But how right were they? To conduct the survey someone visited schools and measured the height of many children with the same age. The results were then compared statistically with the circumstances of their parents. Presumably they found a statistically significant negative correlation between height and indicators of family conflict such as divorce, thus proving the link. Fine so far, but can we conclude that the conflict caused children to be smaller? Would it not have been equally valid to conclude that having small children leads to divorce? The scientist in charge speculated that stress may reduce the amount of growth hormone that young children produce.
In fact he applied his prejudices and drew a conclusion which sounds reasonable without realising that the converse was also a possible explanation of the survey results. It is not difficult to believe his theory but there was nothing from the survey which proved it. In fact the real reason behind the correlation may have been one or more third factors such as wealth. Children of poorer families may have worse standards of nutrition resulting in slower growth, and lack of money might also lead to higher divorce rates. Another cause may have been a genetic trait which shows up in both the growth and temperament of family individuals. Such effects are equally likely to show up as a correlation in the survey but the news article said nothing about such possibilities.
The difference between the possible conclusions from the survey is not just one of semantics. People reading the article could blame their frequent family rows for having a small child. Such feelings of guilt are unlikely to help the situation. They may have been right but I suspect they would have been wrong. Surveys such as this are common and are often reported in the media by people who do not appreciate the traps that statistics can lead us into.
When responsible scientists wish to establish causal links between different effects they are more careful. For example, when a new drug is tested it is necessary to know how effective it is and what side effects it may produce. To do this a group of volunteers is selected for trials. The group is divided in two at random and one half is given the drug. The other half is given a placebo pill which is known to have no effect. Nobody taking part knows which group they are in. Both groups are then monitored for possible effects. The effect is known to be real if it is significantly more noticeable amongst those who took the drug than those who took the placebo. It is then certain that taking the drug really caused the effect. The difference between this example and the survey is that the choice of who got the drug and who did not was controlled. In the survey which claimed to link height and family strife there was no control over whose parents were divorced which were not so it was impossible to distinguish cause from effect or rule out other factors with certainty.
Philosophers such as David Hume have been sceptical about these notions of causality. In 1740 Hume questioned the basic idea of causation. It is sometimes thought that his rejection of causation implies a rejection of scientific laws but it does not. What it really implies is a rejection of free will. Compare the case of the broken vase with the survey and the medical trials. Which does it more closely resemble? You might argue that it is more like the medical trials because a person has control over whether or not they do something like bump into a table. They have free will. The vase breaking is a response to an action of free will, even if it was an accident. If an action is controlled then it must be the cause rather than the effect. If we accept the contention of Hume we deny any distinction between cause and effect so we must also deny our free will.
Causality was not always characterised so simply as it is today. In ancient Greece, at the Lyceum in Athens, Aristotle taught that there were four types of cause: the material, the formal, the efficient and the final. If you build a boat he would have said that the causes were the materials you used, the plans you drew up, the labour you put into it and what you wanted to do with it. If any of these four things were not there, the boat would not be made.
In terms of modern physics we would regard the efficient and final cause as the two extremes of temporal causality, that is, causality related to time. The efficient cause is the initial set of conditions and the final cause is the final set of conditions. Likewise we can regard the material and formal causes as two opposite views of ontological causality, that is causality related to the way in which something is formed.
Let us imagine another example. You are very proud because you have successfully grown a good crop of potatoes in your back garden. You bring a handful in to show your daughter saying "Look, I grew some potatoes!" "Why did they grow?" she inquires, as children do. How would you respond? Just suppose that you are rather philosophical in your ways and you respond according to which types of causality you believe in. The conversation might continue as follows:
"They grew because of biological processes such as photosynthesis."
"Why are there biological processes like photosynthesis?"
"because of atoms and the laws of chemistry which make biological processes work"
"Why are there atoms and laws of chemistry?"
"because of nuclear physics and electromagnetic forces which make atoms out of protons, and electrons?"
"because of more elementary particles and laws of physics which we don't know everything about yet!"
These answers characterise a reductionist or atomist who believes that all explanation can be reduced to underlying laws of physics which may one day be explained through some deep principle of mathematics. Aristotle would say that you had invoked the material cause.
In another mood you might answer differently:
"They grew because I planted them"
"Why did you plant them?"
"because I knew they would be good to eat when they were ready"
"Why did you know?"
"because I learnt such things in school"
"Why did you go to school?"
"because a long time ago people realised that having an education was useful"
... and so on.
This time the conversation might continue through the history of humanity, life on Earth and cosmology until you explain that everything is a result of what happened at the big bang. Of course we are stuck again because we cannot say what caused the big bang. This may be a strange way to explain why potatoes grow but it is exactly how conventional wisdom describes causality in physics. Aristotle would have called it the efficient cause.
Since the 17th century scientists have replaced Aristotle's four causes with just those two: The efficient or prior cause and the material cause, or physical laws. The final and formal cause is gone. Descartes's mechanistic causality is the most widely accepted today. We would say that a cause of an event is any preceding event without which it would not have happened. In addition to this temporal causality many physicists believe that there are fundamental laws of physics to which are other phenomena can be reduced. This reductionism is the material cause and it is what is left of the ontological causality.
If your mind is opened a little by my story of the survey in the news article, then you may also be ready to reconsider your notions of causality in physics. How would you explain the growth of your potatoes if you believed in a final cause?
"They grew to become potatoes"
"Why did they become potatoes?"
"So that we could eat them and grow ourselves"
"Why do we grow?"
