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.

Of all obstacles to understanding the foundations of physics, it is difficult to point to one more challenging than the question, “How Come the Quantum?” unless it be the twin question, “How Come Existence?” Stuck, but studying every available clue, (Box 1), from the papers of Bohr, Einstein, Planck and Schrödinger to the thoughts of the presocratic philosophers, (Box 2), I remember one of the great messages I have received from sixtyfive years of research: Why does a university have students? To teach the professsors! Not least in convincing me of that lesson is the wealth of learning that I owe to Charles W. Misner, graduate student at Princeton University from 1953 to 1957.

Already from the time Misner dropped into my office to talk about a conceivable thesis topic, I gained a vivid impression of what it was to see his active mind at work comparing researchable issues in elementary particle physics and in general relativity. “What is timely and tractable?” That is the proper criterion of choice, according to John R. Pierce, that great guide of productive research at Bell Telephone Laboratories and animating spirit of the travelling-wave tube and the Tel Star satellite.

Charles Misner, so far as I could see, used the same criterion in making his decision. It led to a Ph.D. thesis and a 1957 paper in the Reviews of Modern Physics, entitled “Feynman quantization of general relativity,” forerunner to the great and influential 1962 paper of R. Arnowitt, S. Deser and Misner on the “Dynamics of General Relativity.”

Abstract

An example is presented which points to a certain basic difficulty in the “already unified” approach to unified field theory. It is shown that one can construct a pair of solutions of the combined Einstein-Maxwell equations for which the two space-times are identical in the neighbourhood of an initial spacelike hypersurface (and in fact they may also be identical at all earlier times), but the time-development of the equations leads to space-times which are essentially different in their futures. The construction of such examples requires the electromagnetic field to be null (or zero) in some regions. The example given here represents a collision between two gravitational-electromagnetic waves.

Introductory preamble

This paper was written in late 1959 or early 1960, while I was at Princeton University in the early part of my research career in general relativity. It was at a time when I knew Charlie Misner best, since he was also in Princeton then, and I learnt a great deal from him about issues of general relativity, such as the initial value problem etc. As far as I can recall, it was discussions with him, and also with John Wheeler, that led to the ideas described in this paper.

I had completed the paper, and gave it to John Wheeler for his comments. Unfortunately, unforseen circumstances intervened, and it was not until several months later that the paper resurfaced, at which time my own interests had moved elsewhere. The celebration of Charlie's 60th birthday seemed an ideal occasion on which to resurrect the paper, and I searched through old files in order to locate it.

Abstract

Four-dimensional Euclidean spaces that solve Einstein's equations are interpreted as WKB approximations to wavefunctionals of quantum geometry. These spaces are represented graphically by suppressing inessential dimensions and drawing the resulting figures in perspective representation of threedimensional space, some of them stereoscopically. The figures are also related to the physical interpretation of the corresponding quantum processes.

Introduction

Understanding General Relativity means to a large extent coming to terms with its most important ingredient, geometry. Among his many contributions, Charlie has given us new variations of this theme [1], fascinating because geometry is so familiar on two-dimensional surfaces, but so remote from intuition on higher-dimensional spacetimes. The richness he uncovered is shown nowhere better than in the 137 figures of his masterful text [2].

Today quantum gravity [3] leads to new geometrical features. One of these is a new role for Riemannian (rather than Lorentzian) solutions of the Einstein field equations: such “instantons” can describe in WKB approximation the tunneling transitions that are classically forbidden, for example because they correspond to a change in the space's topology. In order to gain a pictorial understanding of these spaces we can try to represent the geometry as a whole with less important dimensions suppressed; an alternative is to follow the ADM method and show a history of the tunneling by slices of codimension one.

We can readily go from equation to picture thanks to computer plotting routines, from the simpler ones as incorporated in spreadsheet programs [4] to the more powerful versions of Mathematica.

It is a pleasure to contribute this paper in honor of Charlie Misner for his many contributions to gravitational theory and for his warm friendship.

INTRODUCTION

The discovery of general relativity by Einstein and its early experimental verification excited at that time both the scientific and lay public alike. However, during the 1930s and 1940s the hope that gravity would be a unifying principle of nature faded. The discovery of the self-energy infinities of Lorentz covariant quantum field theory indicated the insufficiency of the quantum theoretical framework. Subsequently, it was realized that these infinities were even more virilant in the non-renormalizable general relativity. Most significant was the experimental discovery during this period of the weak and strong interactions, implying that the original ideas of Einstein and Weyl to unify gravity with electromagnetism were premature. Perhaps the one idea from this era that has remained in present day efforts to unify interactions was the most radical: the suggestion by Kaluza and Klein that there might exist additional compactified dimensions in space-time. Most remarkable was the work of Oscar Klein who, using dimensional reduction, discovered non-abelian gauge theory and applied it to construct a precursor of present day electro-weak theory. This was a spectacular theoretical tour-de-force which unfortunately did not appear to stimulate further work at that time.

