(2) (p. 64) In the main body of the text, I mention the importance of the fortunate circumstances of the world in enabling physicists to avoid worrying about foundations. Another very important factor is the clarity of the notion of empty space, developed so early by the Greek mathematicians, which deeply impressed Newton. He felt that he really could see space in his mind’s eye, and regarded it as being rather like some infinite translucent block of glass. He and many other mathematicians pictured its points as being like tiny identical grains of sand that, close-packed, make up the block. But this is all rather ghostly and mysterious. Unlike glass and tiny grains of sand, which are just visible, space and its points are utterly invisible. This is a suspect, unreal world.
We are not bound to hang onto old notions. We can open our eyes to something new. Let me try to persuade you that points of space are not what mathematicians would sometimes have us believe. Imagine yourself in a magnificent mountain range, and that someone asked, ‘Where are you?’ Would you kneel down with a magnifying glass and look for that invisible ‘point’ at which you happen to be in the ‘space’ that the mountain range occupies? You would look in vain. Indeed, you would never do such a silly thing. You would just look around you at the mountains. They tell you where you are. The point you occupy in the world is defined by what the world looks like as seen by you: it is a snapshot of the world as seen by you. Real points of space are not tiny grains of sand, they are actual pictures. To see the point where you are in the world, you must look not inward but outward.
The plaque near the grave of Christopher Wren in St. Paul’s Cathedral says simply: ‘If you seek a monument, look around you.’ The point where you are is a monument too, and you see it by looking around you. It is this sort of change of mindset that I think we need if we are to understand the universe and time.
To conclude this note, a word about what is perhaps the most serious problem in my approach. It is how to deal with infinity. As so far defined, each place in Platonia corresponds to a configuration of a finite number of objects. Such a universe is like an island of finite extent. One could allow the configurations to have infinite extent and contain infinitely many objects. That is not an insuperable problem. The difficulty arises with the operations that one needs to perform. As presented in this book, the operations work only if the points in Platonia, the instants of time, are in some sense finite. There may be ways around this problem—Einstein’s theory can deal beautifully with either finite or infinite universes—but infinity is always rather difficult. There is something ‘beyond the horizon’, and we can never close the circle of cause and effect. In short, we cannot build a model of a completely rational world. Precisely for this reason Einstein’s first and most famous cosmological model was spatially finite, closed up on itself. The constructions of this book are to be seen as a similar attempt to create a rational model of the universe in which the elusive circle does close.
In fact, if the work with Niall O Murchadha mentioned at the end of the Preface, which suggests that absolute distance can be eliminated as a basic concept (see Box 3), can be transformed into a complete theory, the problem of infinity may well be solved in the process. If size has no meaning, the distinction between a spatially finite or infinite universe becomes meaningless.
CHAPTER 5: NEWTON’S EVIDENCE
The Aims of Machian Mechanics (1) (p. 71) In creating the beautiful diagrams that form such an important part of this section, Dierck Liebscher was able to draw on initial data devised by Douglas Heggie (University of Edinburgh), using software written by Piet Hut (Institute for Advanced Study), Steve McMillan (Drexel University) and Jun Makino (University of Tokyo). Dierck has written a very interesting book (alas, as yet published only in German) on the connection between different possible geometries and Einstein’s relativity theory (Liebscher 1999). It contains many striking computer-generated diagrams.
(2) Poincaré’s discussion is contained in his Science and Hypothesis, which, along with the writings of his contemporary, Mach, became a popular-science best-seller. In fact, in this book I am actually revisiting many of the themes discussed by Poincaré and Mach, but with the advantage of hindsight. How are the great issues they raised changed by the discovery of general relativity and quantum mechanics? I have adapted Poincaré’s discussion somewhat to match the requirements of a timeless theory (he considered only the possibility of eliminating absolute space).
(3) Since writing Box 3, which draws attention to the present unsatisfactory use of absolute dislance in physics, I have discovered a way to create dynamical theories in which distance is not absolute. This is achieved by a very natural extension of the best-matching idea described later in the book. The new insights that I mention in the Preface are in part connected with this development. One of the most exciting is that, if such theories do indeed describe the world, gravitation and the other forces of nature are precisely the mechanism by means of which absolute distance is made irrelevant. Since this work is still in progress, I shall make no attempt to describe it in detail, but I shall keep my website (www.julianbarbour.com) up to date with any progress (see also p.358).
CHAPTER 6: THE TWO GREAT CLOCKS IN THE SKY
The Inertial Clock (p. 99) Tait’s work, which I feel is very important, passed almost completely unnoticed. This is probably because two years later the young German Ludwig Lange introduced an alternative construction for finding inertial frames of reference, coining the expression ‘inertial system’. Lange deserves great credit for bringing to the fore the issue of the determination of such systems from purely relative data, but Tait’s construction is far more illuminating. Lange’s work is discussed in detail in Barbour (1989) and Tait’s in Barbour (forthcoming).
The Second Great Clock (p. 107) A very nice account of the history of the introduction of ephemeris time was given by the American astronomer Gerald Clemence (1957).
CHAPTER 7: PATHS IN PLATONIA
Nature and Exploration (p. 109) For physicists and mathematicians who do not know the book, a wonderful account of the variational principles of mechanics, together with much historical material, is given by Lanczos (1986).
Developing Machian Ideas (p. 115) Translations of the papers by Hofrnann, Reissner and Schrödinger, along with other historical and technical papers on Mach’s principle, can be found in Barbour and Pfister (1995).
Exploring Platonia (p. 115) The special properties of Newtonian motions with vanishing angular momentum were discovered independently of the work of Bertotti and myself by A. Guichardet in the theory of molecular motions and by A. Shapere and E Wilczek in the theory of how micro-organisms swim in viscous fluids! A rich mathematical theory has meanwhile developed, and is excellently reviewed in the article by Littlejohn and Reinsch (1997), which contains references to the original work mentioned above. All mathematical details, as well as references to the earlier work by Bertotti and myself, can be found in Barbour (1994a).
CHAPTER 8: THE BOLT FROM THE BLUE
Historical accidents (p. 123) Poincaré’s paper can be found in his The Value of Science, Chapter 2. Pais’s book is in the Bibliography.
Background to the Crisis (p. 124) The best (moderately technical) historical background to the relativity revolution that I know of is the book by Max Born. It is available in paperback.