"So that we can become strong enough to do our jobs"
Eventually it seems that this will lead towards some ultimate unknown destiny of humanity. These days most scientists do not believe in destiny but Aristotle would defend the final cause. A seed grows because it is destined to become a plant and produce more seeds. His error is easily exposed if we tear up the plant before it matures. It grew just the same to begin with even though the final cause was taken away. The same would not be true if we intervened before the seeds grew. Prior cause seems to be more right than final cause but notice that we have invoked our free will again to prove it.
It could be harder to explain growth in terms of the formal cause. We would have to suppose that the potatoes grew because it had a design purpose. You might say:
"They grew because if they didn't we would have nothing to eat. Then we would not be here to ask such questions!"
This may sound like an invalid explanation at first. Yet it is an explanation which might be given by someone who advocates the anthropic principle. Such people claim that the laws of physics and other aspects of our world are the way they are because they must be that way for us to be here. Reductionism and the anthropic principle are opposing philosophies of ontological causality. They correspond to Aristotle's material and formal cause respectively. Aristotle accepted both types of explanation but most people prefer one or the other.
Let us put ontological causality aside for now and consider temporal causality in more detail. Do the laws of physics justify Descartes who threw out final cause in favour of prior cause?
To keep things simple, let us start by considering just classical Newtonian mechanics. The form which the laws of physics take is crucial to our understanding of causality. Newton's laws take the form of a set of differential equations describing the motion of particles under forces that act between them. If we know the initial positions and velocities of all the particles at an initial time then their positions are determined at any future time. So does this form for the laws of physics allow us to justify our concept of temporal causality, that cause comes always from the past and precedes its effect? It would seem so because the initial conditions seem to be causing all that happens in the future.
There is a catch. The laws of physics in this form can be made to work identically in reverse. If we know the final state of a system we can just as easily determine its past. Furthermore, the classical laws of mechanics do not allow any room for free will. All actions are predetermined by any complete past state. They are also postdetermined by any future state. Newton's laws do not explain why past events are the cause of future events.
To appreciate the physics without being misled by intuition we must imagine ourselves separated from space and time. We need to imagine space-time as a single entity which does not evolve. Like a block of existence, the universe just is. Our lives are worldlines through the block stretching between birth and death. We might equally well say that they stretch between death and birth. On close examination we can tell which way our lives went from past to future because we recognise the symptoms of ageing but there are no time stamps built in to space-time. The block universe has no past, present or future. It is just a collection of events.
If the universe is finite and closed with a beginning at the big bang and an end at the big crunch you can think of it as a kind of rugby ball shaped surface which narrows at either end. Space-time is four-dimensional and has nothing outside or inside but we have to visualise it as a two-dimensional surface sitting in space. This limitation of our minds does not matter. We do not have to visualise something to understand it.
People often discuss what came before the big bang. Some think that there must have been something. Others say there was nothing. When we think about the block universe we see that there was no "before". The surface of the sausage is all that there is to the universe and time is part of it. We should not think of an empty space around it since that space does not exist. Of course we do not know that the universe is really rugby ball shaped and there could have been something before the big bang, but it is not necessarily so. We should not let our experience influence our reasoning since our experience is limited to a small part of the universe and prejudices our judgement. It is not easy to imagine a universe which is curved but which has no outside, no before and no after, but we can describe the shape of space-time mathematically without referring to anything outside, so an outside is not necessary. Asking about what came before the big bang is like asking what comes before the letter A in the alphabet. Asking about what is outside the universe or where it is, is like asking what is outside the alphabet or where it is. It is nowhere or everywhere. It just is.
Nevertheless, we can imagine that we are examining the universe from outside as a psychological crutch to support out thoughts. We look closely to see if there are signs of causality but if we are outside we have no control over events. We are in the place of someone who does a survey and tries to establish causal relationships between things we observe. Without control any judgement about causality is subjective. We may be able to measure a correlation between certain sets of events but we have no definitive way of knowing which is cause and which is effect unless we could draw from our experience of how we think past influences future.
Does such a view of a block universe from outside make sense? It is a classical view which ignores the quantum nature of the world. In quantum mechanics it is impossible to separate observer from observed. It is difficult to know what is the significant of quantum theory to causality. There are many different interpretations of quantum mechanics and some would suggest a different answer to others. Time is an infamous problem when applied to quantum mechanics and general relativity. Without a theory of quantum gravity we cannot be sure of any response to the question.
I will adopt a position on quantum mechanics which extends the block universe metaphor. Our space-time can be cut like a sliced sausage. Each thin slice represents the universe at one moment in time and records the state of everything classically at that instant. According to physicist Julian Barbour, the quantum multiverse is a heap of slices. The heap contains all possible slices from all possible universes and is not ordered. Time and change have no absolute meaning and just represent the different ways that the slices can be put back together to make histories of the universes. Our passage through the quantum world is just one of many possible sequences which can follow from each instant. A different analogy of the same notion has been described by David Deutsch. Each slice is a snapshot of the universe. They can be put together as frames in a sequence of film which tells the story of a universe. Indeed, this is the film version of the storyteller's paradigm. Our experience of the universe is like a showing of the film, but even when the film lies in the can the universe still exists without any frame singled out as the present moment. The unordered heap of all possible frames is the multiverse.
