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’).

The aims of quantum cosmology

Our observations of the world give us specific facts. Here, there is a galaxy; there, there is none. Today, there is a supernova explosion; yesterday, there was a star. Here, there are fission fragments; before, there was a uranium nucleus. The task of physics is to bring order to this great mass of facts which constitutes our experience. In the language of complexity theory, the task is to compress the message which describes these facts into a shorter form — to compress it, in particular, to a form where the message consists of just a few observed facts together with simple universal laws of nature from which the rest can be deduced.

In the past, physics, for the most part, has concentrated on finding dynamical laws which correlate facts at different times. Such laws predict later evolution given observed initial conditions. However, there is no logical reason why we could not look for laws which correlate facts at the same time. Such laws would be, in effect, laws of initial conditions.

I believe it was the limited nature of our observations which led to our focus on dynamical laws. Now, however, in cosmology, in the observations of the early universe and even on familiar scales, it is possible to discern regularities of the world which may find a compressed expression in a simple, testable, theory of the initial conditions of the universe as a whole.

Introduction

At the Padova GRG conference, a new avenue to non-perturbative, canonical quantum gravity was suggested (Newman (1984)). By the time the Stockholm conference was held, these preliminary ideas had blossomed into a broad program aimed at analysing the structure of classical and quantum general relativity from a somewhat unusual standpoint (Ashtekar (1986), (1987)). By now, over two dozen individuals have contributed to this program. The purpose of this chapter is to present a brief status report of this body of ideas. Although I will try to be objective, it is inevitable that not everyone who is working in this field will agree with all the views expressed here. Also, since my space is limited, I will have to leave out several interesting results; I apologize in advance for these omissions.

The key idea underlying this program is to shift the emphasis from geometrodynamics to connection dynamics. In the classical theory, the new viewpoint merely complements the traditional one in which the metric, rather than a connection, is taken as the fundamental variable. We do obtain a fresh perspective that simplifies certain issues and suggests new ways of tackling unresolved problems. However, as far as the basic features of the theory are concerned, nothing is really altered conceptually. It is in the quantum regime that the shift of emphasis plays a major role. More precisely, there are indications that connection dynamics is indeed a better tool to analyze the micro-structure of space-time in a non-perturbative way.

Recent suggestions of a “fifth force” have stimulated many experiments to search for new macroscopic interactions arising from the exchange of ultra-low mass fundamental bosons. The experiments fall into two categories: searches for violation of the inverse square law, or of the universality of free fall. The principles of both classes of experiments are described and their results are summarized. Because some groups claim positive effects considerably larger than the upper limits established by others, subtle systematic errors that could masquerade as a “fifth force” are briefly discussed. I conclude that there is, at present, no credible evidence for new macroscopic interactions.

Introduction

One feature common to essentially all extensions of the standard model is the prediction of additional fundamental scalar or vector bosons. While these particles are ordinarily expected to be very massive (mbc2 > 1015 eV), the possibility that some have them have such a low mass, mbc2 < 10−3 eV, that they produce macroscopic forces between unpolarized test bodies, has been considered in a variety of contexts. For example, such speculations have been inspired by Kaluza-Klein theories, quantum gravity ideas, scale invariance, CP-violating pseudo-Goldstone bosons, etc. Some of these would have profound astrophysical consequences: ultra-low mass bosons have been invoked to explain the “vanishing” of the cosmological constant, the anomalous rotation curves of galaxies, and the observed “cumpiness” of the universe.

The workshop on mathematical cosmology was devoted to four topics of current interest. This report contains a brief discussion of the historical background of each topic and a concise summary of the content of each talk.

The observational cosmology program

The standard approach for analyzing cosmological observations is to assume that space-time is isotropic and spatially homogeneous (i.e. that the cosmological principle holds). It then follows that the universe can be described by a Friedmann-Lemaitre-Robertson-Walker (FLRW) model, and the aim is to use the observations to determine the free parameters that characterize such models. A fundamental question, however, is whether ideal cosmological observations on our past null cone can be used to actually determine the geometry of the cosmological space-time, without introducing a priori assumptions about the geometry. This question provides the rationale for the observational cosmology program, as described by Ellis et al. (1985). In this paper it was shown that ideal observations alone do not determine the geometry of spacetime. For example, even if all observations are isotropic about our position, it does not follow that the spacetime is spherically symmetric about our position. However, if the Einstein field equations (EFEs) are assumed to hold, then ideal observations do determine the spacetime geometry off our past null cone.

In the workshop, W. R. Stoeger reported on work in progress with S. D. Nel and G. F. R. Ellis concerning the observational cosmology program.