Introduction

One of Charles Misner's most influential contributions to the mathematical development of general relativity was his proposal, along with R. Arnowitt and S. Deser, of the ADM program of (Hamiltonian) reduction of Einstein's equations [1, 2]. By ADM reduction we mean not simply the elegant but essentially elementary reexpression of Hilbert's action in terms of canonical variables but rather the complete reduction of Einstein's equations to an unconstrained Hamiltonian system defined over an appropriate phase space of “true degrees of freedom.” This program, though developed over the years by many people, beginning of course with Arnowitt, Deser and Misner themselves, remains largely unfinished even today. Nevertheless many model problems have been worked out, to a large extent by Misner and his former students and other collaborators, and the reduction program has proven itself of value, within these models, for the study of both classical and quantum dynamics. In this article we shall describe some recent developments of the ADM program and discuss some of the problems which remain to be solved.

Some of the work described below in Sect. II is based in part on a Yale senior physics project by Juan Lin, whereas much of that described in Sect. Ill is based upon the Yale Mathematics Ph.D. thesis of John Cameron (1991). The work described in Sect. IV represents research in progress in collaboration with Arthur Fischer (University of California, Santa Cruz). Section V mentions some recent joint work with Yvonne Choquet-Bruhat (Universite Paris VI) and James Isenberg (University of Oregon).

Abstract

Gravitational radiation antennas have been operating since 1965. A large number of pulses have been observed, coincident on widely separated antennas. These data and the Supernova 1987A observations are reviewed. It is concluded that some of these pulses may have a gravitational radiation origin.

Introduction

The theory of elastic solid, and interferometer gravitational radiation antennas has been under development at the University of Maryland since 1957. Aluminum bar systems have been operating continuously since 1965.

It is very important to stress that the output of a gravitational antenna (and a neutrino detector as well) differs in fundamental ways from the output of an optical telescope.

When an optical telescope collects light from a star, it can be concluded that most of the light came from the star.

A gravitational antenna - bar or interferometer - has electrical output pulses. For a single antenna, there is no way to guarantee that observed pulses are not noise of internal origin or noise from local disturbances such as lightning. If statistically significant numbers of coincident pulses are observed on widely separated antennas, this is evidence that the pulses have a common origin. Directive information is useful but not conclusive, in identifying the source.

There is no way to be certain that such pulses are or are not due to gravitational radiation.

Therefore the statement which is frequently made that gravitational radiation has not yet been observed is meaningless.

Abstract

We trace the development of ideas on dissipative processes in chaotic cosmology and on minisuperspace quantum cosmology from the time Misner proposed them to current research. We show

1. 1) how the effect of quantum processes like particle creation in the early universe can address the issues of the isotropy and homogeneity of the observed universe,

2. 2) how viewing minisuperspace as a quantum open system can address the issue of the validity of such approximations customarily adopted in quantum cosmology, and

3. 3) how invoking statistical processes like decoherence and correlation when considered together can help to establish a theory of quantum fields in curved spacetime as the semiclassical limit of quantum gravity.

Dedicated to Professor Misner on the occasion of his sixtieth birthday, June 1992.

Introduction

In the five years between 1967 and 1972, Charlie Misner made an idelible mark in relativistic cosmology in three aspects.

First he introduced the idea of chaotic cosmology. In contrast to the reigning standard model of Friedmann-Lemaitre-Robertson-Walker universe where isotropy and homogeneity are ‘put in by hand’ from the beginning, chaotic cosmology assumes that the universe can have arbitrary irregularities initially. This is perhaps a more general and philosophically pleasing assumption. To reconcile an irregular early universe with the observed large scale smoothness of the present universe, one has to introduce physical mechanisms to dissipate away the anisotropies and inhomogeneities. This is why dissipative processes are essential to the implementation of the chaotic cosmology program. Misner (1968) was the first to try out this program in a Bianchi type-I universe with the neutrino viscosity at work in the lepton era.

Abstract

Misner space (i.e. Minkowski spacetime with identification under a boost) is a remarkably rich prototype for a variety of pathologies in the structure of spacetime—some of which may actually occur in the real Universe. The following examples are discussed in some detail: traversable wormholes, violation of the averaged null energy condition, chronology horizons (both compactly and noncompactly generated) at which closed timelike curves are created, classical and quantum instabilities of chronology horizons, and chronology protection.

