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 [email protected]: 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.

Inflationary supercomputing

There are many problems in numerical relativity. From a view point of asymptotic behavior of space-times they are divided into three categories as:

On the other hand from a view point of dimension of the problem they are divided into another three categories as:

ID: A problem in which all the physical quantities depend on one spatial coordinate x1 and the time t;

2D: A problem in which all the physical quantities depend on two spatial coordinates x1 and x2 and the time t;

3D: A problem in which all the physical quantities depend on three spatial coordinates x1, x2 and x3 and the time t.

Every problem can be characterized according to these two classifications. For example, the two black hole collision calculated by Smarr belongs to V-2D. There are also many numerical methods for each problem. They include the characteristic method, Regge calculus, finite element method, finite difference method, spectral method, particle method and so on. Moreover the choice of coordinate conditions is not usually unique for each method. Therefore there are a tremendous variety of possibilities for each author's problem. In reality there are up to 30 papers on numerical relativity in this conference which are reviewed by Centrella in this volume.

One of the main factors, however, which determines progress in numerical relativity is the power of available computers.

Introduction

After 20 years of careful and innovative experimenting most of the researchers in the field of gravitational wave detection believe that success is on the horizon. Theoretical predictions of source strengths and source types have been steadily evolving and it is now clear that we should be aiming to build gravitational wave detectors which have strain sensitivities of ∼ 10-22 over kilohertz bandwidths. Such sensitivity should allow the detection of signals from, for example, supernova events at distances out to the Virgo cluster, coalescing compact binary systems and continuous and stochastic background sources.

One of the most promising ways of achieving the required sensitivity and bandwidth is to use laser interferometry between freely suspended test masses placed several kilometers apart, and the majority of contributions to the workshop on laser interferometer gravitational wave detectors were related to the large laser interferometer projects currently well advanced in planning. These instruments rely on searching for changes in the relative length of two paths, usually at right angles to each other, and formed between the test masses suspended as pendulums.

As will be mentioned again later, a number of interferometers are required around the world to obtain useful astrophysical information from the strength, polarization and timing of signals detected, and currently the belief is that at least three separate detector systems at different sites are necessary.

The present status of the new variables program for canonical quantum gravity is discussed. A summary is given of the papers which were presented at the New Variables Workshop at GR-12, and particular attention is given to those issues which are crucial at the present stage of development of this program. Chief among these issues is whether a theory can be quantized nonperturbatively in the absence of any information about the physical observables algebra of the corresponding classical theory. Finally, a wild speculation about the relationship between nonperturbative quantum general relativity and perturbative string theory is made.

Introduction

Abhay Ashtekar introduced his new variables in the Fall of 1985. In the intervening four years this new formalism has been developed and applied to both classical and quantum general relativity. On both sides significant new results have been achieved, and these developments are being actively pursued by a growing number of people. It was thus appropriate to have a workshop at GR-12 dedicated to this subject. In this, a summary of that workshop, I will try to describe briefly several of the directions that this work has taken, with an emphasis on the present status and open problems. In doing so I will touch on each of the six presentations that were given in the workshop, but I will not strictly follow the format of the workshop itself.

The hot big-bang cosmology is based upon the Friedmann-Robertson-Walker (FRW) solution of general relativity. It is a remarkably successful model, providing a reliable and tested accounting of the history of the universe from about 10-2 sec after the bang until today, some 15 Gyr later. It is so successful that it is known as the standard model of cosmology. It accommodates—and in some instances explains—most of the salient features of the observed universe, including the Hubble expansion, the 2.74 K cosmic microwave background radiation (CMBR), the abundance of the light elements D, 3He, 4He, and 7Li, and the existence of structures likes galaxies, clusters of galaxies, etc. In the splendor of its success, it has elevated our thinking about the evolution of the universe, and we have been able to ask a new set of even more profound questions about the universe. These questions include: What is the origin of the baryon number of the universe? What is the origin of the primeval inhomogeneities that gave rise to the structure we see today? Why is the part of the universe we can see (our present Hubble volume) so isotropic and homogeneous—as evidenced by the uniformity of the CMBR temperature—and spatially flat? What is the structure of the universe beyond our Hubble volume? What is the nature of the ubiquitious dark matter? Are there other significant cosmological relics to be discovered? Why is the cosmological constant (equivalently the present vacuum energy density) so small compared to its natural scale: ∧ ≲ 10-122G-1?

Anisotropies of the temperature of the Cosmic Background Radiation (CBR) give us unique information on the universe at redshifts of about 1000. By using observational limits on these anisotropies we can constrain the parameters of cosmological models and the spectrum and amplitude of the initial perturbations. Additional information on the thermal history of the universe is provided by distortions of the CBR spectrum. In this paper we give an overview of physical processes leading to anisotropies in popular cosmological models.

Introduction

The remarkable isotropy of the CBR in a universe containing so many structures is one of the most fascinating cosmological observations. Strong limits put on possible anisotropies of the CBR let us impose important constraints on models of global structure of the universe, and on models of galaxy and cluster formation.

The difference between the CBR and all the other radiations investigated in astronomy is that the former existed from the beginning of the universe, and the latter were created only after first objects were formed. Also, the CBR has now a 2.74K blackbody spectrum, and the radiation of astronomical objects is usually not blackbody.

Following the discovery of the CBR (Penzias and Wilson (1965)), progress in observations has been rapid. Currently, anisotropies on the level of 10−4 – 10−5 are detectable on a wide range of angular scales and frequencies. (Observations of anisotropy are usually made by comparing intensities of radiation detected by two antennas, separated by an angle θ, the angular scale.

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.