Einstein and Minkowski taught us that space and time cannot be separated. A universe can be sliced up in different ways just as a sausage could be sliced at different angles. A natural development of the time slice analogies is to break each slice down further into small morsels. If space-time is minced up finely enough the multiverse is reduced to a heap of events. The rules which tell us how they can be put back together are the Feynman rules of quantum gravity, whatever they may be. Just like a story broken down into sentences and then words and then letters, there are fewer components each time. The finer the universe is chopped, the smaller is the heap, but each bit can be used many times and combined in an infinity of different permutations. Such a view of the universe seems to demand event symmetry. The heap is unordered and shuffling its contents has no consequence to the multiverse. It should follow that event symmetry, the symmetric group acting to permute space-time events, should be part of the universal symmetry of nature.
Where does this leave the present? At some time we all ask ourselves "why now?" What distinguishes this moment from others? Given that the universe lasts many billions of years it seems a fantastic coincidence that the present even falls within our lifetime. Of course this is nonsense. It could be no other time than "now". When we view the block universe we see all moments at a glance. There before us are all the moments when we asked "why now?" It becomes a stupid question, a trick of our psychology which has a need to know something it calls consciousness. Within the universe it is a hotly debated subject. From outside the question loses its meaning and we judge it differently. It is fortunate that we do not need to apply our philosophy of physics to our everyday lives otherwise we would lose all sense of purpose.
Indeed, the continual increase of entropy is intimately linked to our perception of causality. Entropy is a measure of disorder in a system and defines a thermodynamic arrow of time which can be linked to the psychological arrow of time. There is, however, a catch again. The second law of thermodynamics is inexplicable in terms of the underlying laws of physics which, as far as we know, are reversible. This is enshrined in a theorem of relativistic quantum field theory which proves the necessity of CPT conservation.
The increase of entropy can be understood in certain idealised experiments. For example, take two closed containers filled with gases which are each in thermal and chemical equilibrium, and allow them to mix by connecting the two systems without allowing any energy to escape or enter. When the system comes back into equilibrium the entropy of the final state can be shown theoretically to be higher than the combined entropies in the two original systems. This seems to be theoretical evidence for increasing entropy and it is confirmed by experiment, but we must not be misled. The assumption that prepared systems tend towards equilibrium has been justified, but theory would tell us that they tend towards equilibrium in the past as well as the future. We are victims of our prejudices about causality again and have devised an argument with circular reasoning to support it.
Such attempts to prove the second law of dynamics originated in the 19th century with the work of physicists such as Ludwig Boltzmann. Such a feat can never be achieved because the laws of physics are time symmetric and it is impossible to derive a time asymmetric result from time symmetric assumptions. Boltzmann slipped in some time-asymmetric assumptions in order to derive the result. Physicists have devised many other arguments for why entropy always increases, trying to get round the problem of CPT symmetry.
Here are a few possibilities:
Fault: Electromagnetic radiation cannot be distinguished from its antimatter image, and yet it obeys the second law of thermodynamics.
Query: Does this mean that the third law of thermodynamics is not valid for classical statistical mechanics?
This could be true but can the laws of thermodynamics be a result of quantum gravity whose effects are normally thought to be irrelevant except in the most extreme physical regimes?
But then why were initial conditions set rather than final or mixed boundary conditions?
Entropy might be better understood in terms of information. It can be linked to the number of bits which are needed to describe a system accurately. In a hot disordered system you need to specify the individual state of each particle, while a cold lattice can be described in terms of its lattice shape, size and orientation. Far less information is needed for the low entropy system.
The claim that entropy increases because it started low in the big bang is perhaps the one which has fallen into conventional wisdom, even if it is admitted that we do not understand why it started low. Perhaps it is because of some huge unknown symmetry which was valid at the high temperatures of the big bang and broken later. This is also my opinion but I think that if the universe were closed we would have to apply the argument in reverse at the big crunch too.
In a completely deterministic system the evolution of the system is equally well determined by its final state as by its initial so we could argue that the amount of information in the system must be constant. The difficulty there is that we are assuming an exact knowledge of state which is impossible. In any case, quantum mechanics is not deterministic. If we make a perfect crystal with an unstable isotope, as time passes some of the atoms will decay. The amount of information needed to track the decayed atoms increases. Perhaps, then, it really is quantum mechanics and the collapse of the wave function which is responsible.
If physicists used to think they understood entropy then their faith was deeply shaken when Stephen Hawking and Jacob Bekenstein discovered that the laws of thermodynamics could be extended to the quantum mechanics of black holes. The entropy is given by the area of the black hole but its temperature can only be understood through quantum mechanical effects. This shows that classical understanding of thermodynamics is indeed incomplete and perhaps only a complete theory of quantum gravity can explain the laws fully.
At present the universe is certainly expanding, as demonstrated by Hubble in 1929 when he started measuring the red-shifts of far away galaxies and correlating them to their distance. This defines a cosmological arrow of time which distinguishes past from future. In 1962 J. E. Hogarth suggested the possibility that this cosmological arrow could be linked to the thermodynamic arrow of time. Thomas Gold proposed that when the universe starts to contract the increase of entropy might reach a turning point. As the universe collapses history would run in reverse.
Needless to say, Gold's model of the universe is quite controversial. Intuition suggests that the arrow of time cannot change direction. It would be a complete reversal of causality with events being determined by the future instead of the past. In 1985, Stephen Hawking unexpectedly came out in support of Gold. He published a paper demonstrating that a time reversal was to be expected because the physics of the final crunch must be the same as the physics of the big bang. We might try to understand the quantum state of the entire universe by using Feynman's path integral formulation of quantum mechanics. We must form a sum over all possible space-time manifolds allowed in general relativity. Hawking has argued that we can understand entropy in this way if the universe is an entirely closed system, finite in both time and space but with no boundary. There would be no initial or final conditions to worry about, and both the end and start of the universe would be a consequence of the same laws of physics which are obeyed at all times. If the laws of physics are time reversal invariant we should then expect the end to be like a reversed playback of the beginning.
