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.

Solar models

Here we describe only a few representative main sequence stellar models. One is for the zero age sun, that is, the sun as it was when it had just reached the main sequence and started to burn hydrogen. We also reproduce a model of the present sun, a star with spectral type G2 V, i.e. B − V ∼ 0.63 and Teff ∼ 5800 K, after it has burned hydrogen for about 4.5 × 109 years. In the next section we discuss the internal structures of a B0 star with Teff ∼ 30 000 K and an A0 type main sequence star with Teff ∼ 10 800 K. There are several basic differences between these stars. For the sun the nuclear energy production is due to the proton–proton chain, which approximately depends only on the fourth power of temperature and is therefore not strongly concentrated towards the center. We do not have a convective core in the sun, but we do have an outer hydrogen convection zone in the region where hydrogen and helium are partially ionized. The opacity in the central regions of the sun is due mainly to bound-free and free-free transitions, though at the base of the outer convection zone many strong lines of the heavy elements like C, N, O and Fe also increase the opacity.

In Table 13.1 we reproduce the temperature and pressure stratifications of the zero age sun. In Table 13.2 we give the values for the present sun as given by Bahcall and Ulrich (1987). The central temperature of the sun was around 13 million degrees when it first arrived on the main sequence;…

Available energy sources

So far we have only derived that, because of the observed thermal equilibrium, the energy transport through the star must be independent of depth as long as there is no energy generation. The energy ultimately has to be generated somewhere in the star in order to keep up with the energy loss at the surface and to prevent the star from further contraction. The energy source ultimately determines the radius of the star.

Making use of the condition of hydrostatic equilibrium we estimated the internal temperature but we do not yet know what keeps the temperature at this level. In this chapter we will describe our present knowledge about the energy generation which prevents the star from shrinking further.

First we will see which energy sources are possible candidates. In Chapter 2 we talked about the gravitational energy which is released when the stars contract. We saw that the stars must lose half of the energy liberated by contraction before they can continue to contract. We might therefore suspect that this could be the energy source for the stars.

The first question we have to ask is how much energy is actually needed to keep the stars shining. Each second the sun loses an amount of energy which is given by its luminosity, L = 3.96 × 1033 ergs−1, as we discussed in Chapter 1. From the radioactive decay of uranium in meteorites we can find that the age of these meteorites is about 4.5 × 109 years. We also find signs that the solar wind has been present for about the same time.