The development of the Glashow-Weinberg-Salam model of electro weak interactions combined with the QCD theory of strong interactions to form the Standard Model, has led to the recent approaches to build models of unified interactions.

The formulation of the Einstein field equations admitting two Killing vectors in terms of harmonic mappings of Riemannian manifolds is a subject in which Charlie Misner has played a pioneering role. We shall consider the hyperbolic case of the Einstein-Maxwell equations admitting two hypersurface orthogonal Killing vectors which physically describes the interaction of two electrovac plane waves. Following Penrose's discussion of the Cauchy problem we shall present the initial data appropriate to this collision problem. We shall also present three different ways in which the Einstein-Maxwell equations for colliding plane wave spacetimes can be recognized as a harmonic map. The goal is to cast the Einstein-Maxwell equations into a form adopted to the initial data for colliding impulsive gravitational and electromagnetic shock waves in such a way that a simple harmonic map will directly yield the metric and the Maxwell potential 1-form of physical interest.

*for Charles W. Misner on his 60th birthday

Introduction

Charlie Misner was the first to recognize that the subject of harmonic mappings of Riemannian manifolds finds an important application in general relativity. In a pioneering paper with Richard Matzner [1] he found that stationary, axially symmetric Einstein field equations can be formulated as a harmonic map. Eells and Sampson's theory of harmonic mappings of Riemannian manifolds [2] provides a geometrical framework for thinking of a set of pde's, in the same spirit as “mini-superspace” that Charlie was to introduce [3] for ode Einstein equations a little later.

Charlie Misner's contributions, characterized by profound physical insight and brilliant mathematical skill, have left an indelible mark on general relativity during its course of development for more than the past three decades. Equally important has been his influence on his colleagues and coworkers. To his students he has been a gentle guide, a model mentor and a source of inspiration. Charlie's curriculum vitae included at the end of this volume offers a glimpse of his scholarship and achievements. At the same time, the excerpts from the messages gathered for him on the occasion of his sixtieth birthday, June 13, 1992, are an eloquent testimony to the affection, respect and gratitude of his friends, colleagues and students.

The articles that follow have been written by experts in their respective fields. The areas covered range over a wide spectrum of topics in classical relativity, quantum mechanics, quantum gravity, cosmology and black hole physics. The latest developments in these subjects have been presented, often with reference to the perspective of the past and with indications of future directions. One can discern in most of these articles the influence of Charlie Misner in one form or another.

A novel feature of this Festschrift is that it represents the time-reversed version of the proceedings of an international symposium on Directions in General Relativity organized at the University of Maryland, College Park, May 27–29, 1993, at which the contents of some of these articles and related topics will be discussed in detail. The symposium is in honour of Charles Misner as well as Dieter Brill whose sixtieth birthday falls on August 9, 1993.

From the time that Newton first proposed that there was a universal force of gravity inversely proportional to the square of the distance between two point masses, there have been recurrent investigations of how far that rule was correct, and many different alternative forms have been suggested. The other assumption that Newton made, that the force of gravity did not depend on the chemical composition of bodies, has also been questioned from time to time; Newton himself carried out the first experimental test of what has become known as the weak principle of equivalence. It has often been suggested that some apparently anomalous behaviour in celestial mechanics should be ascribed to a failure of the inverse square law; indeed Clairaut developed the first analytical theory of the motion of the Moon because of discrepancies between Newton's theory and observation that might have been due to an inverse-cube component of the force. As with all subsequent studies before general relativity, careful analysis showed that the effects were consistent with the inverse square law. General relativity predicts a small deviation from the inverse square law close to very massive bodies, a deviation that has been confirmed by careful observation.

The motions of celestial bodies about each other are, with very minor exceptions, unable to reveal any departure from the weak principle of equivalence; if such departures are to be detected, they must be sought in laboratory experiments or geophysical observations.

Three hundred years after Newton published Philosophiae Naturalis Principia Mathematica the subject of gravitation is as lively a subject for theoretical and experimental study as ever it has been (Hawking & Israel, 1987). Theorists endeavour to relate gravity to quantum mechanics and to develop theories that will unify the description of gravity with that of all other physical forces. Experimenters have looked for gravitational radiation, for anomalies in the motion of the Moon that would correspond to a failure of the gravitational weak principle of equivalence, for deviations from the inverse square law and for various other effects that would be inconsistent with general relativity. The cosmological implications of general relativity continue to be elaborated and various ways of using space vehicles to test notions of gravitation have been proposed. In particular, the last three decades have seen a considerable effort devoted to applying modern techniques of measurement and detection of small forces to experiments on gravitation that can be done within an ordinary physics laboratory, and it is those that are the subject of this book.

Our scope is indeed quite restricted. It is concerned with experiments where the conditions are under the experimenter's control, in contrast to observation, where they are not. It is concerned with experiments that can be done within a more or less ordinary-sized room, that is to say, the distances between attracting body and attracted body do not usually exceed a few metres and may often be much less, while the masses of gravitating bodies are of the order of kilograms or much less.