Before Hawking's paper had passed through the publishing process he was already under pressure to change his mind. His colleagues Laflamme and Page set out to convince him that he had made an error. Before the paper went to press they succeeded and he added a note to the paper admitting his mistake. He now claims that there are two possible ways a universe could start or end. One has low entropy the other high. The only consistent picture is one in which it is low at one end and high at the other hence temporal symmetry is broken.
If this argument could be made solid then it would be a powerful one. The path integral formulation avoids problems of time since it is a sum over all possible universes rather than an evolution with separate boundary conditions. However, Hawking's method uses an incomplete semi-classical description of quantum gravity. The argument could only be made complete when we understand quantum gravity better. Until then it is an open question whether or not a closed cosmological model will have a time reversal at half time or not.
There remain very few scientists who have argued in favour of a Gold universe and stuck to it. Most cosmologists have sought reasons to rule it out and have often claimed success. As the philosopher Huw Price has shown, most of those arguments are based on double standards of reasoning. Often time asymmetric conclusions are drawn from time symmetric assumptions. This is just about impossible unless there is some spontaneous symmetry breaking such as that proposed by Hawking.
Intuition suggests that the arrow of time could never reverse. If we could meet other intelligent life-forms who are evolving in reverse, many paradoxes would present themselves. Their past would be our future. What would there be to prevent them from telling us about events in our future? Suppose we decided they were a threat and decided to destroy them. If we succeeded they would cease to exist in their own past. What is to prevent us from bringing about such a paradoxical situation?
The only reasonable answer must be that the arrow of time will only reverse when we are long gone and other time-reversed life-forms are not there either. In other words, the epoch in which the universe will reverse its collapse must be lifeless. Some people already find it hard to accept that the human race must be extinguished at the big crunch. To suggest that we cannot even survive for half as long even when there is no such catastrophe to wipe us out seems almost unthinkable. After many trillions of years the stars will have faded. The universe will be a cold place, hard to live in with so few sources of energy. Could we not at least hope to build a powerful computerised automaton which could be programmed to hibernate through the aeons, using the least power possible to steer away from black holes and other places where it would be destroyed? If so it would be able to take a message of our past into the future? In the collapsing universe it might revive and deliver a message to the anti-thermodynamic inhabitants of the other half of space-time. Sadly the answer must be no since it would create unresolvable paradoxes, but unless we can explain what would stop it we must give up the possibility of a Gold universe.
Conversely, the time-reversed stars of the future will absorb photons because they are time reversed. Those photons should be around now. Could we see them?
Gell-Mann believes that if they are there they could be detected. He says that they would add to the background light of the universe which could be measured. If the light is not there a Gold universe might be ruled out. Huw Price pointed out that it is not so simple. The light from future stars cannot be detected simply by looking at the night sky with a telescope. These photons would be heading for a time reversed star in the future. If you block their passage with any kind of detector such as a photographic film they will simply not be there because they would not then be around in the future. Their behaviour is distinctly acausal. According to Price they would be invisible by ordinary means.
If you hold up a piece of paper in space. Photons of future starlight would not be absorbed. Instead they would be emitted as if they were being drawn out of the paper by a future cause. You are probably thinking that all this is already just too absurd to be possible anyway, but you must suspend your disbelief until a contradiction with either logic or observation has been reached. Light drawn off a surface like this would not register in the ordinary way. It is actually quite difficult to predict what would really happen because the photons are acausal and the paper is not. Would the effect of the photons be detected before or after they are emitted? Despite such logical difficulties we know that energy must be conserved what ever happens. This means that energy will be drawn off the paper. It should be detectable in principle.
It is not absolutely clear whether or not observation can already rule this out but I think they can. The anti-thermodynamic radiation would be present at many wavelengths. Light photons may be difficult to detect in this way but radio waves would be likely to affect radio telescopes and gamma rays would also surely leave their mark. Above all an anti-thermodynamic cosmic background radiation destined for the big crunch would be similar in energy and temperature to the cosmic background radiation from the big bang. Instead of imparting heat to a detector it would take it away. The net effect of both the big bang and big crunch radiation would be no heating. Yet the heat of the cosmic background was detected by Andrew Mckellar in cosmic cyanogen as long ago as 1941 even before its significance was recognised.
When you hold up your hand to light it casts a shadow behind it. Even faint starlight casts such a shadow. What about our anti-thermodynamic light from the future? If you could expose your hand to anti-thermodynamic radiation you expect it to have photons drawn off it destined for some anti-thermodynamic star in the distant future as the universe collapses. Radiation would surround your hand but instead of casting a shadow behind the direction the light is travelling, there would be a kind of anti-shadow in front of it from the direction the radiation is coming. This is simply because light in front of your hand is blocked in its passage towards its destiny.
If you move your hand in front of a lamp, the shadow moves with it. Because of the finite speed of light there is always a slight delay and the movement of the shadow lags behind the movement of your hand by an imperceptible amount. The anti-shadow cast by anti-thermodynamic light behaves differently. It is not difficult to see that it must move ahead of the hand, anticipating every move by the instant of time it would take the light to travel from the shadow to the hand.
