Big Bang to Big Crunch

The Nature of Space and Time contains six lectures-three by Stephen Hawking, three by Roger Penrose – and a closing Hawking-Penrose debate. As Penrose indicates, it might be viewed as continuing the famous Bohr-Einstein exchange of some seventy years ago. Against the background of new cosmological theories, Hawking defends Bohr’s thesis that quantum theory has no radical incompleteness. Those who think it incomplete are wrongly treating its formulas as describing reality, rather than as predictive tools. Like Einstein, Penrose disagrees. Things then get somewhat tense. Hawking uses terms like ‘magic’ in dismissing theories which Penrose brought to the general public in The Emperor’s New Mind. Penrose throws some courteous cold water on the universe-creating mechanism made famous by Hawking’s A Brief History of Time.

The book contains a few cartoons: a full-bearded God throwing dice towards a black hole, for example. Also some jokes: miniature black holes gobbling up ‘all those odd socks’, four-dimensional slide projector screens becoming unavailable through ‘government cuts’. A three-part companion videotape can be had. Still, when the Foreword by Michael Atiyah remarks that ‘some of the presentation requires a technical understanding of the mathematics and physics’ this can seem a grave understatement. Very little of the volume will be intelligible to the ‘broader audience’ to which Atiyah commends it for its ‘argument at a higher (or deeper) level’: the philosophical level. So what might Atiyah have had in mind?

In the Sixties Hawking and Penrose joined in constructing ‘singularity theorems’. These seemed to show that everything now visible to us must have originated from a single point, or from something much like one: some ‘singularity’ where gravitationally induced curvature was indefinitely high, and at which particle histories had their first moments. Hawking’s opening lecture (Chapter I: ‘Classical Theory’) tells us that ‘this led to the abandonment of attempts to argue that there was a previous contracting phase and a nonsingular bounce into expansion. Instead, almost everyone now believes that the universe, and time itself, had a beginning at the big bang. This is a discovery far more important than a few miscellaneous unstable particles, but not one that has been so well recognised by Nobel prizes.’

The Nobel prizes may be delayed for a while because the ‘discovery’ is far from secure. For a start, some have argued that String Theory, currently popular as a Theory of Everything, would ‘smear out’ any singularity. (Hawking notes this, but has a distaste for String Theory: ‘so far, its performance has been pretty pathetic’.) Again, the Hawking-Penrose singularity theorems assumed that gravity never acted repulsively. A cosmological constant – Einstein called it his ‘greatest blunder’, yet it’s taken seriously nowadays – could change matters, however. Penrose explains in his first lecture (Chapter 2: ‘Structure of Spacetime Singularities’) that such a constant could make ‘naked singularities’ respectable, these being singularities into which material can enter from the past, then exit to the future. Moreover, a central theme of Hawking’s third lecture (Chapter 5: ‘Quantum Cosmology’) is that a ‘singular’ beginning of things could be disastrous to science. Its consequence would be that ‘the laws of physics could break down anywhere’ so that ‘predictability would disappear completely’: something of an exaggeration, one might think, yet perhaps containing enough truth to make singularity theorems unpopular with prize-givers. In any case, Hawking himself avoids a singular beginning of things when making the ‘noboundary proposal’, which Chapter 5 defends.

According to this, our universe is ‘compact yet unbounded’, like a circle. At the start it was four-dimensional but Euclidean. Time didn’t exist and, declares Hawking, the universe could have appeared by ‘quantum tunnelling’, not just from a background space (as when particles fluctuate into existence through quantum uncertainties) but ‘out of absolutely nothing at all’. What could then be expected is calculated in a supposedly natural manner by combining all the possibilities. It’s claimed that this would result in fields whose fluctuations were the smallest compatible with quantum uncertainties. The universe would thus start off very smooth – ‘starting off’ being a term which would make sense if the originally timeless space were glued to ordinary spacetime, in a way describable with complex numbers. These are the square roots of negative numbers: the square root of minus one, for instance.

The idea that the universe originated ‘from absolutely nothing’, yet in a manner governed by physical laws, could be thought unacceptably odd. Although myself notorious for defending the Neo-Platonic theory that the mere ethical need that there be something, perhaps a divine person or a good universe, could have created the something in question, even I can find little sense in the notion that a quantum-physical need for a universe could call such a universe into existence – there being some fair probability that, so to speak, the non-existent ‘quantum dice’ would fall in a universe-creating fashion. Penrose concentrates his attack on other points, though. He insists that Hawking hasn’t accounted for ‘time’s arrow’, the distinction between the smoothness (‘low entropy’) of the past and the disorderliness (‘high entropy’) of the future – a distinction which underlies our experience of time as progressing from past to future.