Introduction

In August 1965, at the Fourth Summer Seminar on Applied Mathematics at Cornell University, Charles Misner gave a lecture titled “Taub-NUT space as a counterexample to almost anything” [Mis67]. Near the end of his lecture, Misner introduced an exceedingly simple spacetime that shares some of Taub-NUT's pathological properties. This spacetime has come to be called Misner space.

Twenty-three years later, when I became intrigued by the question of whether the laws of physics forbid traversable wormholes and closed timelike curves, and if so, by what physical mechanism the laws prevent them from arising, Bob Wald and Robert Geroch reminded me of Misner space and its relevance to these issues. Since then, I and others probing these issues have found Misner space and its variations to be fertile testing grounds and powerful computational tools.

In this paper, I shall describe the remarkable pathologies that Misner space encompasses, and what we have learned about wormholes and closed timelike curves with the aid of Misner space. My discussion will be quite elementary, requiring little more than a basic understanding of special relativity.

Abstract

The functional Schrodinger approach to canonical quantum gravity requires the construction of time and frame variables from the canonical data. I review the difficulties of basing such variables purely on the geometric data. I describe different ways of introducing matter variables for this purpose. I argue that the simplest phenomenological medium, an incoherent dust, does more than one can reasonably expect. The comoving coordinates of the dust particles and the proper time along their worldlines become canonical coordinates in the phase space of the coupled system. The Hamiltonian constraint can easily be solved for the momentum canonically conjugate to the dust time. The ensuing Hamiltonian density has an extraordinary feature not encountered in other systems: it depends only on the geometric variables, not on the dust frame and the dust time. This has three important consequences. Firstly, the functional Schrödinger equation can be solved by separating the dust time from geometric variables. Secondly, the Hamiltonian densities strongly commute and can therefore be simultaneously defined by spectral analysis. Thirdly, the Schrodinger equation can be solved independently of the supermomentum constraint, which is then satisfied by parametrizing this solution.

Canonical quantum gravity

One can never pay off old debts; the most one can do is to acknowledge them. Not that I lacked opportunity to acknowledge those which I owe Charlie Misner. He helped to set the entire framework of canonical gravity, and there is hardly a paper which I wrote since coming to the United States in which I did not have the chance – and the pleasure – to refer to his contributions.

Introduction

It was an honour to be a student of Charles Misner. And it was a pleasure to work with him. It is once again an honour and a pleasure to write an article in celebration of his sixtieth birthday.

The phenomenon of rotation generates interesting physical effects and involves intriguing basic concepts. One of the earliest classic examples of this is Newton's water-pail experiment. Within the framework of the general theory of relativity rotation displays rather unusual features. These are incorporated, for instance, in the spacetime structure such as that of a rotating black hole. Dragging of inertial frames and the occurrence of the ergosphere, to name two examples, are the outcome of the rotation built into the spacetime. Rotational effects also show up in the characteristics of particle motion and associated phenomena like gyroscope precession. These effects are revealed in an elegant manner by the invariant geometrical description of particle trajectories following the directions of spacetime symmetries, assuming that the spacetime under consideration admits such symmetries. In three dimensions, this geometrical characterization involves the specification of the curvature κ and torsion τ of the curve as functions of some parameter that varies along the curve, normally the arc-length. This can be extended to higher dimensions including time and the relevant geometrical parameters would then be κ, τ1, τ2…τn-1 for n-dimensions. These parameters fit naturally into the Frenet-Serret formalism that is well known to the geometers. When the formalism is applied to timelike integral curves of space symmetries many interesting features emerge. In the major part of this article we shall briefly review the results that have been obtained in this area of investigation.

This summary of Charles Misner's publications only hints at the richness of his research. Misner's work is characterized by a fascination with geometry in its broadest sense and by a desire to probe the physical manifestations of gravitation. Many of his papers initiated new areas of study in general relativity. These areas either provide continuing research interest, have experienced one or more revivals, or have developed from an essential ingredient he provided. This review will emphasize those aspects of his research which have become part of the essential background of our subject. References [n] are to Misner's list of publications near the end of this volume.

