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.

ABSTRACT

The problem of the origin of rotational inertia is examined within the framework of the relativistic theory of gravitation. It is argued that gravitomagnetic effects cannot be interpreted in terms of the relativity of rotation. Absolute and relative motion are discussed on the basis of the hypothesis that these are complementary classical manifestations of movement.

What is the origin of inertia? For instance, with respect to what does the Earth rotate around its axis? The rotation of a body does not generate any basic new gravitational effect in the Newtonian theory. This is not the case, however, in Einstein's theory of gravitation. The striking analogy between Newton's law of gravitation and Coulomb's law of electricity has led to a description of Newtonian gravity in terms of a gravitoelectric field. Any theory that combines Newtonian gravity with Lorentz invariance is expected to contain a gravitomagnetic field in some form; in general relativity, the gravitomagnetic field is usually caused by the angular momentum of the source of the gravitational field. The first gravitomagnetic effects were described by de Sitter soon after Einstein's fundamental work on general relativity. The question of relativity of rotation was also discussed by de Sitter following his investigation of the astronomical consequences of Einstein's relativistic theory of gravitation; de Sitter concluded that the problem of inertia did not have a solution in the general theory of relativity.

Abstract

Introduction

For past years many 2-dimensional field theories have been intensively studied as laboratories for many theoretical issues, due to great mathematical simplicities that often exist in 2-dimensional systems. Recently these 2-dimensional field theories have received considerable attention, for different reasons, in connection with general relativistic systems of 4-dimensions, such as self-dual spaces [1] and the black-hole space-times [2, 3]. These 2-dimensional formulations of self-dual spaces and blackhole space-times of allow, in principle, many 2-dimensional field theoretic methods developed in the past relevant for the description of the physics of 4-dimensions.

ABSTRACT

Measurement is a fundamental notion in the usual approximate quantum mechanics of measured subsystems. Probabilities are predicted for the outcomes of measurements. State vectors evolve unitarily in between measurements and by reduction of the state vector at measurements. Probabilities are computed by summing the squares of amplitudes over alternatives which could have been measured but weren't. Measurements are limited by uncertainty principles and by other restrictions arising from the principles of quantum mechanics. This essay examines the extent to which those features of the quantum mechanics of measured subsystems that are explicitly tied to measurement situations are incorporated or modified in the more general quantum mechanics of closed systems in which measurement is not a fundamental notion. There, probabilities are predicted for decohering sets of alternative time histories of the closed system, whether or not they represent a measurement situation. Reduction of the state vector is a necessary part of the description of such histories. Uncertainty principles limit the possible alternatives at one time from which histories may be constructed. Models of measurement situations are exhibited within the quantum mechanics of the closed system containing both measured subsystem and measuring apparatus.

Abstract

There is a special talent in being able to ask simple questions whose answers reach deeply into our understanding of physics. Dieter is one of the people with this talent, and many was the time when I thought the answer to one of his questions was nearly at hand, only to lose it on meeting an unexpected conceptual pitfall.

Abstract

A maximal slice in a spacetime (M, gab) is a closed, embedded, spacelike, submanifold of co-dimension one whose trace, of extrinsic curvature vanishes. The issue of whether maximal slices exist in certain classes of spacetimes in general relativity has arisen in many analyses. One of the most prominent early examples of the relevance of this issue occurs in the positive energy argument given by Dieter Brill in collaboration with Deser [5], where the existence of a maximal slice in asymptotically flat spacetimes was needed in order to assure positivity of the “kinetic terms” in the Hamiltonian constraint equation. The existence and properties of maximal slices has remained a strong research interest of Brill, and he has made a number of important contributions to the subject.

This review of Dieter Brill's publications is intended not only as a tribute but as a useful guide to the many insights, results, ideas, and questions with which Dieter has enriched the field of general relativity. We have divided up Dieter Brill's work into several naturally defined categories, ordered in a quasi-chronological fashion. References [n] are to Brill's list of publications near the end of this volume. Inevitably, the review covers only a part of Brill's work, the part defined primarily by the areas with which the authors of the review are most familiar.

GEOMETRODYNAMICS—GETTING STARTED

In a 1977 letter to John Wheeler, his thesis supervisor, Brill recalled that after spin 1/2 failed [1] to fit into Wheeler's geometrodynamics program he asked John “for a ‘sure-fire’ thesis problem, and [John] suggested positivity of mass.” Brill's Princeton Ph.D. thesis [A, 2] provided a major advance in Wheeler's “Geometrodynamics” program. By studying possible initial values, Brill showed that there exist solutions of the empty-space Einstein equations that are asymptotically flat and not at all weak. Moreover, in the large class of examples he treated, all were seen to have positive energy. Although described only at a moment of time symmetry, these solutions were interpreted as pulses of incoming gravitational radiation that would proceed to propagate as outgoing radiation.

This book is an expanded version of a public lecture delivered at the meeting of the International Astronomical Union at Cambridge (Massachusetts) in September 1932. It also furnished the subject-matter of a series of three addresses which were broadcast in the United States shortly afterwards.

I deal with the view now tentatively held that the whole material universe of stars and galaxies of stars is dispersing, the galaxies scattering apart so as to occupy an ever-increasing volume. But I deal with it not as an end in itself. To take an analogy from detective fiction, it is the clue not the criminal. The “hidden hand” in my story is the cosmical constant. In Chapter iv we see that the investigation of the expanding universe falls into line with other methods of inquiry, so that we appear to be closing down on the capture of this most elusive constant of nature.

The subject is of especial interest, since it lies at the meeting point of astronomy, relativity and wave-mechanics. Any genuine progress will have important reactions on all three.

I am treating of very recent developments; and investigations both on the theoretical and on the observational side are still in progress which are likely to teach us much more and may modify our views.

I have explained in the previous chapters that theory led us to expect a systematic motion of recession of remote objects, and that by astronomical observation the most remote objects known have been found to be receding rapidly. The weak point in this triumph is that theory gave no indication how large a velocity of recession was to be expected. It is as though an explorer were given instructions to look out for a creature with a trunk; he has brought home an elephant—perhaps a white elephant. The conditions would equally well have been satisfied by a fly, with much less annoyance to his next-door neighbour the time-grabbing evolutionist. So there is great argument about it.

I think the only way to remove the cloud of doubt is to supplement the original prediction, and show that physical theory demands not merely a recession but a particular speed of recession. The theory of relativity alone will not give any more information; but we have other resources. I refer to the second great modern development of physics—the quantum theory, or (in its most recent form) wave-mechanics. By combining the two theories we can make the desired theoretical calculation of the speed of recession.

This is a new adventure, and I do not wish to insist on the accuracy or finality of the first attempt.