I shall describe here a few subjects which in my opinion were the most interesting among those presented orally or at the poster session during the Workshop.

1) The (still hypothetical) discovery of a half-millisecond pulsar in the Supernova SN 1987A attracted a lot of attention. It could drastically change our understanding of neutron star physics and in particular our understanding of the equation of state at nuclear densities. In this context models of compact stars involving strange (e.g. bosonic) matter are interesting and important.

2) The classical problem of test particle motion in a given gravitational field experienced a surprising new development: it was claimed that the centrifugal force can be attractive to the axis of rotation and some repulsive phenomena may be connected with gravity!

3) Gravitational radiation found a new astrophysical application: it was suggested that energy and angular momentum losses due to gravitational waves can be equivalent to viscous stresses in thick accretion disks around supermassive black holes. This may be relevant for quasars.

The optical variability with frequency 1968.629 Hz, discovered recently by Middleditch et al. (1989) in the supernova SN 1987A, is now generally interpreted as due to rotation of a neutron star. An alternative possibility, that the reported frequency represents a radial oscillation of the neutron star was discussed at the Workshop by J.R. Ipser and L. Lindblom. They made significant progress in the numerical treatment of the normal-mode pulsation equations by re-expressing these equations in terms of a single potential function.

It would all be a lot easier (and more satisfying) if one were reporting on the discoveries being made and the new astrophysical information being observed through the gravitational wave channel but unfortunately this can not yet be done. Instead this talk as many others given by workers in this promising but not yet started field has to dwell on the current technical state and prospects. The prospects are now better than ever and one can only hope that in one of the next GR conferences the groups working in this area will be able to talk about the waveforms they are observing and the physics of the sources that are being uncovered.

The active search for gravitational waves from astrophysical sources has been in process for the past two and one half decades. The search began with J. Weber's initial experiments using resonant acoustic bar detectors. These detectors now much improved by advances in transducer technology, better seismic isolation and operation at cryogenic temperatures have attained rms strain sensitivities of h ≈ 10−18 in few Hertz wide bands in the 1 kHz region. More important three such detectors (Stanford, Louisiana State University and CERN/Rome) have made triple and paired coincidence measurements, thereby setting new upper limits on the gravitational wave flux incident on the Earth.

Introduction

The workshop consisted of an introductory overview and of seven specialized talks. The talks were selected among twenty-five submitted abstracts and were presented in an order consistent with the three subsequently discussed categories, i.e. from rigourous mathematical results about approximation methods, to definition of approximation methods, to physical results obtained by approximation methods. This order was chosen to emphasize the following easily forgotten fact. Ideally, physics should connect the mathematical axioms defining our theories to the results of observations or experiments by means of a tight chain of deductions. However, in practice, this chain of deductions often contains gaps, and one of the main sources of gaps lies in the use of mathematically ill justified approximation methods as substitutes for the exact theories. In order to bridge the latter gap one needs, on the one hand, some clear algorithmic definition of the approximation methods used together with a formal study of the structure of its successive terms, and, on the other hand, mathematical theorems proving that the formal sequence defined by the ‘approximation method’ is either convergent, or asymptotic, to some exact solution. Progress on both aspects has recently been obtained, and has been reported, or quoted, during the workshop.

Talks presented at the workshop fell into three categories. Talks on mathematical results about approximation methods showed instances where the conceptually important gap between mathematics and the use of approximation methods by physicists can be closed, or narrowed.

Introduction

The methods invented so far to develop conjectures on and to work out the details of the time evolution of gravitational fields fall into two classes. The first class comprises “physical considerations,” techniques of differential geometry and topology and other ideas which do not take into account the field equations while the remaining methods like: study of explicit solutions, of the Cauchy problem local in time (with its important related notions like “domain of dependence,” “Cauchy stability,” etc.), of formal expansion type solutions, of approximation procedures, etc., have been used to derive information about the evolution more or less directly from the field equations. The division above is somewhat artificial, as is illustrated e.g. by the use of certain positivity assumptions together with Raychaudhuri's equations in the proof of the Hawking-Penrose singularity theorems. However, since the field equations are used in this case in a very weak way one obtains quite general results about the occurrence of a non-complete geodesic but almost no information about the expected “singularity.”

In spite of all the ingenuity with which the methods indicated above have been employed, the important open problems of classical general relativity, e.g. the development and (causal) structure of singularities and the formation of horizons etc. for space-times arising from regular data, the asymptotic behavior of gravitational fields and the relation between the far fields and the structure of the sources etc., remained to a large extent unsolved.

I apologize for uttering this talk in English if that is not your native tongue. Perhaps, at future conferences you will be wearing little earphones and can listen to the translation of such a speech into Hindustani, Japanese, American and other idioms — or, even better, switch to Chopin or Bach.

