Abstract

Introduction

It is with great pleasure that we dedicate this paper to Dieter Brill, our teacher, advisor, and colleague, on the occasion of his 60th birthday. Our contribution concerns the initial value problem for general relativity, which is amongst Dieter's many areas of expertise. As is the case with almost all research activity developed around the general relativity group at the University of Maryland, the ideas we will present have benefitted from Dieter's always kind and sometimes maddening insightful questioning. Of course it is our wish that this paper will prompt some more such questioning.

Abstract

Abstract

Dedicated to Professor Brill on the occasion of his sixtieth birthday, August 1993

Introduction

Interpretations of Quantum Mechanics and Paradigms of Statistical Mechanics

This paper attempts to bring together two basic concepts, one from the foundations of statistical mechanics and the other from the foundations of quantum mechanics, for the purpose of addressing two basic issues in physics:

1. the quantum to classical transition, and

2. the quantum origin of stochastic dynamics.

Both issues draw in the interlaced effects of dissipation, decoherence, noise, and fluctuation. A central concern is the role played by coarse-graining —the naturalness of its choice, the effectiveness of its implementation and the relevance of its consequences.

Abstract

Introduction

“I just can't understand it. All the young men I know are retiring.” So exclaimed Mrs. Niels Bohr in a post-war visit to Princeton on seeing Paul Dirac look from floor to ceiling and back again to floor in a desperate effort to answer her question, “Who is there now at Cambridge? Is Robert Frisch still there?”

“Frisch is retiring. I cannot remember who else is there, except me.”

Any thought of Brill retiring is foreign to anyone who sees him in action, as vigorous now as he was in his Princeton undergraduate (A.B. 1954) and graduate (Ph. D. 1959) days.

INTRODUCTION

Of the many influential contributions made by Dieter Brill to the mathematical development of general relativity, one of particular significance was his discovery together with Stanley Deser, of the linearization stability problem for Einstein's equations [1]. Brill and Deser showed that the Einstein equations are not always linearization stable (in a sense we shall define more precisely below) and they initiated the long (and still continuing) technical program to deal with this problem when it arises.

Our aim in this article is not to review the extensive literature of positive results on linearization stability but rather simply to introduce the reader to this subject and then to discuss some recent research that has developed out of the study of linearization stability problems. These latter include the relationship of linearization stability questions to the problem of the Hamiltonian reduction of Einstein's equations and lead one directly to the study of a number of recent results in pure Riemannian geometry (e.g., the solution of the Yamabe problem by Schoen, Aubin, Trudinger and Yambe, the Gromov-Lawson results on the existence of metrics of positive scalar curvature and the still unfinished classification problem for compact 3-manifolds). They also include a study of the quantum analogue of the linearization stability problem which has been significantly advanced recently by the work of A.

Abstract

Brill waves are the simplest (non-trivial) solutions to the vacuum constraints of general relativity. They are also rich enough in structure to allow us believe that they capture, at least in part, the generic properties of solutions of the Einstein equations. As such, they deserve the closest attention. This article illustrates this point by showing how Brill waves can be used to investigate the structure of conformal superspace.

INTRODUCTION

From time to time I amuse myself by mentally assembling a list of articles I would like to have written. The candidates for this list have to satisfy a number of criteria. Naturally, they have to be both important and interesting to me. Equally, they have to contain results that I can convince myself, however unreasonably, that I could have obtained. Every time I make my list I am struck again by the number of articles by Dieter Brill appearing on it. At first glance, this is explained by the large overlap between our interests. In reality, however, the explanation is to be found by considering the kind of article that Dieter has written over the years and the way in which he manages to convey major insights in a deceptively simple fashion.

Abstract

Introduction

In discussing solutions to the linearized Einstein equations, it is important to make sure that they extend to exact solutions. It was found by Professors Brill and Deser [2, 3] that there are spurious solutions in linearized gravity in static flat spacetime with the topology of the 3-torus (T3).

