Twenty six abstracts were submitted for this workshop, seven of which were selected for oral presentation. The main topics covered were gravitational lensing (9 abstracts), large-scale structure (6 abstracts) and cosmic strings (4 abstracts), all of these topics being represented in the talks. These are not the only areas which have seen important advances recently but they are perhaps the most interesting ones from the perspective of a relativist. In this report I will summarize the contents of the talks, referring to some of the posters where appropriate. Whenever distance scales arise, the Hubble parameter is assumed to be 50 km/s/Mpc.

Gravitational lensing

Blandford's plenary contribution (Chapter 5, this volume) illustrates the increasing usefulness of gravitational lensing as a cosmological tool in recent years and this is reflected in the large number of abstracts on the topic. Besides confirming light-bending itself, gravitational lensing can provide evidence of the existence and distribution of dark matter in galaxies, identify the presence of objects on scales from jupiters to supermassive black holes, probe features of large-scale structure, and perhaps even measure the Hubble constant. The posters of Ho Tenlin and Yakimov illustrated how particular instances of gravitational lensing can provide cosmological information. The talks focussed on more general mathematical issues.

The interpretation of observations of distant objects is complicated by the fact that a beam of light may suffer many weak gravitational encounters rather than a single strong one as it propagates through an inhomogeneous background.

Introduction

The gravitational interaction between waves is a phenomenon in which the richness and the originality of the theory of general relativity are explicitly manifested. It became apparent in 1970-71 when Khan, Penrose and Szekeres found the first exact solutions describing the collision of pure gravitational waves: it was shown that when two plane gravitational waves with collinear polarization, and with a step or an impulsive profile collide, their subsequent interaction culminates in the creation of a curvature singularity, an event unpredicted by any linearized version of the theory of gravity. As we shall see, this is only a particular result, although probably the most remarkable, of the interaction of gravitational waves. Similar behaviors are also manifested when waves of a different nature collide. This is due to the fact that any kind of energy generates a gravitational field. As a consequence, when two arbitrary waves collide, a gravitational interaction will accompany, as a side effect, the interaction which is peculiar to the particular fields considered. These gravitational effects, though negligible to some extent, are nevertheless relevant from a theoretical point of view. In this lecture we shall investigate the main features of the scattering of plane waves in terms of exact solutions of Einstein's equations. Therefore, let us start by explaining what gravitational plane waves are and how to find exact solutions of Einstein's equations describing their interaction.

Though historically, the solar system has been the principal area for testing theories of gravitation, we seem to be at the end of the golden age of solar-system tests (Reasenberg (1987)). The classical effects have now been measured within the limits of today's technology and further significant improvements cannot be expected in the near term. Future space-based experiments such as GPB, Gravity Probe B; LAGOS, Laser Gravitational-Wave Observatory in Space; and POINTS, the (proposed) Astrometric Optical Interferometer, await further technological as well as engineering developments and logistic (launch) support to deliver them to the laboratory of space.

Recent years have seen, however, ground-based gravitational experimentation undergo a resurgence, driven by new experimental capabilities and by new theoretical work. The question that was raised by Fischbach et. al. (1986) of a possible short-range gravity force, dubbed the “Fifth Force,” has been particularly important. Though the experiments that gave rise to this suggestion have, in retrospect, turned out to be less compelling than was originally thought, the gravitational physics community has been forced to recognize the possibility of a short-range gravitational interaction.

This suggestion lent itself to fairly straightforward testing and the experimental community responded with great enthusiasm and ingenuity. Now some five years later, though it appears that this quite plausible theoretical suggestion has been ruled out by experiments at the level which initially was suggested. Nevertheless it gave rise to a rather exciting period in gravitational physics.

Introduction

Although the roots of string theory go back to the late 1960's, the first connection between string theory and gravity was noticed in 1974 independently by Yoneya (1974) and by Scherk and Schwarz (1974). By the early 1980's it became clear that the recently developed superstring theory was an excellent candidate for our first perturbatively finite quantum theory of gravity. One loop calculations were shown to be finite and general arguments suggested that this should hold to all orders. (For a review of what was known at that time see Schwarz (1982).) Since then, an enormous amount of work has been done and progress made in our understanding of string perturbation theory. The evidence for finiteness has grown stronger and stronger (see e.g. D'Hoker and Phong (1988); Atick, Moore and Sen (1988); La and Nelson (1989), and references therein). Although there is always a chance of some unexpected results, no one who works on this subject doubts that it is true.