This effect could be used in principle to send messages back in time. To do it effectively the distance from the hand to where the shadow was cast would have to be made large. A mirror could be used to reflect the shadow from a long distance away back to a point near where the hand is moving. By detecting the anti-shadow you could see what your hand is about to do. You could literally use hand signals to send messages into the past. It is difficult to see how the paradoxes presented by such a phenomenon could be avoided unless anti-thermodynamic light is invisible, but as I have already argued, it should be detectable. Either anti-thermodynamic light is not available to us or we will have to face up to these paradoxes.
There is an alternative way in which a Gold universe might work. It could be that thermodynamic and anti-thermodynamic matter and radiation never mix. Instead they might meet in the middle of time when thermodynamic matter may slowly transform into anti-thermodynamic matter. Thermodynamic matter would only be present in the expanding half of the universe and anti-thermodynamic matter would only be present in the collapsing half .
Think again about the electromagnetic radiation. Remember it was argued that light left over from the stars in the expanding universe and the cosmic background radiation would survive into the collapsing half. It was assumed that this radiation would be randomly dispersed so that it would strike any objects that are around during the collapse. However, this assumes that each photon is causally influenced only by its dim and distant past, never the future. On reflection this is not what would be most probable. It is more likely that the radiation would fall under the influence of its destiny if the collapse is anti-thermodynamic. In that case the photons which are radiated from stars now and pass into the collapsing phase of the universe will be the same photons which are anti-thermodynamically absorbed by the anti-thermodynamic stars in the collapse. If this were to be the case then there would likewise be no anti-thermodynamic radiation from the future around now and we would not be able to send paradoxical messages back in time. There would be no inconsistency.
You might think that a huge coincidence would be required for all the photons emitted by stars now to conspire to fall onto anti-thermodynamic stars in the future, but the whole point is that a low entropy phase of the universe already appears as a fantastic statistical fluke. This comes about because the initial and final state force it to happen and the rest of time has to cope with it. It drives evolution and other acts of the universe which would otherwise seem highly improbable. A fluke such as photons travelling through the aeons and hitting an anti-thermodynamic star must be weighed against the equally unlikely events which must happen if it hits a cold anti-thermodynamic surface.
Substance made out of protons, electrons and neutrons is a different matter. Time reversal (T) alone is not an exact symmetry of nature but if we combine it with charge conjugation (C) and parity inversion (P) we do get an exact symmetry called CPT. This operation effectively exchanges matter and anti-matter. In 1967, Andrei Sakharov found a way to account for why the universe is dominated by matter with very little anti-matter. It is due to the slight CP violating effects in the nuclear forces. In the heat of the big bang these would have been significant enough to account for the imbalance left over from the first instants. If this is correct then a similar effect must apply in reverse at the big crunch which we are assuming is anti-thermodynamic. The alarming consequence is that the collapsing phase shall be dominated by anti-matter.
It is going to be more difficult to explain how thermodynamic matter can transform into anti-thermodynamic anti-matter somewhere around the middle of time because CP violating effects are improbable at low temperatures. If the universe lasts long enough the problem will be resolved because protons can decay to produce positrons and then the electrons can anti-decay to make anti-protons. But the half life of this process is at least 1032 years, so unless the universe is set to live much longer than that there is a problem. Proton decay could be forced to happen as the statistically least costly way of making the transformation but if so it would probably be happening already. Experiments which try to detect proton decay say otherwise.
A second possibility is that all the matter falls into black holes where matter is indistinguishable from anti-matter. The anti-matter would then have to emerge from white holes in the reverse fashion. This brings us to the next problem. Where are the white holes? Unless the universe is going to go on long enough for all the protons to decay we will need them. Even if it is going to go on long enough for the protons to decay, there are other particles such as neutrinos which may never reach an equilibrium state with an equal mix of particles and anti-particles. Only photons and other particles which are their own anti-particles can be guaranteed to carry over from the expanding phase to the collapsing phase without spoiling the time symmetry.
The gravitational field equations of classical general relativity are symmetric under time reversal just as for all the other forces. To complement black holes there can also be white holes which are the time reversal of black holes. Just as black holes swallow matter, always get bigger and can never be destroyed, white holes can release matter, always get smaller and can never be created. If black holes survive after the first half of the history of the universe as the classical theory says they must, then a Gold universe must likewise contain white holes which are their time reversal. Those white holes would have to be out there now and must have been already there at the big bang, even though the true cause of their creation is in the future.
The white holes would be dormant, waiting for the distant future when their destiny will be to release all the anti-thermodynamic anti-matter which makes up the anti-stars of the collapsing universe.
There seem to be some probable inconsistencies in this scenario. We should be able to detect those white holes because they will act as gravitational lenses even if they are alone in deep intergalactic space. Astronomers are increasingly finding that black holes are common and that they range in size from a few solar masses up to billions of solar masses. The white holes would have to be at least as common and as big. We do see gravitational lenses but they appear to be due to ordinary galaxies and it already seems unlikely that we can account for so many white holes in the universe. There are other conceptual problems if white holes are around. What if they were to collide with ordinary stars, galaxies or even dust. White holes must attract ordinary matter yet it is not supposed to be able to fall in. Dormant white holes would be very paradoxical objects, especially if we could locate them. The difficulties would be even greater in the early universe where they would inevitably have had a significant influence.
