He asked how probable a state should be. Imagine a grid of 100 holes into which you drop 1000 marbles at random. It is hugely improbable that they will all finish up in one hole. I am not going to give numbers, but it is simple to work out the probability that all will land in one hole or, say, in four adjacent holes. In fact, one can list every possible distribution of the marbles in the grid, and then see in how many of these distributions all the marbles fall in one hole, in four adjacent holes, eight adjacent holes, and so on. If each distribution is assumed to be equally probable, the number of ways a particular outcome can happen becomes the relative probability of that outcome, or state. Boltzmann had the inspired idea that, applied to atoms, this probability (which must also take into account the velocities of the atoms) is a measure of the entropy that had been found through study of the thermodynamics of steam engines.
There is no need to worry about the technical details. The important thing is that states with low entropy are inherently improbable. Boltzmann’s idea was brilliantly successful, and much of modern chemistry, for example, would be unthinkable without it. However, his attempt to explain the more fundamental issues associated with the unidirectionality of physical processes was only partly successful.
He wanted to show that, matching the behaviour of macroscopic entropy, his microscopic entropy would necessarily increase solely by virtue of Newton’s laws. This seems plausible. If a large number of atoms are in some unlikely state, say all in a small region, so that they have a low entropy, it seems clear that they will pass to a more probable state with higher entropy. However, it was soon noted that there are exactly as many dynamically possible motions of the atoms that go from states of low probability to states of high probability as vice versa. This is a straight consequence of the fact that Newton’s laws have the same form for the two directions of time. Newton’s laws alone cannot explain the arrow of time.
Only two ways have ever been found to explain the arrow: either the universe was created in a highly unlikely special state, and its initial order has been ‘degrading’ ever since, or it has existed for ever, and at some time in the recent past it entered by chance an exceedingly improbable state of very low entropy, from which it is now emerging. The second possibility is entirely compatible with the laws of physics. For example, if a collection of atoms (which obey Newton’s laws) is confined in a box and completely isolated, it will, over a sufficiently long period of time, visit (or rather come arbitrarily near) all the states that it can in principle ever reach, even those that are highly ordered and statistically very unlikely. However, the intervals of time between returns to states of very low entropy are stupendously long (vastly longer than the presently assumed age of the universe), and neither explanation is attractive.
The fact is that mechanical laws of motion allow an almost incomprehensibly large number of different possible situations. Interesting structure and order arise only in the tiniest fraction of them. Scientists feel they should not invoke miracles to explain the order we see, but that leaves only statistical arguments, which give bleak answers (only dull situations can be expected), or the so-called anthropic principle that if the world were not in a highly structured but extremely unlikely state, we should not exist and be here to observe it.
One of my reasons for writing this book is that timeless physics opens up new ways of thinking about structure and entropy. It may be easier to explain the arrow of time if there is no time!
The Next Revolution in Physics (p. 14) The possible non-existence of time has just begun to be discussed in authoritative books for the general public. Both Paul Davies, in his About Time, and Kip Thorne, in his Black Holes and Time Warps, devote a few pages to the topic. In apocalyptic vein, Thorne likens the fate of space-time near a black hole singularity to
a piece of wood impregnated with water . . . the wood represents space, the water represents time, and the two (wood and water, space and time) are tightly interwoven, unified. The singularity and the laws of quantum gravity that rule it are like a fire into which the water-impregnated wood is thrown. The fire boils the water out of the wood, leaving the wood alone and vulnerable; in the singularity the laws of quantum gravity destroy time . . . (p. 477)
However, Thorne’s magnificent book is devoted to other topics, and nothing prepares the reader for this dramatic and singular end of time. Moreover, the evidence, as I read it, is that timelessness permeates the whole universe, not just the vicinity of singularities. Paul Davies, for his part, repeatedly expresses a deep mystification about time. His book is almost a compendium of conundrums, and he candidly consoles the reader with ‘you may well be even more confused about time after reading this book than you were before. That’s all right; I was more confused myself after writing it’ (p. 10). In fact, I think Paul’s subtitle, Einstein’s Unfinished Revolution, is the key to a lot of the puzzles. As we shall see in Part 3, there are aspects of physical time which Einstein did not address.
Among the popular books that I know, the two that undoubtedly give most prominence to the problem of time in quantum gravity are Lee Smolin’s The Life of the Cosmos, which contains some discussion of my own ideas, and David Deutsch’s The Fabric of Reality. There is considerable overlap between my book and Deutsch’s chapter ‘Time: the first quantum concept’. One technical book, now going into a third edition, that from the start has taken timelessness very seriously is Dieter Zeh’s The Physical Basis of the Direction of Time.
It may be that the reason why a book like this one, devoted exclusively to the idea that time does not exist, has not hitherto been published by a physicist has a sociological explanation. For professionals working in institutes and dependent on the opinions of peers for research funding, such a book might damage their reputation and put further research in jeopardy. After all, at first it does seem outrageous to suggest that time does not exist. It may not be accidental that I, as an independent not reliant on conventional funding, have been prepared to ‘come out’.
In this connection, my experience at a big international conference in Spain in 1991 devoted to the arrow of time was very interesting. The following is quoted from my paper in the conference proceedings (available in paperback as Halliwell et al., 1994):
During the Workshop, I conducted a very informal straw-poll, putting the following question to each of the 42 participants:
Do you believe time is a truly basic concept that must appear in the foundations of any theory of the world, or is it an effective concept that can be derived from more primitive notions in the same way that a notion of temperature can be recovered in statistical mechanics?
The results were as follows: 20 said there was no time at a fundamental level, 12 declared themselves to be undecided or wished to abstain, and 10 believed time did exist at the most basic level. However, among the 12 in the undecided/abstain column, 5 were sympathetic to or inclined to the belief that time should not appear at the most basic level of theory.
Thus, a clear majority doubted the existence of time. When I took my straw-poll, I said that I intended to publish the names with their opinions, which was why two people abstained, to remain anonymous. As it happens the conference generated immense media interest in Spain, not least because of the presence of Stephen Hawking and Nobel Laureate Murray Gell-Mann, and the reporter from El Pais got hold of a copy of my results. One of the participants (neither of the above), finding his own opinion quoted in a big article the day after the conference, was none too pleased and greeted me when we met six months later at a conference in Cincinnati with ‘You and your damned straw-poll!’ I then realized why the editors had meanwhile asked me to withhold the names in my paper, which I happily did.