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.

When a physicist refers to curvature of space he at once falls under suspicion of talking metaphysics. Yet space is a prominent feature of the physical world; and measurement of space—lengths, distances, volumes —is part of the normal occupation of a physicist. Indeed it is rare to find any quantitative physical observation which does not ultimately reduce to measuring distances. Is it surprising that the precise investigation of physical space should have brought to light a new property which our crude sensory perception of space has passed over?

Space-curvature is a purely physical characteristic which we may find in a region by suitable experiments and measurements, just as we may find a magnetic field. In curved space the measured distances and angles fit together in a way different from that with which we are familiar in the geometry of flat space; for example, the three angles of a triangle do not add up to two right angles. It seems rather hard on the physicist, who conscientiously measures the three angles of a triangle, that he should be told that if the sum comes to two right angles his work is sound physics, but if it differs to the slightest extent he is straying into metaphysical quagmires.

In using the name “curvature” for this characteristic of space, there is no metaphysical implication.

A spherical world, closed but continually expanding, is a new playground for thought. Let us play in it a little to familiarise ourselves with it. In this chapter I shall mix together results which may prove to be of scientific importance and results that are probably no more than mathematical curiosities. The plan is to set down anything that seems worthy of note, even though we cannot see that it has any ultimate importance in nature.

For a model of the universe let us represent spherical space by a rubber balloon. Our three dimensions of length, breadth and thickness ought all to lie in the skin of the balloon; but there is only room for two, so the model will have to sacrifice one of them. That does not matter very seriously. Imagine the galaxies to be embedded in the rubber. Now let the balloon be steadily inflated. That's the expanding universe.

The galaxies are supposed to be scattered more or less evenly over the surface; our observational knowledge, however, is limited to a portion which corresponds roughly to the size of France on a terrestrial globe. The galaxies have individual motions, i.e. motions with respect to the material of the balloon, but these are comparatively small; in the main they recede from one another simply by the stretching of the rubber.

The first hint of an “expanding universe” is contained in a paper published in November 1917 by Prof. W. de Sitter. Einstein's general theory of relativity had been published two years before, but it had not yet attained notoriety; it was not until the eclipse expeditions of 1919 obtained confirmation of its prediction of the bending of light that public interest was aroused. Meanwhile many investigators had been examining the various consequences of the new theory. Prominent among them was de Sitter who was interested especially in the astronomical consequences. In the course of a highly technical discussion he found that the relativity theory led to an expectation that the most remote celestial objects would be moving away from us, or at least that they would deceive the observer into thinking that they were moving away.

De Sitter was perhaps a tipster rather than a prophet. He would not promise anything definitely; but he suggested that we ought to keep a look out for the recession as a rather likely phenomenon. Theory was at the cross-roads, and desired guidance from observation as to which of two possible courses should be pursued.

Sir arthur (stanley) eddington om, frs (1882–1944), pioneer of stellar dynamics, tester and expounder of general relativity theory, father of modern theoretical astrophysics, explorer of the foundations of physics, was one of the greatest and most influential scientists of this century.

He had long been fascinated by questions about the constants of physics, the significance of large dimensionless numbers given by certain combinations of these, and their possible association with large-scale properties of the astronomical universe (e.g. The Mathematical Theory of Relativity (1923), page 167). Then about 1930 he rather stumbled upon a possible relation between these quantities and the rate of expansion of the Universe. When he wrote this in what seemed to be its simplest form, it yielded very closely the empirical rate that E.P. Hubble had just claimed to have discovered. At the same time, Eddington was seeking some ‘meeting’ of relativity and quantum theory. He claimed to see a connection with the foregoing relation such that, with minimal elaboration, it served to yield a theoretical prediction of the masses of the proton and electron. He was engaged in a highly sophisticated development of all this, which in due course resulted in the monumental monographs Relativity Theory of Protons and Electrons (1936) and Fundamental Theory (1946). Clever as these are in exposing fundamental problems for the whole of physics, no one has ever claimed to be convinced of the validity of the solutions they offer. No one, that is, except Eddington.