It begins to look like we have finally found a likely contradiction which would rule out a Gold universe, but once again we have only considered the mixing solution for black and white holes. Could there be a better meeting solution as there seems to be for radiation and matter? The only way out would be if black holes could somehow transform gently into white holes. Then there would be no need to account for white holes in the universe now. The black holes which are being discovered all over the universe now, would transform into the complementary white holes which will have to be around in the collapsing universe.
The transformation of black holes into white holes is not easy to understand. In classical physics it simply cannot happen. In quantum mechanics the situation is a little different. According to Hawking, black holes radiate and can lose mass. When Hawking considered the possibility of a Gold universe he considered whether it would be possible for the transformation to happen. A black hole would become quiet when all the matter around it had been pulled in. It could gently radiate but any black hole of the size we have found them to be would be too large to radiate significantly. How could it switch to throwing out matter like a white hole?
As a matter of fact, a dormant black hole would be virtually indistinguishable from a dormant white hole from an external point of view. The gravitational field around them is the same. Only internally are they different. Hawking argued that if a black hole comes into thermal equilibrium with the radiation that surrounds it, so that it radiates the same as it takes in, then it should be in a time symmetric state. This leaves open the possibility that the transformation could take place. From outside the black hole Hawking radiation would just appear to get stronger until what was a black hole is behaving just like a white hole. What would happen internally? A black hole has an internal singularity which lies in the future of anyone who falls in, whereas a white hole must have one in the past from which any outgoing matter originates.
H- Dieter Zeh is one physicist who has continued to study this possibility. Matter which fell into the black hole would seem to be frozen on the event horizon from the point of view of someone who stays outside. Zeh has suggested that quantum effects could simply cause it to turn round and come back out again. The black hole singularity would never form. Unfortunately it is difficult to envisage how the dynamics could work. The curvature at the event horizon of a large black hole is slight and quantum effects should be small. From the point of view of what we are trying to imagine here there is an even worse problem. We were going to claim that the matter which fell into the black hole would later re-emerge as anti-matter from the white hole, but if it is the same matter which turns round and heads back out it cannot change from matter to anti-matter.
It is interesting that Stephen Hawking still believes that black holes and white holes are identical when they are very small. Such virtual quantum black/white holes must be part of the vacuum but they are very different from the macroscopic ones which form from collapsed stars. They would be more like elementary particles and may even turn out to be the same thing as particles when we understand quantum gravity. It would be extraordinary if large black holes could also be identified with white holes. They would have to have both a future and past singularity.
As it happens, the classic static model of a black hole found by Schwarzschild does have a future and past singularity, but a more realistic model of a black hole which formed from a collapsing star cannot have such time symmetry in classical general relativity. If it is possible for it to happen when quantum gravity effects are taken into account it will be very different from what we expect classically. Yet, despite the strangeness of the idea, the possibility cannot be ruled out. The closest description of what it would be like in the language of physics we understand would be that the inside of a black hole would be a quantum supposition of the wave functions of a black hole and a white hole.
The black hole complementarity principle proposed by physicists considering the information loss problem gives further hope to the possibility that a black hole can transform into a white hole. The principle asserts that there is no inconsistency between the point of view of an observer who falls past the event horizon of a black hole towards its singularity and another observer outside who sees him stop at the horizon and eventually return as thermal Hawking radiation. If this is true then we should also accept that there is no inconsistency if there is a third observer who emerges from the event horizon as if it were a white hole too. It is as if the event horizon were a cross-roads in time.
The sources of low entropy are both the initial and the final singularity of the universe. Thus it has two origins. Entropy follows its natural statistical tendency to increase away from those origins where some unknown principle of quantum gravity must be responsible for the extraordinary low entropy. Although life evolves backwards, intelligent life in the collapsing phase will have experiences similar to ours. Their future is our past and they can find no record of it.
The light radiation from our thermodynamic stars, as well as the cosmic background radiation which fills space today, will survive into the collapsing phase. It will gradually transform from being thermodynamic to being anti-thermodynamic. All matter made of particles with mass is most likely to fall into the black holes which are the dead remnants of stars and galaxies. Even neutrinos must follow such a fate, which may only be possible for them if they have a small mass. The black holes themselves will slowly transform to white holes from which the anti-thermodynamic matter of the collapsing phase emerges.
Perhaps the most difficult part of this vision for us is the fate of ourselves and other life. It cannot survive until the collapse or even leave any reminder of its past. Otherwise there might be a paradoxical mixing of thermodynamic and anti-thermodynamic life. The universe will see to it that this does not happen and its job will certainly be made easier if the universe grows to a very old age before the expansion stops.
The question of homogeneity has always been a controversial one in cosmology. In 1933 just a few years after Hubble had shown that the universe is expanding, Arthur Milne proposed homogeneity as a cosmological principle. It is certainly a convenient principle because homogeneous models of the universe are much easier to analyse, but why should we believe it is true? Even in the 1930s Fritz Zwicky was arguing for the presence of galactic clusters in the cosmos, evidence for less homogeneity than others wanted. In 1953 Gérard de Vaucouleurs also produced evidence for large scale structure but still most were sceptics. In the 1980s when detailed maps of the distribution of galaxies were produced the doubters had to concede. There are huge voids and walls on scales which extend to a significant fraction of the size of the observable universe.
Our measurements of the cosmic microwave backgrounds show a high degree of isotropy and this is taken as proof that the universe is homogeneous on larger scales. Our observation is limited by a horizon defined by the age of the universe and the speed of light. Thus we cannot observe anything beyond about 15 billion light years distance. Why should we imagine that the size of the universe is a similar order of magnitude to its current age? We have been unable to measure the extent to which space is curved and cannot place limits on its size.
