Does God Play Dice? (2 page)

The wave function contains all that one can know of the particle, both its position, and its speed. If you know the wave function at one time, then its values at other times are determined by what is called the Schroedinger equation. Thus one still has a kind of determinism, but it is not the sort that Laplace envisaged. Instead of being able to predict the positions and speeds of particles, all we can predict is the wave function. This means that we can predict just half what we could, according to the classical 19th century view.

Although quantum mechanics leads to uncertainty, when we try to predict both the position and the speed, it still allows us to predict, with certainty, one combination of position and speed. However, even this degree of certainty, seems to be threatened by more recent developments. The problem arises because gravity can warp space-time so much, that there can be regions that we don't observe.

Interestingly enough, Laplace himself wrote a paper in 1799 on how some stars could have a gravitational field so strong that light could not escape, but would be dragged back onto the star. He even calculated that a star of the same density as the Sun, but two hundred and fifty times the size, would have this property. But although Laplace may not have realised it, the same idea had been put forward 16 years earlier by a Cambridge man, John Mitchell, in a paper in the Philosophical Transactions of the Royal Society. Both Mitchell and Laplace thought of light as consisting of particles, rather like cannon balls, that could be slowed down by gravity, and made to fall back on the star. But a famous experiment, carried out by two Americans, Michelson and Morley in 1887, showed that light always travelled at a speed of one hundred and eighty six thousand miles a second, no matter where it came from. How then could gravity slow down light, and make it fall back.

This was impossible, according to the then accepted ideas of space and time. But in 1915, Einstein put forward his revolutionary General Theory of Relativity. In this, space and time were no longer separate and independent entities. Instead, they were just different directions in a single object called space-time. This space-time was not flat, but was warped and curved by the matter and energy in it. In order to understand this, considered a sheet of rubber, with a weight placed on it, to represent a star. The weight will form a depression in the rubber, and will cause the sheet near the star to be curved, rather than flat. If one now rolls marbles on the rubber sheet, their paths will be curved, rather than being straight lines. In 1919, a British expedition to West Africa, looked at light from distant stars, that passed near the Sun during an eclipse. They found that the images of the stars were shifted slightly from their normal positions. This indicated that the paths of the light from the stars had been bent by the curved space-time near the Sun. General Relativity was confirmed.

Consider now placing heavier and heavier, and more and more concentrated weights on the rubber sheet. They will depress the sheet more and more. Eventually, at a critical weight and size, they will make a bottomless hole in the sheet, which particles can fall into, but nothing can get out of.

What happens in space-time according to General Relativity is rather similar. A star will curve and distort the space-time near it, more and more, the more massive and more compact the star is. If a massive star, which has burnt up its nuclear fuel, cools and shrinks below a critical size, it will quite literally make a bottomless hole in space-time, that light can't get out of. Such objects were given the name Black Holes, by the American physicist John Wheeler, who was one of the first to recognise their importance, and the problems they pose. The name caught on quickly. To Americans, it suggested something dark and mysterious, while to the British, there was the added resonance of the Black Hole of Calcutta. But the French, being French, saw a more risqué meaning. For years, they resisted the name, trou noir, claiming it was obscene. But that was a bit like trying to stand against le weekend, and other franglais. In the end, they had to give in. Who can resist a name that is such a winner?

We now have observations that point to black holes in a number of objects, from binary star systems, to the centre of galaxies. So it is now generally accepted that black holes exist. But, apart from their potential for science fiction, what is their significance for determinism. The answer lies in a bumper sticker that I used to have on the door of my office: Black Holes are Out of Sight. Not only do the particles and unlucky astronauts that fall into a black hole, never come out again, but also the information that they carry, is lost forever, at least from our region of the universe. You can throw television sets, diamond rings, or even your worst enemies into a black hole, and all the black hole will remember, is the total mass, and the state of rotation. John Wheeler called this, 'A Black Hole Has No Hair.' To the French, this just confirmed their suspicions.

As long as it was thought that black holes would continue to exist forever, this loss of information didn't seem to matter too much. One could say that the information still existed inside the black hole. It is just that one can't tell what it is, from the outside. However, the situation changed, when I discovered that black holes aren't completely black. Quantum mechanics causes them to send out particles and radiation at a steady rate. This result came as a total surprise to me, and everyone else. But with hindsight, it should have been obvious. What we think of as empty space is not really empty, but it is filled with pairs of particles and anti particles. These appear together at some point of space and time, move apart, and then come together and annihilate each other. These particles and anti particles occur because a field, such as the fields that carry light and gravity, can't be exactly zero. That would mean that the value of the field, would have both an exact position (at zero), and an exact speed or rate of change (also zero). This would be against the Uncertainty Principle, just as a particle can't have both an exact position, and an exact speed. So all fields must have what are called, vacuum fluctuations. Because of the quantum behaviour of nature, one can interpret these vacuum fluctuations, in terms of particles and anti particles, as I have described.

These pairs of particles occur for all varieties of elementary particles. They are called virtual particles, because they occur even in the vacuum, and they can't be directly measured by particle detectors. However, the indirect effects of virtual particles, or vacuum fluctuations, have been observed in a number of experiments, and their existence confirmed.

If there is a black hole around, one member of a particle anti particle pair may fall into the hole, leaving the other member without a partner, with which to annihilate. The forsaken particle may fall into the hole as well, but it may also escape to a large distance from the hole, where it will become a real particle, that can be measured by a particle detector. To someone a long way from the black hole, it will appear to have been emitted by the hole.

This explanation of how black holes ain't so black, makes it clear that the emission will depend on the size of the black hole, and the rate at which it is rotating. But because black holes have no hair, in Wheeler's phrase, the radiation will be otherwise independent of what went into the hole. It doesn't matter whether you throw television sets, diamond rings, or your worst enemies, into a black hole. What comes back out will be the same.

So what has all this to do with determinism, which is what this lecture is supposed to be about. What it shows is that there are many initial states, containing television sets, diamond rings, and even people, that evolve to the same final state, at least outside the black hole. But in Laplace 's picture of determinism, there was a one to one correspondence between initial states, and final states. If you knew the state of the universe at some time in the past, you could predict it in the future. Similarly, if you knew it in the future, you could calculate what it must have been in the past. The advent of quantum theory in the 1920s reduced the amount one could predict by half, but it still left a one to one correspondence between the states of the universe at different times. If one knew the wave function at one time, one could calculate it at any other time.