Conspicuously absent from this brief history is the remarkable explosion of interest in string theory beginning in the fall of 1984 and the almost equally remarkable drop in interest in the past year or so. This mood swing had nothing to do with string theory providing a consistent quantum theory of gravity. Rather, it resulted from the hope that the “uniqueness” of string theory would lead to definite low energy predictions in a simple way. Unfortunately, this hope has not been fulfilled.

The theory of imaging of cosmologically distant point and extended sources, specifically quasars and galaxies by an intervening mass is described. Particular attention is paid to formalisms which allow one to understand the qualitative principles governing image formation. The importance of caustics is emphasized, particularly their role in the formation of highly magnified images. The prospects for measuring the Hubble constant and the cosmological density parameter are reviewed.

Introduction

The history of gravitational lensing is one in which general relativists can take some pride. The basic effect was anticipated by many researchers including Einstein (1936); Refsdal (1964); Press and Gunn (1973); Bour-rassa and Kantowski (1975); long before the discovery by Walsh, Car-swell and Weymann (1981) of the first convincing example of multiple imaging of a background quasar by an intervening galaxy. This is perhaps not too surprising since gravitational lensing is an almost trivial consequence of the general theory. What was surprising was how rich a field the elementary geometrical optics of gravitational lensing has become when stimulated by the observational discoveries reviewed here by Bernard Fort (Chapter 6, this volume). I intend to review some of the theoretical approaches to gravitational lenses that have been developed over the past ten years emphasizing those that are directly relevant to interpreting the observations.

Gravitational lenses have been heralded as important astronomical tools; specifically they are probes of the dark matter found in the outer parts of galaxies, rich clusters of galaxies and perhaps also the universe at large.

In order to enable serious discussions, to allow the speakers to describe their subject in detail and give self-contained reviews, only three talks were scheduled for the workshop. The other submitted papers which partly contained substantial contributions to quantum field theory in curved space-time and its applications, have been presented to the participants through the abstracts and posters.

Coarse-grained effective action

In many studies of quantum fields in dynamic space-times one often needs to treat the high frequency and the low frequency normal modes differently. One familiar example in early universe quantum processes is in the cosmological particle creation backreaction problem (Lukash and Starobinsky (1974); Hu and Parker (1978)), where a division is made at the non-adiabatic limit for each mode to distinguish (quantum) particle creation from (classical) red-shifting effects. Another is the recently proposed stochastic inflation model where a cut-off of the fluctuation field momenta at the horizon introduces a Markovian noise source (Starobinsky (1986); Rey (1987)). However, these simple cut-off procedures cannot account for the mixing of high and low frequency modes due to non-linear interaction or nonadiabatic effects.

In his talk B. L. Hu proposed a coarse-grained effective action for these problems. Most work on coarse-grained free energy is done in critical phenomena physics carried out by condensed matter physicists. After briefly covering these approaches, B. L. Hu turned to his work done in collaboration with Yuhong Zhang.

Dear colleagues and friends, Having to summarize this meeting with its many contributions to a great variety of topics is a questionable honour. Looking through my notes I realize all too well that my wish to understand details and at the same time not to get lost in them, is larger than my ability to do so. I shall nevertheless try and give you a kind of overview of our field as it came to light-or remained in partial darkness-during this meeting.

The development of general relativity and, more generally, the physics of gravitation as it was and is reflected in the GR-meetings, starting with the Bern conference in 1955 as recalled by Engelbert Schucking in his splendid talk Thursday night, has some similarity with that of the universe, or at least with the standard model of it: a rather smooth, uniform beginning, then the evolution of more and more structure, and now a field that looks rather inhomogeneous, expanding here, contracting there, transparant in some regions, opaque (to me) in others, and partly chaotic. Important aspects of the present state are the interconnections of general relativity with gauge theories, particle physics and astrophysics. The most conspicuous and important feature of this evolution, however, is that experimental and observational research on the properties of gravity on the laboratory, terrestrial, solar system, galactic and cosmological scale has grown considerably from very small beginnings and occupies now a sizeable part of the plenary talks and, in particular, of the workshops.