Martin Rees has compared our view of the universe with a seascape as seen from a ship in the middle of the ocean. As far as the eye can see it seems unchanging except for the waves which we see at close range. The view is limited to the horizon and beyond who knows what there is. It seems to be only an application of Occam's razor which justifies the assumption that space is homogenous on scales hundreds of orders of magnitude larger than the observable horizon.
"Pluralitas non est ponenda sine neccesitate"
"Frustra fit per plura quod potest fieri per pauciora"
"Entia non sunt multiplicanda praeter necessitatem"
In fact, only the first two of these forms appear in his surviving works and the third was written by a later scholar. Many scientists have adopted or reinvented Occam's Razor. Isaac Newton stated the rule: "We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances."
The most useful statement of the principle for scientists is, "When you have two competing theories which make exactly the same predictions, the one that is simpler is the better."
In physics we use the razor to cut away metaphysical concepts. The canonical example is Einstein's theory of special relativity compared with Lorentz's theory that ruler's contract and clocks slow down when in motion through the Ether. Einstein's equations for transforming space-time are the same as Lorentz's equations for transforming rulers and clocks, but Einstein and Poincaré recognised that the Ether could not be detected according to the equations of Lorentz and Maxwell. By Occam's razor it had to be eliminated.
But the non-existence of the ether cannot be deduced from Occam's Razor alone. It can separate two theories which make the same predictions but does not rule out other theories which might make a different prediction. Empirical evidence is also required and Occam himself argued for empiricism, not against it.
Ernst Mach advocated a version of Occam's razor which he called the Principle of Economy, stating that "Scientists must use the simplest means of arriving at their results and exclude everything not perceived by the senses." Taken to its logical conclusion this philosophy becomes positivism; the belief that what cannot be observed does not exist. Mach influenced Einstein when he argued that space and time are not absolute but he also applied positivism to molecules. Mach and his followers claimed that molecules were metaphysical because they were too small to detect directly. This was despite the success the molecular theory had in explaining chemical reactions and thermodynamics. It is ironic that while applying the principle of economy to throw out the concept of the ether and an absolute rest frame, Einstein published almost simultaneously a paper on Brownian motion which confirmed the reality of molecules and thus dealt a blow against the use of positivism. The moral of this story is that Occam's razor should not be wielded blindly. As Einstein put it in his autobiographical notes:
"This is an interesting example of the fact that even scholars of audacious spirit and fine instinct can be obstructed in the interpretation of facts by philosophical prejudices."Occam's razor is often cited in stronger forms than Occam intended, as in the following statements...
"If you have two theories which both explain the observed facts then you should use the simplest until more evidence comes along"
"The simplest explanation for some phenomenon is more likely to be accurate than more complicated explanations."
"If you have two equally likely solutions to a problem, pick the simplest."
"The explanation requiring the fewest assumptions is most likely to be correct."
... or in the only form which takes its own advice...
"Keep things simple!"
Notice how the principle has strengthened in these forms which should be more correctly called the law of parsimony, or the rule of simplicity. To begin with we used Occam's razor to separate theories which would predict the same result for all experiments. Now we are trying to choose between theories which make different predictions. This is not what Occam intended. Should we not test those predictions instead? Obviously we should eventually, but suppose we are at an early stage and are not yet ready to do the experiments. We are just looking for guidance in developing a theory.
This principle goes back at least as far as Aristotle who wrote "Nature operates in the shortest way possible." Aristotle went too far in believing that experiment and observation were unnecessary. The principle of simplicity works as a heuristic rule-of-thumb but some people quote it as if it is an axiom of physics. It is not. It can work well in philosophy or particle physics, but less often so in cosmology or psychology, where things usually turn out to be more complicated than you ever expected.
Simplicity is subjective and the universe does not always have the same ideas about simplicity as we do. Successful theorists often speak of symmetry and beauty as well as simplicity. Paul Dirac said that if requirements for simplicity and beauty clash we should strive for mathematical beauty first and simplicity second. The law of parsimony is no substitute for insight, logic and the scientific method. It should never be relied upon to make or defend a conclusion. As arbiters of correctness only logical consistency and empirical evidence are absolute. Dirac was very successful with his method. He constructed the relativistic field equation for the electron and used it to predict the positron. But he was not suggesting that physics should be based on mathematical beauty alone. He fully appreciated the need for experimental verification.
The final word falls to Einstein, himself a master of the quotable one liner. He warned,
"Everything should be made as simple as possible, but not simpler."
Now cosmologists are turning to the open homogeneous cosmologies as the most likely model of our universe. Time starts at a big bang singularity and space is infinite from that moment onwards. The observable universe is a small finite part of the whole universe which lies inside the light cone traced back to the big bang. In the diagram below, the size of the observable universe appears bigger near the singularity but this is not an isometric diagram. In fact the universe is expanding as illustrated by the sequence of fixed length rulers which get smaller with time just as a scale gets smaller on a flat map of the world with increasing distance from the poles.
The net result is that the size of the observable universe shrinks to zero near the horizon even though the whole universe remains infinite.
This model of the universe poses some paradoxes. The singularity appears as a region of infinite extent yet it is everywhere uniform and flat. There is nothing mathematically inconsistent about such a universe and it does not come into contradiction with any known laws of physics, but is it a reasonable model of the universe? The uniformity suggests a difficult horizon problem: How is it co-ordinated over the infinite extent of the universe just an instant after the big bang. In a finite closed universe the horizon problem can be explained away by invoking inflationary theories, but no matter how rapidly the universe may have expanded in the first instants you cannot explain correlations over unlimited distances.