The first of these conferences, GR-0, took place in Berne three months after Einstein's death, thus probably not a contributing factor. The talks were in English, German, and French with Pauli's Schlusswort in German. This was once the language of relativity, Einstein's language. When asked in old age how his English was, he answered: “Immer besser, niemals gut.”

The declining knowledge of German has had the lamentable effect that among the least read authors in relativity is Saint Albert. His works are being published now and you can read the young man's love letters. Optimistically, I expect that by 2155–remember, they are still working on Euler—you might be able to enjoy reading his thoughts on gravitation. But by then the English edition is possibly no longer the appropriate medium when billions of Chinese are steeped in lerativity.

What we need, within our lifetime, is an edition of Einstein's scientific papers translated without comment. It could even be a best seller among physicists who'd shelve it in their study next to the Einstein icon. This is something we relativists owe the man who put us into business.

Superstrings continue to be a source of inspiration for the basic understanding of quantum gravity. They seem to provide a more fundamental arena than quantum field theory. Even though we still do not have a theory of everything, string concepts bring a new theoretical richness to research in quantum and classical gravity.

Based in previous work on general gauge theories, S. Ichinose analyzed in this session 2D conformal gravity centering his attention on gauge fixing, physical quantities, the energy momentum tensor, and renormalization. C. G. Torre in his talk presented a new formulation of Hamiltonian 2D-gravity which is covariant under all the relevant groups: the spacetime diffeomorphism group, the slice diffeomorphism, and the group of conformal isometries. The key ingredients that allow covariance with respect to the above groups are the enlargement of the phase space by the inclusion of the cotangent bundle over the space of embeddings of a Cauchy surface into the spacetime and the extensive use of conformal 2D isometries.

I. Bakas showed that the Sugawara formalism used in 2D conformal field theory to construct the stress-energy tensor of some non-linear σ-models may have a natural geometric interpretation as a gauge fixing mechanism for current algebras. In the simplest case (SL(2,C)) this procedure provides a realization of the diffeomorphism group of the circle in terms of a 2-D non linear σ-model. Its connection with the choice of the variables proposed by Isham, Klauder and others in the context of canonical quantum gravity was discussed.

In the 20 years after Weber's first publications there have been yearly announcements of new, better ways to detect the signals that everyone knows must be there. Drever, Billing, Douglass and Tyson, Garwin and Levine all built detectors that were of better sensitivity than that estimated by Weber in his original papers. None of those experiments were able to verify Weber's results. In almost every meeting of the past 20 years there have been the promises of the potential of the “second generation” detectors: cryogenic detectors using superconducting instrumentation that would be so sensitive that they would be able to see signals from as far away as the Virgo cluster. In many such meetings for the last 10 years these reports of the potential sensitivity have been tempered by descriptions of the experimental difficulties that were temporarily delaying the attainment of that sensitivity.

The exciting thing about this meeting was that–while there were still tales of wonders to come and of the experimental difficulties that some detectors were confronting–we also had reports of coordinated runs between groups looking for coincidental excitation of their antennas. We had informal meetings outside of the workshops to coordinate data taking and exchange of data. In short, this workshop was the first meeting where three or more groups were confident enough about their detectors that they could talk of coordinating their experiments and of coordinating their existing experiments with the upcoming third generation resonant bar detectors.

The contributed papers presented to the GR-12 workshop on “Quantum Cosmology and Baby Universes” have demonstrated the great interest in, and rapid development of, the field of quantum cosmology. In my view, there are at least three areas of active research at present. The first area can be defined as that of practical calculations. Here researchers are dealing with the basic quantum cosmological equation, which is the Wheeler-DeWitt equation. They try to classify all possible solutions to the Wheeler-DeWitt equation or seek a specific integration contour in order to select one particular wave function of generalize the simple minisuperspace models to more complicted cases, including various inhomogeneities, anisotropies, etc. The second area of research deals with the interpretational issues of quantum cosmology. There are still many questions about how to extract the observational consequences from a given cosmological wave function, the role of time in quantum cosmology, and how to reformulate the rules of quantum mechanics in such a way that they could be applicable to the single system which is our Universe. The third area of research is concerned with the so-called “third quantization” of gravity. In this approach a wave function satisfying the Wheeler-DeWitt equation becomes an operator acting on a Wave Function of the many-universes system. Within this approach one operates with Euclidean worm-holes joining different Lorentzian universes. This is, perhaps, one of the most fascinating, although not entirely clear, subjects considered recently.