In Chapter 5 Hawking speaks of his own ‘greatest mistake’. Taking an expanding half-universe in which time’s arrow pointed in the direction familiar to us, he thought he could glue it to a contracting half-universe in which the arrow was reversed, so that ‘cups would mend themselves and jump back on the table.’ Unfortunately, this would work only in the cases of highly ‘unnatural’ universes: ones whose structures were chosen with fantastic care. Penrose considers that something rather similar goes wrong with Hawking’s ‘no-boundary proposal’ universe, at the junction between the original ultra-smooth space and the subsequent spacetime. Trickery involving complex numbers cannot make the junction into something ‘natural’, Penrose suggests. Instead, one needs special principles forcing the world’s beginnings to be immensely smooth whereas its endings (in the depths of black holes, or in a Big Crunch) are extremely disorderly. Only such principles could explain why we detect black holes, which suck light inwards, but no white holes pouring light outwards. The principles are, he surmises, connected with an arrow of time that appears even at the quantum level. Present-day quantum physics is severely incomplete since it cannot account for ‘the collapse of the wave function’: the manner, that’s to say, in which quantum fuzziness – maybe one should say fusedness or ‘superposition’ of alternatives – suddenly gives way to something definite.

For Penrose, the collapse of the wave function doesn’t depend on observers. Instead, what’s in question is interactions which sufficiently amplify the distinction between the various alternatives permitted by quantum physics. Penrose believes that the growing divergence between the alternatives puts regions of spacetime under a stress that gives them an in creasing tendency to ‘decay’, so that one alternative is chosen and the others become unreal. Gravity is involved here. Gravity is a matter of spacetime geometry, and this geometry becomes troubled when divergent alternatives become fused, ‘superposed’, as described by quantum theory.

Hawking is vehemently against all such ideas. He points at the unabashed use of complex numbers in Penrose’s own theory of ‘twistors’. (Surveyed in Penrose’s heavily mathematical Chapter 6, ‘The Twistor View of Space-time’, this is a theory in which we ‘try to regard entire light rays as more fundamental even than spacetime points’.) He goes on to propose that the distinction between black holes and white holes is a matter of language. We label a singularity a ‘black hole’ when it’s large and radiating little, and a ‘white hole’ when it’s little and sending out large amounts of what has come to be called ‘Hawking radiation’. Furthermore, there’s no genuine, sudden change from fuzziness to definiteness: ‘I totally reject the idea that there is some physical process that corresponds to the reduction of the wave function or that this has anything to do with quantum gravity or consciousness. That sounds like magic to me, not science.’

Hawking is sure that Penrose misunderstands Schrödinger’s cat, an animal supposedly in a ‘superposition’ of being-dead and being-alive until jelled into one state or the other by observations. The cat’s being ‘half alive and half dead’ is something which ‘doesn’t bother me’, Hawking comments. ‘I don’t demand that a theory correspond to reality because I don’t know what it is. Reality is not a quality you can test with litmus paper. All I’m concerned with is that the theory should predict the results of measurements.’ What’s more, any alive-dead fuzziness could be in question only when the poor cat was superbly insulated from its environment. Interactions with, for instance, air molecules, would quickly carry away information about the cat’s condition, and the loss of information would ensure that any alive-deadness would terminate. Here we get both to some very difficult philosophical questions and to the core of the Hawking-Penrose exchange.

How could loss of information make a situation more definite? At the heart of quantum theory lies the double-slit experiment described in the second of Penrose’s lectures (Chapter 4: ‘Quantum Theory and Spacetime’). Shine a beam of light at one slit in a barrier. Passing through the slit, photons hit a photographic plate beyond the barrier. Open a second slit and switch on the same beam, perhaps one so faint that only a single photon is in flight at any moment. Your information about the path taken by a given photon is reduced by the opening of the second slit. And yet this increase in your ignorance is accompanied by a new definiteness, for you now know that various areas of the photographic plate – areas which were reached by photons when only the one slit was open – are areas where no photon will ever land. It is this kind of thing which encourages talk of superposed alternatives. It is as if each photon had divided into two possibilities which went through the two slits, then interfered with each other in the manner of waves, so that they totally cancelled out in various zones. When the photographic plate is reached – or, some have (seemingly absurdly) suggested, when an observer looks at the resulting photograph – the wavelike superposition of possible locations is ‘collapsed’ into the photon’s being at a single point in one of the remaining zones

Now, Hawking’s opening lecture had declared that ‘a physical theory is just a mathematical model’ so that ‘it is meaningless to ask whether it corresponds to reality.’ To Hawking there can be nothing faulty or incomplete in the notion that quantum probabilities are just degrees of knowledge and of ignorance. Can adding to ignorance in one respect increase knowledge in another respect? Of course it can, since experiments tell us so. Why on earth should we find it mysterious?