Introduction

Experimental tests of general relativity are difficult. Physicists were well aware that pregnant new conceptual insights came mostly from young minds. Now, experimentalists must be cut from that mold, previously the exclusive preserve of theorists. Why? The time elapsed between a good idea for an experimental test and its execution is fast approaching the normal human lifetime. Experimenters must now start young-very young-to live to see the fruits of their ideas realized.

I divide the remainder of this paper into two parts, corresponding, respectively, to the past light cone and the future light cone. Under the former rubric, I discuss, in order, tests of the principle of equivalence, light deflection, signal retardation, perihelia advances, geodetic precession, and the constancy of the gravitational “constant.” Under the latter, I mention improved reincarnations of some of these tests, as well as proposed redshift and frame-dragging experiments.

With the proper reference frame established, I move on to the review of recent results and present plans. Because of space and time constraints, much of the treatment is perforce superficial, but, in keeping with the Fourth of July spirit, I shall be democratic and treat all experiments with (almost) equal superficiality.

Past light cone

Principle of equivalence

Space tests of the principle of equivalence have been primarily concerned with measuring the equivalence between gravitational and inertial mass in regard to the contribution to each of gravitational self-energy.

Studies of the ˜ 3K cosmic microwave background or “relict” radiation, discovered 25 years ago, have substantially improved our understanding of cosmology and of the formation of large-scale structure in the universe. Here, I look at the implications of these studies for gravity theory.

I will begin by reviewing the observations, particularly the spectrum and the large-scale isotropy of the radiation. Measurements of the spectrum, when combined with other astrophysical data like the abundance of light nuclei, establish the temperature and expansion rate of the universe at early times. These values in turn may be used to tell us whether unmodified general relativity adequately describes the dynamics of the early universe. Upper limits on any large-scale anisotropy sharply restrict the range of possible anisotropic cosmological models, and provide supporting evidence for a period of “inflationary” expansion early in the universe. The preceding paper by Dr. Panek explores some of these consequences further.

Introduction

It is an honor to have been invited to review the cosmic microwave background radiation (CBR) for this audience. As an observer and experimentalist, I feel particularly privileged since general relativity is sometimes viewed as the province of theoreticians, not those of us with hands dirtied in the laboratory or at thy telescope.

As is well known, the CBR was discovered 25 years ago by Penzias and Wilson (1965).

The upper limit set by gravity on the rotation of neutron stars is sensitive to the equation of state of matter at high density. No uniformly rotating equilibrium can have angular velocity greater than that of a particle in circular orbit at its equator, and, for a given baryon mass, the configuration with maximum angular velocity rotates at this Keplerian frequency. The limiting frequency decreases with increasing stiffness in the equation of state, because (for a given mass) models of neutron stars constructed from equations of state that are stiff above nuclear density have substantially larger radii and moments of inertia than models based on the softer equations of state. While for cold neutron stars the Keplerian frequency is the gravitational limit on angular velocity, for hotter stars (T > 10 K), viscosity is apparently low enough that gravitational instability to nonaxisymmetric perturbations sets in slightly earlier. The corresponding constraint on the equation of state is more stringent; and if the 1968 Hz frequency seen in optical emission from SN 1987A is the angular velocity of a newly formed pulsar, neutron star matter must be unexpectedly soft above nuclear density. Too soft an equation of state, however, cannot support a spherical neutron star with mass as large as the observed 1.44 solar mass member of the binary pulsar 1913+16. A rather narrow range of equations of state survives the two observational constraints.

Quark stars and stars with pion or kaon condensates are possible alternatives.

The study of exact solutions and exact properties of Einstein's equations is a rather broad mathematical subfield of general relativity. Of the roughly 80 abstracts submitted to the symposium devoted to this topic, time limitations permitted only a small fraction to be presented orally. Table 1 lists the papers given at the two sessions of this symposium. The 16 presented papers fall roughly within the categories of “exact solutions,” “gravitational energy,” “mathematical results” and “symmetry properties of Einstein's equations” and are briefly discussed under those headings in the following. Since most of the oral presentations described extremely recent research results, they did not, for the most part, include references to published papers concerning these results. For this reason the attached reference list is extremely sketchy.

