half the number for the Cauchy problem. In fact, the evolution equations reduce to one complex equation for one complex variable. The remaining metric variables (2 real and 1 complex) are obtained by a simple radial integration along the characteristics.

Abstract. We present three-dimensional Newtonian and post-Newtonian codes, including the gravitational radiation damping effect, using a finite difference method. We follow the emission of gravitational radiation using the quadrupole approximation. Using these codes we calculate the coalescence of a neutron star binary. For Newtonian calculations the initial configuration is given as a hydrostatic equilibrium model of a close neutron-star binary. Calculations were performed for neutron stars of different masses as well as of the same masses. In order to evaluate general relativistic effects, we compare the results of the calculation of the coalescence of a binary comprising two spherical neutron stars using the post-Newtonian code with results using the Newtonian code.

INTRODUCTION

The most promising sources for laser-interferometric gravitational-wave detectors are catastrophic events such as the gravitational collapse of a star or the coalescence of a black-hole or neutron-star binary. We need to know the characteristics of the waves for design of detectors. It requires general relativistic calculations of stellar collapse and binary coalescence. In the last decade, 2 dimensional (2D) calculations were successfully performed for a head-on collision of two black holes (Smarr 1979) and axisymmetric collapse of a rotating star (Stark and Piran 1986). They found that the efficiency of gravitational wave emission (the ratio of the energy emitted in gravitational radiation to the total rest mass) is less than 0.1%. Nakamura, Oohara and Kojima (1987), on the other hand, pointed out that the efficiency may be much greater in non-axisymmetric black-hole collision.

Abstract. Our project was inspired by the prospect that a new non-electromagnetic astronomy will develop by the end of the century. Projects like Virgo and Ligo will lead to detectors able to detect extra-Galactic and Galactic sources of gravitational radiation. New generation of neutrino detectors like Superkamiokande will be able to detect various Galactic neutrino sources. All these considerations motivated us to study in detail potential Galactic sources of bursts of the gravitational radiation and neutrinos. In this paper, our projects are described in some detail. The advantages and the drawbacks of the numerical technique used in our computer simulations (pseudospectral methods) are discussed. Possible applications of the numerical methods are illustrated by some examples of astrophysical interest: coalescence of two neutron stars, mini-collapse of a neutron star (phase transition) and formation of a black hole due to the collapse of a neutron star.

INTRODUCTION

The main idea, which motivated our project, is that massive stellar cores, involved in supernovae of type II, are not the only collapsing Galactic objects generating bursts of gravitational waves that could be detected by the next generation of gravitational wave detectors. It is quite likely that SNI and SNII are only an optically detectable subset of a larger class of collapse events, which are less spectacular (as far as electromagnetic radiation is concerned) but perhaps quite frequent, and which are able to radiate a conspicuous amount of gravitational radiation.

Abstract. We investigate initial data for localized gravitational waves in space-times with a cosmological constant Λ. By choosing the appropriate extrinsic curvature, we find that the Hamiltonian and momentum constraints turn out to be the same as those of the time-symmetric initial value problem for vacuum space-times without Λ. As initial data, we consider Brill waves and discuss the cosmological apparent horizon. Just as with Brill waves in asymptotically flat space-time, the gravitational “mass” of these waves is positive. Waves with large gravitational mass cause a strong cosmic expansion. Hence, the large amount of gravitational waves do not seem to be an obstacle to the cosmic no-hair conjecture.

INTRODUCTION

The present isotropy and homogeneity of our universe is something of a mystery within the framework of the standard big bang scenario. The inflationary universe scenario, however, is one of the favourable models which may explain the so-called homogeneity problem [1]. In this scenario, when a phase transition of the vacuum occurs due to an inflaton scalar field and supercooling results, the vacuum energy of the scalar field plays the role of a cosmological constant and the space-time behaves like the de Sitter one with a rapid cosmic expansion. This phenomenon is called inflation. As a result, all inhomogeneities go outside the horizon by rapid cosmic expansion. After inflation, the vacuum energy of the scalar field decays into radiation and the standard big bang scenario is recovered. However, there still remains a question in the above scenario.

The launch of Sputnik 1 on 4 October 1957 was a traumatic event for the USA and much of the western world. For years there had been an unspoken assumption that the Russians were dark and backward people, and that all new initiatives in science and technology occurred, almost as a natural law, in ‘the West’. Disbelief was widespread. ‘What I say is truth, and truth is what I say’, that popular saying of the 1980s, had its adherents in the 1950s too, and they assured the world that Sputnik 1 was just propaganda and was not really in orbit at all.

