Abstract

Introduction

Dieter Brill and one of us (JI) used to talk a lot about Mach's Principle. We both were of the Wheeler school, so our Machian discussions often focussed on issues involving the initial value formulation of Einstein's theory. One such issue was the following question: If a spacetime (M, g) is known to be globally hyperbolic, how can one tell (from intrinsic information) if a given embedded spacelike hypersurface Σ is a Cauchy surface for (M, g)? The answer to this question is important if one wants to know what minimal information about the universe “now” is needed to determine the spacetime metric (and hence its inertial frames) for all time in (M, g).

It turns out [1] that if a spacelike hypersurface Σ embedded in a globally hyperbolic spacetime is compact (without boundary), then it must be a Cauchy surface.

Abstract

In dimension eight there are three basic representations for the spin group. These representations lead to a concept of triality and hence lead to the construction of two exceptional commutative algebras: The Chevalley algebra A of dimension twenty-four and the Albert algebra I of dimension twenty-seven. All of the exceptional Lie groups can be described using triality, the octonians, and these two algebras.

On a complex four-manifold, with triality a parallel field, the Dirac and associated twistor operators can be constructed on either bundle of exceptional algebras. The geometry of triality leads to refinement of duality common to four dimensions.

In nine dimensions there is a weaker notion of triality which is related to several additional multiplicative structures on J.

Introduction

Cartan's classification of simple Lie groups yields all Lie groups with some exceptions. In his thesis Cartan described these exceptional groups but, save the one of smallest rank G2, he was unable to describe the geometry of the groups. In 1950 Chevalley and Schafer successfully identified F4 and E6 as structure groups for the exceptional Jordan algebra and the Freudenthal cross product on J, respectively. Later Freudenthal successfully identified the geometry associated with the remaining exceptional groups, E7 and E8.

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.