To appreciate Misner's impact on general relativity one need only recall the state of this field when he began his research at Princeton in the 1950's. Major activities in the previous decades included the then-ignored work by Oppenheimer and Snyder on gravitational collapse and by Alpher and Herman predicting a 5 degree cosmic background radiation. Apart from cosmology, the appreciated work included the Einstein-Infeld-Hoffman equations of motion results from the late 1930's, Bergmann's studies of quantum gravity from the early 1950's, and the studies of the initial value problem by Lichnerowicz and Fourès (Choquet-Bruhat). Active centers with an interest in general relativity as Wheeler started his group at Princeton included those led by Bergmann at Syracuse, Lichnerowicz and Fourès-Bruhat in France, Bondi in London, Klein and Møller in Scandinavia, Synge and Pirani in Dublin (one of Schild's sojourns also), Jordan and Ehlers in Hamburg, Inf eld in Warsaw, and a few others.

Abstract

I discuss current observational limits on the inhomogeneity and the isotropy of the universe. Isotropy observations come from the COBE differential microwave radiometer. COBE results are consistent with prior estimates based on cosmic nucleosynthesis. The COBE results on the present structure can be used to limit the range of background density, in particular the closure of the described universe.

Examples from the literature are given whereby a 30 eV massive neutrino simultaneously fits both the observed structure on small scales, and the level of observed quadrupole anisotropy. Further simulations are needed to verify these theoretical fits to the observations.

This paper is dedicated to Charles Misner on his sixtieth birthday.

Introduction

In 1966, prompted by the apparent anisostropic distribution of the three or four then known QSOs, Charles Misner began investigating the behavior of anisotropic universes. These had been studied before, by Kasner [1], Zel'dovich [2], Thorne [3], Taub [4], but Misner's development was a tour de force combining differential geometry, classical mechanics, and astrophysics. One track of his research led to the Mixmaster universe [5], a closed 3-spherical universe in which the ratios of the principal circumferences oscillate as the universe expands and recollapses. This oscillation can lead to very large horizon lengths in particular directions, and gave the hope of explaining the horizon problem. The Mixmaster results led directly to Hamiltonian cosmology and the Quantum Cosmology research effort. The more astrophysical branch of this research [6] developed into studies of dissipation in anisotropic Bianchi type I cosmologies.

Abstract

In several of the class A Bianchi models, minisuperspaces admit symmetries. It is pointed out that they can be used effectively to complete the Dirac quantization program. The resulting quantum theory provides a useful platform to investigate a number of conceptual and technical problems of quantum gravity.

Introduction

Minisuperspaces are useful toy models for canonical quantum gravity because they capture many of the essential features of general relativity and are at the same time free of the technical difficulties associated with the presence of an infinite number of degrees of freedom. This fact was recognized by Charlie Misner quite early and, under his leadership, a number of insightful contributions were made by the Maryland group in the sixties and the seventies. Charlie's own papers are so thorough and deep that they have become classics; one can trace back to them so many of the significant ideas in this area. Indeed, it is a frequent occurrence that a beginner in the field gets excited by a new idea that seems beautiful and subtle only to find out later that Charlie was well aware of it. It is therefore a pleasure and a privilege to contribute an article on minisuperspaces to this Festschrift –of course, Charlie himself may already know all our results!

In this paper we shall use the minisuperspaces associated with Bianchi models to illustrate some techniques that can be used in the quantization of constrained systems –including general relativity– and to point out some of the pitfalls involved.

Abstract

An important limitation is shown in the analogy between the Aharonov-Bohm effect and the parallel transport on a cone. It illustrates a basic difference between gravity and gauge fields due to the existence of the solder form for the space-time geometry. This difference is further shown by the observability of the gravitational phase for open paths. This reinforces a previous suggestion that the fundamental variables for quantizing the gravitational field are the solder form and the connection, and not the metric.

INTRODUCTION

I recall with great pleasure the discussions which I had with Charles Misner on fundamental aspects of physics, such as the geometry of gravity, gauge fields, and quantum theory. In particular, I remember the encouragement he gave to my somewhat unorthodox attempts to understand the similarities and differences between gauge fields and gravity from their effects on quantum interference, and their implications to physical geometry. It therefore seems appropriate to present here for his Festschrift some observations which came out of this investigation.