We resolve the longstanding paradox that in classical electrodynamics the energy and linear momentum of the Abraham-Lorentz (classical) electron do not transform as 4-vector components under Lorentz transformations. In our treatment these quantities transform properly and remain finite in the point particle limit.

Introduction

In Classical Electrodynamics the energy and linear momentum of the extended classical electron do not transform as components of a 4-vector (Leighton, 1959); also infinities arise in the point charge limit.

Abraham (1905) and Lorentz (1909) proposed the classical model of the electron. Lorentz suggested a model in which the electron consisted of a thin, uniformly charged shell. Poincaré (1909) added a stress in order to stabilize the electron. However, he used an expression for the cohesive stress-energy tensor which led to difficulties (Fermi, 1922; Pais, 1948).

Dirac (1938a, 1938b), Matthison (1931, 1940, 1942) and others have proposed point particle models for the classical electron. The lack of covariance associated with the Abraham-Lorentz model and other extended models of the electron does not arise in such models or in classical models based on non local generalisations of Maxwell electromagnetism; see for example, Bopp (1942), McManus (1948) and Feynmann (1948). Erber (1961) reviews much of the literature on models of the classical electron.

Abstract

It is a pleasure to dedicate this work to Dieter Brill on the occasion of his 60th birthday. I have learned much from him over the years, not least during our old collaborations on general relativity. I hope he will be entertained by these considerations of related theories in another dimension.

Introduction

Perhaps the most paradoxical feature of topologically massive (TM) theories [1, 2] is that their gauge invariance coexists with the finite mass and single helicity, parity violating, character of their excitations. This phenomenon, common to vector (TME) and tensor (TMG) models, is special to 2+1 dimensions because in higher (odd) dimensions the operative Chern—Simons (CS) terms are of at least cubic order in these fields and so do not affect their kinematics; only higher rank tensors could acquire a topological mass there.

ABSTRACT

If string theory describes nature, then charged black holes are not described by the Reissner-Nordström solution. This solution must be modified to include a massive dilaton. In the limit of vanishing dilaton mass, the new solution can be found by a generalization of the Harrison transformation for the Einstein-Maxwell equations. These two solution generating transformations and the resulting black holes are compared. It is shown that the extremal black hole with massless dilaton can be viewed as the “square root” of the extremal Reissner-Nordström solution. When the dilaton mass is included, extremal black holes are repulsive, and it is energetically favorable for them to bifurcate into smaller holes.

INTRODUCTION

It is a pleasure to honor Dieter Brill on the occasion of his sixtieth birthday. Over the years, Dieter has worked on many aspects of general relativity. But two of his recent interests are negative energy (in higher dimensional theories) (Brill and Pfister 1989; Brill and Horowitz 1991), and the possibility that extremal charged black holes can quantum mechanically bifurcate (Brill 1992). I would like to describe some recent work which touches on both of these areas.

For many years, it has been widely believed that static charged black holes in nature are accurately described by the Reissner-Nordström solution.

Abstract

In 1976 Dieter Brill and Frank Flaherty (1976) published an extremely important paper, “Isolated Maximal Hypersurfaces in Spacetime”, establishing that maximal hypersurfaces are unique in closed universes with attractive gravity everywhere. That is, there is only one such hypersurface, if it exists at all. In an earlier paper, Brill had established that in three-torus universes, only suitably identified flat space possessed a maximal hypersurface, so the existence of a maximal hypersurface is not guaranteed. These results by Brill are important because maximal hypersurfaces are very convenient spacelike hypersurfaces upon which to impose initial data; on such hypersurfaces the constraint equations are enormously simplified. Furthermore, in asymptotically flat space, foliations of spacetime by maximal hypersurfaces often exist, and the simplifications of the constraint equations on such a foliation make it easy to numerically solve the full four-dimensional vacuum Einstein equations for physically interesting situations.