Exact solutions

Virtually all of the known exact solutions of Einstein's equations involve some significant element of idealization. One usually imposes a stringent geometrical symmetry upon the solutions to be considered and, in the non-vacuum case, simplifying assumptions upon the matter sources to be included. Goenner and Sippel discussed several classes of exact solutions which, though highly idealized in the geometrical sense (being in fact pp waves), were nevertheless more realistic in their material content. The sources included both Maxwell fields and a viscous, heat-conducting plasma subject to certain natural energy and entropy inequalities. Several classes of solutions were discussed which represented the generation of a gravitational wave by an electromagnetic wave and a temperature wave propagating in the viscous gas.

We note with sadness that GR-12 was the last important conference for two of the significant figures in physics in the last half of the twentieth century: William M. Fairbank and Eduardo Amaldi. Ironically, they were raised in traditions far removed from general relativity but both had made important experimental contributions to the field during the past twenty years. Fairbank, with Schiff, Cannon, and Everitt, started the investigation that we now call the Stanford Relativity Gyroscope experiment and helped bring it to the point that it will be put into orbit as NASA's Gravity Probe B. He then, with Hamilton, started the cryogenic gravity wave detection project at Stanford to further advance the pioneering experiments of Weber. Amaldi joined with Pizzella to build tuned gravity wave detectors in Italy. When the Italian bureaucracy became too difficult he helped move the experimental laboratory to CERN where it has become the world's strongest program.

In the evening of the next to last day of GR-12 both Fairbank and Amaldi attended a small informal meeting of experimentalists representing all of the major tuned bar groups. The focus of the meeting was to establish times when all of the experiments would be operated in coincidence and to establish protocols for exchanging the data that would be generated by these coincidence experiments. They both expressed great confidence that such a coordinated effort would lead us to the discovery of gravitational waves and the development of gravitational wave astronomy.

Introduction

During the last two years a burst of results has come from radio and optical surveys of “galaxy lenses” (where the main deflector is a galaxy). Even if this kind of work were better known and had already been reviewed for this assembly a few years ago, I cannot pass up the main results which have presently emerged. This will be the first part of the presentation.

On the other hand, in September 1985 we pointed out a very strange blue ring-like structure on a Charge-Coupled Device (CCD) image of the cluster of galaxies Abell 370 (Soucail et al. (1987a)). After ups and downs and a persistent observational quest, this turned out to be the Einstein arcs discovery (Soucail et al. (1988); Lynds and Petrosian (1989)), an important observational step in this particular field since the first observation of the double Quasi-Stellar Object (QSO) 0957 + 561 in 1979 (Walsh et al. (1979)).

Following this discovery, new observational results have shown that many rich clusters of galaxies can produce numerous arclets: tangen-tially distorted images of an extremely faint galaxy population probably located at redshift larger than 1 (Fort et al. (1989); Tyson (1989)). This new class of gravitational lenses proves to be an important observational topic (Mellier (1989a); Fort (1989a)). This story will be the second part of this presentation.

The classification between galaxy lenses and cluster lenses is somewhat arbitrary.

Introductory survey

The implementation of a new computer algebra system is time consuming: designers of general purpose algebra systems usually say it takes about 50 man-years to create a mature and fully functional system. Hence the range of available systems and their capabilities changes little between one GR meeting and the next, despite which there have been significant changes in the period since the last report.

I do not believe there is a single “best” system (though like everybody else I am biased towards the systems I actually use), and in particular one should be extremely cautious about any claims about comparative efficiency of systems. These introductory remarks therefore aim to give a very brief survey of capabilities of the principal available systems and highlight one or two trends. The most recent full survey of computer algebra in relativity (as far as I know) is in Ref. 2, while very full descriptions of the Maple, REDUCE and SHEEP applications will appear in a forthcoming lecture notes volume.

The oldest of the still current general purpose systems are REDUCE and Macsyma. REDUCE is a highly portable system available on a very wide range of machines and is sufficiently cheap to have become widespread in most parts of the world. Its main disadvantage until recently has been the rather small range of auxiliary packages for applied mathematics, but this is improving rapidly with the availability of contributed programs through electronic mail (to reduce-netlib@rand.org: an initial mail should contain the one line ‘send index’).