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.

Let us now turn our thoughts beyond the earth and its atmosphere to the phenomena which may properly be described as astronomical. We see a procession of objects moving ceaselessly across the sky—the sun by day, the moon and stars by night. These all appear to cross the sky from east to west, because the rotation of the earth, from which we view the spectacle, causes us to move continually from west to east.

The most conspicuous phenomenon is of course the daily motion of the sun across the heavens, producing the alternations of light and darkness, heat and cold, which we describe as day and night. The rising and setting of the moon and its passage across the sky are only one degree less conspicuous, and must have been not only noticed, but also familiar, since the days when human beings first appeared on the earth.

The sun shews no changes either of shape or brightness, except when our own atmosphere dims its light, but the moon continually varies in both respects. Every month it goes through the complete cycle of changes, which we call its “phases”. It begins as a thin crescent of light, which we describe as the new moon. This increases in size until after about a week we have the semicircle of light we call half moon, and then a week later the complete circle we call full moon.

We know that the moon always looks about the same size in the sky and from this we can conclude that it is always at about the same distance from the earth. And we can measure the distance in the same way as we measure the distance of an inaccessible mountain peak, or the height of an aeroplane.

When an aeroplane is up in the air, people who are standing at different points must look in different directions to see it. If it is directly overhead for one man, it will not be directly overhead for another man a mile away, and its height can be calculated simply by noticing how far its position appears to be out of the vertical for the second man. Using this method, astronomers find that the distance of the moon varies between the limits of 221,462 miles and 252,710 miles, the average distance being 238,857 miles. Thus, in round numbers, we may think of the moon as being a quarter of a million miles away.

At such a distance, we can hardly expect to see much detail with our unaided eyes. Indeed, as we watch the moon sailing through the night sky, we can detect nothing on its surface beyond a variety of light and dark patches, which, with a bit of imagination, we can make into the man in the moon with his bundle of sticks, or an old woman reading a book, or—as the Chinese prefer to think—a jumping hare.

Every year for more than a century, the Royal Institution has invited, some man of science to deliver a course of lectures at Christmastide in a style “adapted to a juvenile auditory”. In practice, this rather quaint phrase means that the lecturer will be confronted with an eager and critical audience, ranging in respect of age from under eight to over eighty, and in respect of scientific knowledge from the aforesaid child under eight to staid professors of science and venerable Fellows of the Royal Society, each of whom will expect the lecturer to say something that will interest him.

The present book contains the substance of what I said when I was honoured with an invitation of this kind for the Christmas season 1933—4, fortified in places with what I have said on other slightly more serious occasions, both at the Royal Institution and elsewhere.

It is a pleasure to acknowledge many courtesies and return thanks for much valuable help. I am indebted to Sir T. L. Heath for permission to borrow largely from his Greek Astronomy and other books; to many Institutions, Publishers and private individuals for the loan of negatives, prints, blocks, etc., and permission to reproduce these in my book—detailed acknowledgment is made in the List of Illustrations.

Let us leave the earth, in which we have burrowed for long enough, and turn our thoughts, and our eyes, upwards.

We all know what we may expect to see—the sun, the blue sky, and possibly some clouds, by day; stars, with perhaps the moon and one or more planets, by night. We see these objects by light which has travelled to us through the earth's atmosphere, and if we see them clearly, it is because the atmosphere is transparent—it presents no barrier to the passage of rays of light.

Perhaps we are so accustomed to this fact that we merely take it for granted. Or perhaps we think of the atmosphere as something too flimsy and ethereal ever to stop the passage of rays of light. Yet we know exactly how much atmosphere there is, for the ordinary domestic barometer is weighing it for us all the time. When the barometer needle points to 30, there is as much substance in the atmosphere over our heads as there is in a layer of mercury 30 inches thick. This again is the same amount as there would be in a layer of lead about 36 inches thick, for mercury is heavier than an equal volume of lead in the ratio of about six to five. To visualise the weight of the atmosphere above us, we may think of ourselves as covered up with 144 blankets of lead, each a quarter of an inch in thickness.