Introduction

Quantum theory and relativity theory, the two great revolutions of twentieth century physics, do not seem to mesh very well. Indeed the only fully consistent relativistic quantum theories seem to be linear free fields. It is not clear whether the difficulty of combining the two theories is one of principle or merely one of practice. On the most pedestrian level, the Hamiltonian formalism of classical mechanics, which is most suitable for quantizing a classical theory, requires an explicit choice of a time-like coordinate and dynamical variable adapted to this choice; relativistic theories, in contrast, treat space and time variables on essentially equal footing and are expressed most naturally in the Lagrangian formalism. Of course, for classical physical theories the two formalisms are equivalent in the sense that one can map from one to the other by means of a Legendre transformation. Thus one can, for example, follow the quantization procedure of a classical theory in the Lagrangian formalism, or demonstrate the relativistic covariance of a theory in the Hamiltonian formalism. But the exercises are decidedly awkward, and, given the ensuing difficulties which occur for non-trivial theories, one does have cause for puzzlement.

On a more fundamental level, the measurement theory for the standard interpretation of the quantum mechanical formalism requires that if an observation of a complete commuting set of local observables is performed by an observer in some finite neighborhood of a space-time point, the state of the system is reduced everywhere, including in regions well outside the forward light cone of that neighborhood.

In this report the current status of numerical relativity is presented. Progress in the field is discussed through reviews of work on gravitational radiation and cosmic strings.

Numerical relativity is a young and vital area, growing in many dimensions. This field has its roots in the effort to calculate the gravitational wave emission from astrophysical sources, and it continues to be energized by the promising new developments in gravitational wave detectors. For example, there are a number of new code-building efforts; several of these are aimed at constructing 3-D codes while others are concerned with putting more realistic physics into 1-D and 2-D codes. There is increased activity in combining analytic and numerical techniques to calculate the gravitational radiation produced. And researchers continue to emphasize the need for improved numerical analysis to insure the accuracy and stability of the codes.

There is also increasing diversity in the types of problems being treated numerically, from cosmic strings to inflation, along with the more traditional stellar collapses and black hole collisions. And new people are joining the community, both those just beginning their research careers as well as more experienced researchers crossing over from other areas. The interaction of these new recruits with the “seasoned veterans” of numerical relativity is producing many new ideas and approaches.

This continued blossoming of numerical relativity is fueled by the growth of computing resources.

In the plenary talk on solar system tests of relativistic gravitation, Irwin Shapiro gave a summary of the present state of the experiments, including a new test of the geodetic precession effect using lunar laser ranging data. The contributed papers at this workshop, by contrast, seemed to look more to the future. What will be the next generation of gravitational experiments in space and what preparations are required now in order that these tests may be successfully accomplished?The workshop began with a mini-workshop which discussed a new NASA program of technology development directed toward future experimental tests of relativistic gravitation. Ron Hellings of JPL began with an overview of the program and discussed two of the studies that are presently underway. Work to improve the accuracy of spacecraft ranging systems could pay immediate dividends of improved accuracy of tests of relativity on many planetary missions. Improvement from the accuracy that was available on the Viking landers (about 10 m) to a limit of 10 cm is the goal of work at JPL by Larry Young and collaborators. Also at JPL there is being pursued by Steve Macenka and Bob Korechoff a study of the capability of full-aperture metrology in optical systems. This technology seeks to use an auxiliary laser signal, sharing the optics in an astrometric telescope, to monitor small changes in the optical path, in order to allow the optics to be actively stabilized.

Abstract. Data on kinematics, spatial distributions, and galaxy morphology in different density regimes within individual galaxy clusters show that many clusters are not in a stationary state but are still in the process of forming.

INTRODUCTION

Paradigms for galaxy clusters are changing. As in all tearing away from secure positions (Kuhn 1970) the process is controversial, yet continuing. Most papers in this volume suggest directions that will probably lead to even stronger new ideas about cluster cosmogony. We are concerned in this review with physical properties that have relevance for the question of whether clusters of galaxies are generally stationary, changing only slowly in a crossing time or if they are dynamically young. We examine if parts of a cluster may still be forming, falling onto an old dense core that would have been the first part of a density fluctuation to collapse even if all galaxies in a cluster are the same age, having formed before the cluster. During the 1930's the stationary nature of clusters seemed beyond doubt. A suggestion that they are dynamically young would have been too radical even for Zwicky who was the model of prophetic radicals. Rather, Zwicky (1937) took the stationary state to be given in making his calculation of a total mass, following an earlier calculation by Sinclair Smith (1936). The justification was that rich clusters such as Coma (1257 +2812; or Abell A1656), Cor Bor (1520 +2754; A2065), Bootis (1431 +3146; A1930), and Ursa Major No.2 (1055 +5702; A1132), known already to Hubble (1936) and to Humason (1936), appear so regular.