My view of the event was different. For several years we had been showing in theory how ballistic rockets could be turned into satellite launchers by adding a small upper stage to produce the necessary extra velocity. The USSR had launched an intercontinental rocket in August 1957, and little extra velocity would be needed to attain orbit. So it would be quite easy for the USSR to launch a small satellite like Sputnik 1, which was a sphere 58 cm in diameter of mass 84 kg with four long aerials (Fig. 2.1). The real surprise was the final-stage rocket that accompanied Sputnik 1 into orbit. The rocket appeared much brighter than the pole star as it crossed the night sky, and seemed likely to be at least 20 m long, far larger than anything contemplated in our paper-studies of satellites: the final-stage rocket for our reconnaissance satellite was less than 5 m long.

It was in 1953 that the metamorphosis of missiles into satellites began. One important new start was the prospect of rockets for upper-atmosphere research. The impetus came from a group of scientists belonging to the Royal Society's Gassiot Committee, particularly Professor Harrie Massey of University College London, and Professor David Bates of the Queen's University, Belfast. The existence of the Gassiot Committee was an extraordinary stroke of luck for space science, as I came to realize much later. The Royal Society covers all science, and until 1935 the one exception to this rule was the Gassiot Committee, the Society's only specialized ‘inhouse’ committee: it had been formed in 1871, to oversee Kew Observatory, and was expanded during the Second World War to cover atmospheric physics in general. The Gassiot Committee was vitally important for two reasons: first, it was a preconstructed official pathway into space; second, the Royal Society was fully committed from the outset, thus making respectable a subject dismissed by many as ‘utter bilge’.

The Gassiot Committee organized an Anglo-American conference on rocket exploration of the upper atmosphere, at Oxford in August 1953, and this can now be seen as the first British step on the ladder into space which we climbed for nearly twenty years. I cannot remember much about the meeting, except that it was held in a dark medieval lecture-room, lit by a few light bulbs with dusty white shades: it seemed paradoxical that these new ventures into space were being planned in such antiquated surroundings.

A rocket fired up the north face of the Eiger towards the summit might serve as a suitable simile for the worldly aspects of my career in science. From 1957 until about 1970 the upward thrust was strong, and the rocket seemed on course for the stratosphere. During the 1970s the propellant seemed to burn out and the momentum decreased. About 1980 the rocket came to rest on a rather precarious shelf, halfway up the cliff: there was a danger of being pushed off into free fall; on the other hand, the position was a commanding one, from which good work might be done. As it turned out, the danger was averted and the decade was most productive.

In 1980 the researches based on orbit analysis seemed to be in good health. The Earth Satellite Research Unit at Aston University, under Dr Brookes, had moved to a spacious modern building at St Peter's College, Saltley, and the prediction service was transferred from the Appleton Laboratory to ESRU in July, because the Appleton Laboratory was being moved and merged with the Rutherford Laboratory. (Pierre Neirinck retired from Appleton but continued as a keen analyst of satellites.) In September 1980, when a meeting of visual observers was held at St Peter's College, ESRU was thriving, with four staff members working on predictions, four more as Hewitt camera observers, and a strong research team that included Philip Moore and two recently-appointed Research Fellows, Graham Swinerd and Bill Boulton, both working on orbit analysis and popularly known as the heavenly twins.

The clouds had obediently unfolded to reveal that ‘chariot of fire’ over the Caribbean on 14 April 1958; but the descent of Sputnik 2 left us without any satellites to predict. The first US satellite, the pencil-shaped Explorer 1, had been launched on 1 February; the grapefruit-like Vanguard 1 followed on 17 March; and Explorer 3 on 26 March. But these three satellites were small and faint, and, with orbits inclined at less than 35° to the equator, they were far to the south and nearly always below the horizon for observers in Britain.

During this welcome respite there was time, on 22 April, for a visit to Herstmonceux, where the moated castle was worlds apart from the hotchpotch of rather ugly buildings at the RAE. (The contrast always startled me, even in later years.) Thus began a secure and friendly cooperation with the Royal Greenwich Observatory that flourished for more than thirty years, with benefit to both sides. The road back from historic Herstmonceux ran through Piltdown, a name redolent of even older times – or so it was thought until the Piltdown Man was exposed as bogus.

The hiatus in prediction did not last long, for Sputnik 3 was launched on 15 May, which was presciently marked in 1958 diaries as Ascension Day. We heard about the launch just before noon, and early that afternoon sent out the first set of predictions, which proved accurate to half a minute.