These are restless days in which everyone travels who can. The more fortunate of us may have travelled outside Europe to other continents—perhaps even round the world—and seen strange sights and scenery on our travels. And now we are starting out to take the longest journey in the whole universe. We shall travelor pretend to travel—so far through space that our earth will look like less than the tiniest of motes in a sunbeam, and so far through time that the whole of human history will shrink to a tick of the clock, and a man's whole life to something less than the twinkling of an eye.

As we travel through space, we shall try to draw a picture of the universe as it now is—vast spaces of unthinkable extent and terrifying desolation, redeemed from utter emptiness only at rare intervals by small particles of cold lifeless matter, and at still rarer intervals by those vivid balls of flaming gas we call stars. Most of these stars are solitary wanderers through space, although here and there we may perhaps find a star giving warmth and light to a family of encircling planets. Yet few of these are at all likely to resemble our own earth; the majority will be so different that we shall hardly be able to describe their scenery, or imagine their physical condition.

We all know now that our sun is a very ordinary star, but it took men a long time to discover this. Perhaps this is not surprising, for certainly it does not look much like an ordinary star to us. The reason is, of course, that it is enormously nearer than any of the other stars.

We have seen how the ancients imagined the earth to be the fixed centre of the universe, round which everything else moved. The stars merely formed a background of light, against which they could map out the motions of the sun, moon and planets. They thought of the stars as attached to the inside of a hollow sphere, which turned round over the earth much as a telescope dome turns round over the floor of a telescope, or “as one might turn a cap round on one's head”. And although a few of the more philosophical of the Greeks gave reasons for thinking that the earth moved round the sun, they had no means of making their opinions or arguments known to a wide circle of people, so that these were forgotten as the world gradually became submerged in the intellectual darkness of the Middle Ages. Then, in 1543, a Polish monk, Copernicus, advanced views and arguments which were very similar to those which Aristarchus of Samos had propounded 1800 years earlier, although the extent to which he was indebted to his Greek predecessors is not clear.

There are nine planets circling round the sun, of which of course the earth is one. Of the other eight, five have been known from pre-historic times, while the remaining three—the three farthest from the sun—are comparatively recent discoveries.

The row of models exhibited in fig. 60 shew how greatly these nine planets differ in size. Those which are nearest to, and farthest away from, the sun are the smallest, while the middle members, Jupiter and Saturn, are the largest. Jupiter, the central member, is largest of all, with a diameter of nearly 90,000 miles, and a volume 1300 times that of the earth. Jupiter stands in the same proportion to the earth as a football to a marble, while on the same scale Mars would be hardly larger than a pea.

If we wish to complete our model by placing the objects shewn in fig. 60 at their proper distances, the nearest planet, Mercury, must describe an orbit which is not quite circular, but is such that, even at its nearest approach to the sun, the planet would be 20 feet away. The earth must keep at a distance of 50 feet from the sun, while Pluto, the farthest planet of all, must describe an orbit nearly half a mile in radius.

We see that the solar system consists mainly of empty space, and yet the emptiness of the solar system is as nothing compared to the emptiness of space itself.

The moon and planets look very conspicuous objects in the sky, but we know that these are very near neighbours which only look bright and big because they are near. For the rest our unaided eyes can see nothing of the universe except stars.

A small telescope or field-glass will shew us more stars in abundance, but it will shew us something else as well. A new class of object comes within our ken, the fuzzy indefinite patches of faint light which we describe as “nebulae”.

The word “nebula” is of course the Latin word for a mist or cloud. In the early days of astronomy it was used indiscriminately to describe any object of misty or fuzzy appearance—any object, indeed, which did not exhibit a clear outline. Since then it has been found that the nebulae fall into three distinct classes.

The first consists of objects known as planetary nebulae, which lie entirely within our system of stars. It is now known that these are themselves stars which, for reasons not altogether understood, have become surrounded by very extensive atmospheres. Examples are shewn in fig. 98 (facing p. 204). We described the red giant stars as large, but when their atmospheres are counted in, these stars are beyond all comparison larger. Our rocket, travelling at 5000 miles an hour, would take 9 years to travel through the biggest of red giants, but about 90,000 years to travel through one of these planetary nebulae.