Penrose thinks this won’t do. Hawking hasn’t explained why observations always show definite entities, instead of such things as an alive-dead cat. The fact that air molecules carry away information about cats could in no way transform them all into sharply focused objects. For one thing, it would simply raise the same problems as before, because quantum theory – at least until made more complete, for instance with the Penrosian ‘decay’ mechanism – assures us that those air molecules are them selves fuzzy, ‘in superpositions’, until they are observed.

Penrose would seem to have a strong case. That a physical theory ‘is just a mathematical model’, and that it’s ‘meaningless to ask whether it corresponds to reality’, are slogans with little plausibility when applied to comparatively simple theories: the theory, for instance, that heated iron expands and glows because its particles move faster. And while the slogans may sound better when applied to complex theories, the difference between simplicity and complexity is a mere matter of degree. There’s no magic point of increased complexity at which good theories, predictivcly fruitful, suddenly come to say things which in no way ‘correspond to reality’. If there were such a point, then the predictive fruitfulness would itself become magical.

Besides, Hawking often shows signs that he sees himself as describing realities. In particular, his universe-creation story is meant to describe quantum-physical realities important for understanding the existence of the universe, instead of tools useful for (whatever could this mean?) predicting the universe’s existence. Again, there’s surely quite a difficulty in understanding how missing information, or ignorance, could help us to see the world in sharp focus, so that we detected dead cats instead of alive-dead superpositions.

This last point is reinforced when one looks at Hawking’s explanation of why we don’t see entire superposed galaxies. The limited speed of light restricts how much of the universe we can observe. There are regions so distant that no light from them could yet have reached us, during the roughly ten billion years since the Big Bang. There are even, in the universe as Hawking pictures it, regions which could never become visible to our descendants. ‘The no-boundary proposal implies that the universe is spatially closed,’ he writes, and ‘a closed universe will collapse again before an observer has time to see all the universe.’ Now, how does Hawking account for the truth that we never see galaxies in superpositions? He does so by saying there are cosmic regions of which we must for ever remain ignorant. ‘People normally try to account for decoherence by interactions with an external system that is not measured. In the case of the universe there is no external system, but I would suggest that the reason we observe classical behaviour’ – rather than, for example, superposed galaxies – ‘is that we can see only part of the universe.’

To me, this makes little sense. For how could the existence of distant regions, too far away to enter into causal contact with us, cause us to see definite objects, rather than superpositions? Yet if unable to cause this, how could it explain it? ‘By showing there is something whereof we must remain ignorant’ doesn’t look like an answer.

At a famous point in the Bohr-Einstein controversy, Bohr protested that Einstein had failed to pay proper attention to Einsteinian relativity. Something rather similar occurs in the Hawking-Penrose exchange, during Chapter 7 (‘The Debate’). A recent biography of Einstein states that he might as well have gone fishing from 1925 onwards, instead of struggling with the quantum world. Penrose comments: ‘I believe that the reason why Einstein didn’t continue to make big advances in quantum theory was that a crucial ingredient was missing. This missing ingredient was Stephen’s discovery, fifty years later, of black hole radiation. It is this information loss, connected with black hole radiation, which provides the new twist.’ Penrose’s idea is that the loss can occur only if there is a complementary gain of information when superpositions come to an end, a process which he thinks can be understood only by bringing the arrow of time into quantum physics.

Just what’s going on here? Well, many pages of The Nature of Space and Time are devoted to the apparent loss of information when black holes ‘evaporate’ thanks to their Hawking radiation. Black holes used to be thought of as regions from which not even light could escape. Hawking showed that quantum effects make them radiate. This leads them to shrink, at first extremely slowly but later very fast. It seems they must eventually vanish. Now, an observer outside a black hole may have no way of telling just how it originated. The hole can be considered as having swallowed vast amounts of information about the particles which went into making it. The Hawking radiation emitted by the hole takes a random form, so cannot give back this information. When the hole vanishes, isn’t the information lost entirely?

Many physicists are unhappy with such a prospect. Some have tried to escape it by saying that quantum theory breaks down when the hole becomes very small. But in his second lecture (Chapter 3: ‘Quantum Black Holes’) Hawking develops a beautiful thought-experiment in which two large black holes annihilate each other. He thereby gets around the objection that seemingly lost information may always be recoverable when black holes become tiny, or that tiny black holes may never completely evaporate. Penrose’s hunch is that although the information really is lost, it is elegantly balanced by new information gained through the decay of superpositions. In resisting this, he suggests, Hawking is resisting the importance of Hawking’s best idea.