Geometry is a part of mathematics which can be visualized, and is intimately related to symmetries. This may explain the tremendous usefulness of geometry in physics. In section 2, I shall make some basic remarks about the similarities and differences between the geometries of gravity and gauge field. Then I shall illustrate, in section 3, an important difference between them that arises due to the existence of the solder form for gravity, using the Aharonov-Bohm (AB) effect and parallel transport on a cone.

Abstract

We extend the argument that spacetimes generated by two timelike particles in D=3 gravity (or equivalently by parallel-moving cosmic strings in D=4) permit closed timelike curves (CTC) only at the price of Misner identifications that correspond to unphysical boundary conditions at spatial infinity and to a tachyonic center of mass. Here we analyze geometries one or both of whose sources are lightlike. We make manifest both the presence of CTC at spatial infinity if they are present at all, and the tachyonic character of the system: As the total energy surpasses its tachyonic bound, CTC first begin to form at spatial infinity, then spread to the interior as the energy increases further. We then show that, in contrast, CTC are entirely forbidden in topologically massive gravity for geometries generated by lightlike sources.

Among the many fundamental contributions by Charlie Misner to general relativity is his study of pathologies of Einstein geometries, particularly NUT spaces, which in his words are “counterexamples to almost everything”; in particular they can possess closed timelike curves (CTC). As with other farsighted results of his which were only appreciated later, this 25-year old one finds a resonance in very recent studies of conditions under which CTC can appear in apparently physical settings, but in fact require unphysical boundary conditions engendered by identifications very similar to those he discovered. In this paper, dedicated to him on his 60th birthday, we review and extend some of this current work. We hope it brings back pleasant memories.

Introduction

Originally constructed by Gödel [1], but foreshadowed much earlier [2], spacetimes possessing CTC in general relativity came as a surprise to relativists.

Abstract

We discuss the relationship between geometry, the renormalization group (RG) and gravity. We begin by reviewing our recent work on crossover problems in field theory. By crossover we mean the interpolation between different representations of the conformal group by the action of relevant operators. At the level of the RG this crossover is manifest in the flow between different fixed points induced by these operators. The description of such flows requires a RG which is capable of interpolating between qualitatively different degrees of freedom. Using the conceptual notion of course graining we construct some simple examples of such a group introducing the concept of a “floating” fixed point around which one constructs a perturbation theory. Our consideration of crossovers indicates that one should consider classes of field theories, described by a set of parameters, rather than focus on a particular one. The space of parameters has a natural metric structure. We examine the geometry of this space in some simple models and draw some analogies between this space, superspace and minisuperspace.

Introduction

The cosmopolitan nature of Charlie Misner's work is one of its chief features. It is with this in mind that we dedicate this article on the occasion of his 60th birthday. There are several recurring leitmotifs throughout theoretical physics; prominent amongst these would be geometry, symmetry, and fluctuations. Geometry clarifies and systematizes the relations between the quantities entering into a theory, e.g. Riemannian geometry in the theory of gravity and symplectic geometry in the case of classical mechanics. Symmetry performs a similar role, and in the case of continuous symmetries is often intimately tied to geometrical notions.

ABSTRACT

We describe a method for the numerical solution of Einstein's equations for the dynamical evolution of a collisionless gas of particles in general relativity. The gravitational field can be arbitrarily strong and particle velocities can approach the speed of light. The computational method uses the tools of numerical relativity and N-body particle simulation to follow the full nonlinear behavior of these systems. Specifically, we solve the Vlasov equation in general relativity by particle simulation. The gravitational field is integrated using the 3 + 1 formalism of Arnowitt, Deser, and Misner. Our method provides a new tool for studying the cosmic censorship hypothesis and the possibility of naked singularities. The formation of a naked singularity during the collapse of a finite object would pose a serious difficulty for the theory of general relativity. The hoop conjecture suggests that this possibility will never happen provided the object is sufficiently compact (≲M) in all of its spatial dimensions. But what about the collapse of a long, nonrotating, prolate object to a thin spindle? Such collapse leads to a strong singularity in Newtonian gravitation. Using our numerical code to evolve collisionless gas spheroids in full general relativity, we find that in all cases the spheroids collapse to singularities. When the spheroids are sufficiently compact the singularities are hidden inside black holes. However, when the spheroids are sufficiently large there are no apparent horizons. These results lend support to the hoop conjecture and appear to demonstrate that naked singularities can form in asymptotically flat spacetimes.