The authors have introduced recently a “microcanonical functional integral” which yields directly the density of states as a function of energy. The phase of the functional integral is Jacobi's action, the extrema of which are classical solutions at a given energy. This approach is general but is especially well suited to gravitating systems because for them the total energy can be fixed simply as a boundary condition on the gravitational field. In this paper, however, we ignore gravity and illustrate the use of Jacobi's action by computing the density of states for a nonrelativistic harmonic oscillator.

DEDICATION

We dedicate this paper to Dieter Brill in honor of his sixtieth birthday. His continued fruitful research in physics and his personal kindness make him a model colleague. JWY would especially like to thank him for countless instructive discussions and for his friendship over the past twenty—five years.

INTRODUCTION

Jacobi's form of the action principle involves variations at fixed energy, rather than the variations at fixed time used in Hamilton's principle. The fixed time interval in Hamilton's action becomes fixed inverse temperature in the “periodic imaginary time” formulation, thus transforming Hamilton's action into the appropriate (imaginary) phase for a periodic path in computing the canonical partition function from a Feynman functional integral (Feynman and Hibbs 1965).

Abstract

INTRODUCTION: IDEAL MEASUREMENTS AND QUANTUM FIELD THEORY

Whatever may be its philosophical limitations, the textbook interpretation of nonrelativistic quantum mechanics is probably adequate to provide the quantum formalism with all the predictive power required for laboratory applications. It is also self-consistent in the sense that there exist idealized models of measurements which allow the system-observer boundary to be displaced arbitrarily far in the direction of the observer. And the associated “transformation theory” possesses a certain formal beauty, seemingly realizing the “complementarity principle” in terms of the unitary equivalence of all orthonormal bases.

It is Dieter Brill's gentle insistence on clarity of vision and depth of perception that has so influenced the development of general relativity and the scholarship of his colleagues and students. In both research and teaching, he is always searching for simpler descriptions with a deeper meaning. Ranging from positive energy and the initial value problem to linearization stability, from Mach's principle to topology change, Dieter's unique style has left its mark. The collection of essays here dedicated to Dieter Brill is a fitting tribute and clear testimony to the impact of Dieter's contributions.

This Festschrift is the second volume of the proceedings of an international symposium on Directions in General Relativity organized at the University of Maryland, College Park, May 27–29, 1993 in honour of the sixtieth birthdays of Professor Dieter Brill, born on August 9, 1933, and Professor Charles Misner. The first volume is a Festschrift for Professor Misner, whose sixtieth birthday was on June 13, 1992.

Ever since we announced a symposium and Festschrift for these two esteemed scientists in the Fall of 1991, we have been blessed with enthusiastic responses from friends, colleagues and former students of Charlie and Dieter all around the globe. Without their encouragement and participation this celebration could not have been realized.

Abstract

Recent studies of topology change and other topological effects have been typically initiated by considering semiclassical amplitudes for the transition of interest. Such amplitudes are constructed from riemannian or possibly complex solutions of the Einstein equations. This simple fact limits the possible transitions for a variety of possible matter sources. The case of riemannian solutions with strongly positive stress-energy is the most restrictive: no possible solution exists that mediates topology change between two or more boundary manifolds. Restrictions also exist for riemannian solutions with negative or indefinite stress-energy sources: all boundary manifolds must admit a metric with nonnegative curvature. This condition strongly restricts the possible topologies of the boundary manifolds given that most manifolds only admit metrics with negative curvature. Finally, the ability to construct explicit examples of topology changing instantons relies on the existence of a symmetry or symmetries that simplify the relevant equations. It follows that initial data with symmetry cannot give rise to a nonsymmetric solution of the Einstein equations. Moreover, analyticity properties of the Einstein equations strongly suggest that in general, complex solutions encounter the same topological restrictions. Thus the possibilities for topology change in the semiclassical limit are highly limited, indicating that detailed investigations of such effects should be carried out in terms of a more general construction of quantum amplitudes.