One possible way to explain this homogeneity would be Penrose's Weyl curvature hypothesis. This suggests that there is some physical law which applies to singularities and ensures that the Weyl part of the curvature tensor must be zero there. That would be sufficient to resolve the problem and it is quite possible that it could be a consequence of the unknown theory of quantum gravity which is significant at the singularity. However, the singularities which form in black holes cannot be subject to the same law since black holes are finite in size. The only known distinction between black hole singularities and the big bang is that the former always sits in the future light cone of all observers while the latter is in the past. A law which applies to one and not the other would have to break CPT invariance. Penrose has conjectured this possibility but the favourite theories of quantum gravity like superstring theory are all CPT invariant. What is the solution to this puzzle?
In truth there are several acceptable resolutions, but which is the most reasonable? How should Occam's razor be applied here? We could postulate two physically different types of singularity for the big bang and black holes to keep the simplest homogeneous model alive, or we can break CPT, or we can discard the homogeneous universe. In my opinion the last of these is the preferred but this is just my philosophical prejudice. I would like the universe to be symmetrical in time. It does not have to be so clearly symmetrical in shape as the Gold universe. It may have a random distribution of regions where time's arrow points in different directions and others where the absence of matter or thermal equilibrium makes the direction of flow indeterminate. All this must be happening far beyond our currently observable horizon. This description of reality fits best the storyteller's paradigm since it means the universe is more diverse. Of course the universe has no obligation to satisfy anyone's philosophical preferences but it is at least worth while exploring this possibility. A future unified theory may be able to tell us what the universe is like on very large scales, but it might equally well remain an unanswerable question.
The time reversal of Lemaître's models can also describe the formation of a black hole from a pressureless, spherically symmetrical, non-rotating cloud of dust. A particular case of this was studied by Oppenheimer and Snyder in 1939. A sphere of dust is uniform in density with empty space outside. The dust sphere collapses to form a black hole. The interesting thing about this solution of the equations of gravity is that the geometry inside the sphere is identical to the standard homogenous cosmology of Friedmann except that it runs in reverse. The lesson to be learnt from this is that the same model in reverse is a possible model of the big bang. It looks identical to the standard homogeneous big bang within a region which might cover the whole observable universe.
In other words, the big bang could be a white hole which is indistinguishable from the standard cosmological models for restricted observers such as us. Lemaître's solutions were more general than this. The density of the dust could vary gradually away from the centre, but so long as the variation was gradual this could describe the universe with our observable universe being one small region well within the event horizon.
The idea that the big bang may be a white hole is not popular with many serious cosmologists. One reason is that classically white holes cannot form. Since I have discounted causality I can accept the possibility of a white hole as easily as I can a black hole. Indeed, the white hole could also be a black hole in accordance with Hawking's complementarity. Once it was thought that the universe consisted of just our galaxy which had a centre and no stars outside a certain limit. Now I am suggesting that the big bang could be a similarly isolated object on a much larger scale. Just as our galaxy turned out to be one of many, so too may the big bang.
It is quite possible, as far as we can tell, that the big bang is actually just a huge white hole which formed in a larger universe. Perhaps on some huge scale there is a population of black and white holes of vastly different sizes. What does that say about the laws of thermodynamics? We can expect that inside a very large white hole time's arrow is flowing away from the singularity as we observe in our neighbourhood of the universe. The opposite can be expected in a very large black hole. The big bang is represented by a large object which is both a black hole and a white hole with time flowing outwards in both directions which we would call past and future. There might be many such objects in the universe. Within them there are smaller black holes which form from collapsing stars. These will eventually emerge from the large white hole and may subsequently fall into another large black hole. Then their arrow of time will reverse as they become white holes.
According to this model black holes always become white holes as the arrow of time reverses yet there are two distinct possibilities. For small black and white holes the arrow of time always flows in, while for large ones it always flows out. This is not inconsistent. The arrow of time must be most strongly influenced by the largest singularity in the past light cone. The full explanation will have to await a more unified theory of physics. The effects of quantum gravity near a singularity must determine the extent of its homogeneity and low entropy. Over all the universe is not governed by temporal causality. Time flows in both directions. For example, the near flatness of the universe near the big bang is due to influence from the future, not the past.
Occam's razor does not have a very good track record in cosmology. Usually space turns out to contain more complexity than we imagined before we looked. It will be billions of years before we are able to see beyond the current horizon defined by the speed of light. In the shorter term, theory is our only hope to know what the structure of space-time is like on very large scales.
Traditionally physicists have simply said that such universes must be ruled out because if we could travel back to our past we could change our history, which seems to raise contradictions. Recently some physicists have started to question this assumption. It seems possible that quantum mechanics may allow closed time like curves through space-time wormholes to be constructed, at least in principle. The contradictions which were thought to be a consequence of time travel do not stand up to close examination.
Perhaps it would be possible to travel back to the past and see our parents but some chance event would prevent us from being able to change their lives in ways which we know never happened. If that is a correct interpretation then it attacks our faith in our own free will.
There is perhaps little that we can conclude reliably about causality from our current understanding of physics. Only when we have found and understood the correct theory for quantum gravity will we be able to know the truth. We may be prevented from finding that theory if we hold fast to our prejudices.