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.

Introduction

The discovery of γ-ray bursts was serendipitous, as was that of pulsars, which were discovered at about the same time. Pulsars were first detected in 1967 in an experiment designed to study interplanetary scintillation of compact radio sources, and the discovery paper (Hewish et al. 1968) was subsequently published; the first γ-ray burst (GRB) was also seen in the year 1967 (although not reported until six years later; see Strong and Klebesadel 1976 for an account of the chronology) in a satellite-borne detector intended to monitor violations of the nuclear explosion test ban treaty. The publication of the discovery of GRBs was first made in 1973 by Klebesadel, Strong and Olson (1973). The detector comprised six caesium iodide scintillators, each of 10 cm3, mounted on each of the four Vela series of satellites (5A, 5B, 6A and 6B), these vehicles being arranged nearly equally spaced in a circular orbit with a geocentric radius of ∼ 1.2 × 105 km. The detectors were sensitive to individual γ-rays in the approximate energy range 0.2−1.5 MeV and the detector efficiency ranged from 17 to 50%. The scintillators had a passive shield around them; background γ-ray counting rates were routinely monitored. A statistically significant increase in the counting rates initiated the recording of discrete counts in a series of quasi-logarithmically increasing time intervals. The event time was also recorded. Data were telemetered down to the ground-based receiving stations.

The popularity of the subject of gamma-ray astronomy has led to the need to update the material presented in the first edition, and this we are pleased to do.

The subject is in an exciting state in the lower energy region, below some tens of GeV, with the successful launch of the Gamma Ray Observatory in April, 1991. Already, sufficient data have appeared to show that, barring unforseen accidents, the subject will march forward at these energies. It is unfortunate that the Soviet GAMMA 1 satellite did not meet its design specifications – a reminder of the difficulties still inherent in satellite experiments.

The supernova SN 1987A continues to provide data of interest to the gamma-ray astronomer, and the results achieved so far have been included in this edition.

At the higher energies, advances have been less spectacular; indeed, there is some disappointment that many of the claimed TeV and PeV sources have still not been confirmed. Our view is that time variability of genuine sources married with some spurious signals probably accounts for the situation. Nevertheless, the subject is so important that continued, indeed enhanced, effort is needed.

The rate of publications in the field of gamma-ray astronomy at all energies is several times higher now than in 1985, when the manuscript for the first edition was turned in to the editors. Although we have made every effort to make the presentation in the second edition up to date (till the end of July, 1991), we apologise for inadvertent omission of any important results prior to that date.

Gamma-ray astronomy comprises the view of the Universe through what is essentially the last of the electromagnetic windows to be opened. All other windows from radio right through to X-rays have already been opened wide, and as is well known their respective astronomies are quite well developed – and the views there are very rich. Gamma-ray astronomy promises to be likewise; the strong link of γ-rays to very energetic processes and the considerable penetration of the γ-rays see to that.

Admittedly one deals with a small number of photons in this new window and yet a considerable amount of progress has already been made; hopefully this progress will shine through in what follows.

It is usually necessary to make a selection of topics when writing a book, and the present one is no exception. The selection made here reflects both the interests of the authors (both of whom are cosmic ray physicists) and the perceived needs of the subject. The authors' interests and, no doubt, biases show through in the areas in which they have themselves contributed (Chapters 4 and 5). There appears to be a contemporary need for a comprehensive review of γ-ray bursts and this is the reason for an extended Chapter 3. We have not included in Chapter 2 any material relating to γ-ray lines in solar flares – a very important subject in its own right – as we felt that it was outside the character of this book, dealing as it does with source regions exclusively beyond the solar system.

Introduction

The spectroscopy of γ-ray astronomy is, understandably, an area where important advances are to be expected, an expectation born of similar previous experience with other regions of the electromagnetic spectrum. Technical difficulties are considerable at present, however, due to low line fluxes aggravated by serious background problems; nevertheless, a promising start has been made and several interesting observations have already appeared.

As with astronomy in general, a distinction can be made between observations of ‘discrete’ objects (such as stars, supernovae, other galaxies, etc.) and signals from more extended regions, in particular the interstellar medium (ISM).

In the first category, γ-ray lines from the Sun – due to energetic protons and heavier nuclei interacting with the solar atmosphere – provide interesting and important information about a variety of solar phenomena. This subject of solar γ-ray spectroscopy is distant from the main stream of topics discussed here, and the reader is directed to a number of useful reviews by Ramaty and Lingenfelter (1981), Trombka and Fichtel (1982), Ramaty and Murphy (1987), and the books by Chupp (1976) and Hillier (1984).

In the non-solar region, which is of main concern here, only a few γ-ray lines have been detected from non-transient celestial sources so far. These include the lines at 1809 keV from the Galactic Equatorial Plane, the line at 511 keV from the Galactic Centre region and the one at 1369 keV from the object SS 433; these will be described in Sections 2.2, 2.3 and 2.4, respectively.

Introduction

Studies of ultra high energy gamma-rays (UHEGR) i.e. γ-rays at energies greater than 100 GeV, provide us with information on the conditions existing in remote celestial regions, such as magnetic and electric fields, matter and radiation densities, and on the acceleration mechanisms of charged particles. Additionally such studies have an important bearing on the problem of the origin of the cosmic radiation. There is, as yet, no universally accepted identification of either the sources or the mechanisms of production of cosmic rays, though, as was pointed out in Chapter 4, there are strong arguments made in favour of some. The problem is confounded by the fact that cosmic rays, almost all of which are charged particles, undergo frequent deflections in the interstellar magnetic fields, making it impossible to know the source directions. Thus, even a primary cosmic ray proton of energy as high as 1015 eV has a Larmour radius in the ISM of only ∼ 0.3pc and has its initial direction almost isotropised. Electrically neutral radiation is free from this problem. The more commonly occurring neutral particles are neutrons, neutrinos and γ-rays. Neutrons are unstable; they would not survive in most cases from source to Earth even after allowing for relativistic time dilatation, with a decay mean free path of only 9.2 (E/1015 eV) pc. Neutrinos, being weakly interacting, are not easy to detect, γ-rays, on the other hand, are ideal as their production and interaction cross sections are rather high and they are stable.

Introduction

For reasons concerned with the availability of contemporary γ-ray data, the lower limit for ‘medium energy’ quanta can be taken as 35 MeV (this is the lower limit for the important SAS II satellite experiment). The upper limit again comes from satellite data availability and is rather arbitrarily taken as 5000 MeV, the upper limit of the highest COS B satellite energy band; in fact, the photon flux falls off with energy so rapidly that our knowledge about γ-rays above 1000 MeV from satellite experiments is virtually nil. As will be discussed in Chapter 5, however, knowledge blooms again above 1011 eV, where Cerenkov radiation produced by γ-ray-induced electrons in the atmosphere allows detections to be made.

Although there are some who still believe that unresolved discrete sources contribute considerably to the diffuse γ-ray flux, the majority view is that the sources are responsible for only 10−20% of the γ-ray flux and that the predominant fraction arises from cosmic ray (CR) interactions with gas and radiation in the interstellar medium (ISM). In fact, some 30 years ago, both Hayakawa (1952) and Hutchinson (1952) had made estimates of the CR–ISM-induced γ-ray flux and had shown it to be within the scope of experimental measurement.

The foregoing is not to say that the discrete sources are unimportant, indeed the reverse is true, and there is considerable interest in ways of explaining the observed γ-ray flux from identified sources (the Crab and Vela pulsars) and the unidentified but definite sources such as Geminga (2CG 195 + 04 in the COS B source catalogue of Hermsen 1980, 1981).