INTRODUCTION

In this article we want to describe a long-term research project that the authors plan to carry out over a period in the immediate future. We would like to outline the basic ideas of the project and give a few preliminary calculations the bear on the validity of our ideas as well as some speculations on where the research will lead. We chose this volume to do this because it occured to us that each of the components of our plan lies in a field that Charlie has touched on at some point in his varied career. These components are the Hamiltonian formulation of the gravitational field equations, path integral quantization, and quantum cosmology. Such a wide-ranging list shows how much we all owe to Charlie as a scientist and we hope that our efforts will demonstrate our debt to his lead in these fields.

The genesis of our paper lies in a long-term concern that both of us have had about the structure of quantum mechanics that has been the subject of years of blackboard discussions between us (the speculations of one of us [M.R.] go all the way back to fondly-remembered discussions with Charlie and other Charlie-students in Maryland). The focal point of our ideas has always been the cookbook nature of quantum mechanics as based on a Hamiltonian action functional for the classical equations of motion and canonical commutation relations derived from the Poisson bracket relations of the variables in the action.

Abstract

Misner [1] suggested kinetic theory ideas could explain the smoothing of the universe at early times. Recently [2], [3] the inverse has been investigated: the generation of Bianchi I anisotropy in cosmologies with an exact FRW geometry, again for kinetic theory reasons. At the time it was stated that more general anisotropy and inhomogeneity could be generated by the same methods. In this paper the more general case is addressed to show how the model can be produced and to identify some open questions connected with it.

Introduction

One of the major contributions Charles Misner has made to cosmology was the first serious investigation of the chaotic cosmology idea [1], namely that the universe started off in a very chaotic state and then developed (by physical processes) towards a smooth state. This idea has since been taken up with a vengeance by the Inflationary Universe school of thought, utilising Guth's insight [4] that the vast expansion associated with the false vacuum (a scalar field) could provide the required smoothing mechanism.

The resulting theory is very interesting but perhaps over-stated [5]. In particular it does not overcome the Stewart remark [6] that no matter what smoothing mechanisms one might find there are always initial data that will lead to universes more lumpy than the observed universe (simply run the equations backwards from any considered present state to find the needed initial conditions). Furthermore entropy arguments have been adduced by Penrose [7] to suggest that the universe must have started off in a smooth state rather than the very chaotic state suggested by many inflationists.

Introduction

Developments in physics on the smallest and largest scales (elementary particle theory and cosmology) over the past several decades have provided strong evidence for the hot Big Bang theory of the early Universe and for gauge field models of elementary particle interactions, particularly as embodied in the Standard Model of elementary particle physics.

In adopting these two theories it is very difficult to avoid the phenomenon of cosmological vacuum phase transitions. Such phase transitions arise because of the temperature dependence of the Higgs field potential. This potential plays several roles within gauge field theories, from providing a mechanism for spontaneous symmetry breaking to providing a possible vacuum energy density. The Higgs field potential (which we shall henceforth label as U(φ), where φ represents the field) at high temperature possesses a single vacuum state, which we may arbitrarily choose to be located at φ = 0. The Higgs potential undergoes a continuous evolution as the temperature is changed and at low temperatures the potential may possess multiple vacuum states (i.e., multiple minima). The nature of the transition from the high temperature vacuum state to the low temperature vacuum state will depend upon the shape of the potential during the course of its evolution. It is possible that the Higgs field will undergo a classical “rolling” evolution between the initial and final vacuum states.

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.