The End of Time: The Next Revolution in Physics

It was recognized in the late nineteenth century that this unidirectionality of observed processes was in sharp conflict with the fact that Newton’s laws should work equally well in either time direction. Why do natural processes always run one way, while the laws of physics say they could run equally well either way? For four decades, from 1866 until his suicide on 5 September 1906 in the picturesque Adriatic resort of Duino, the Austrian physicist Ludwig Boltzmann attempted to resolve this conflict. He introduced a theoretical definition of entropy as the probability of a state. He firmly believed in atoms – the existence of which remained controversial until the early years of the twentieth century – conceived of as tiny particles rushing around at great speed in accordance with Newtonian laws. Heat was assumed to be a measure of the speed of atoms: the faster the atoms, the hotter the substance. By the second half of the nineteenth century, physicists had a good idea of the immense number of atoms (assuming that they existed) there must be even in a grain of sand, and Boltzmann, among others, saw that statistical arguments must be used to describe how atoms behave.

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!

CHAPTER 2

Time Capsules