The Time Illusion

The arrow that points but does not move

John Gribbin

Copyright © 2016 by John Gribbin

People like us, who believe in physics, know that the distinction between past, present and future is only a stubbornly persistent illusion.

— Albert Einstein

Once Upon a Time

The mystery of time can be summed up in a so-called “paradox” attributed to the Ancient Greek philosopher Zeno of Elea. Elea was a community on the south-western coast of Italy, and the leading Eleatic philosopher was Parmenides, born around 539 BC. He argued that multiplicity, motion and change are mere appearances, and that what we would now call the Universe is unchanging, only our impression changes. Zeno was one of Parmenides’ students, and offered several “paradoxes” to highlight these ideas. Unfortunately, Zeno’s own writings have not survived, and all we have are second-hand accounts of what he said, from people such as Plato and Aristotle (who were not fans of Eleatic philosophy). It has even been suggested that Zeno intended his examples as satires of the arguments used by opponents of Eleatic philosophy. We shall never know for sure what was in his mind, but nevertheless the arrow paradox sums up the mystery of time and motion.

Zeno starts from the straightforward statement that in order for motion to occur, an object must change its position. Then he discusses an arrow in flight. At any instant of time, the arrow occupies a space equal to itself, and is not moving (no less an intellect than Bertrand Russell called this “a plain statement of an elementary fact”). The arrow cannot move from the space it occupies to where it is not, because in any instant there is literally no time for it to move. At every instant the arrow is motionless. So motion is an illusion. Or, as Aristotle summed up Zeno for us, “if everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless”.

For more than two thousand years, the conventional wisdom has been that Eleatic philosophy is a poor description of reality and that Zeno’s “paradoxes” are no more than clever word-play. If you had the same sort of education as me, you encountered Zeno’s puzzles at an early age, and then ignored them. I hope to persuade you that this was a mistake. Time’s arrow certainly does point in one direction, from what we call the Past towards what we call the Future. But does it move? Before I answer that question, I need to sum up the conventional wisdom, starting with Isaac Newton’s image of a predictable “clockwork Universe”.

The world view that held sway from the late seventeenth century to the early twentieth century is known as “Newtonian” physics (sometimes as “classical” physics). Isaac Newton was far from being the only contributor to this package of ideas – for example, Galileo Galilei carried out key experiments investigating inertia, and Robert Hooke came up with the insight now often referred to as “Newton’s First Law”. But Newton put everything together in one package, with mathematical equations to describe precisely what was going on, and published it in an influential book, the Principia, in 1687. So that is often regarded as the date when classical physics took centre stage, and Newton’s name is indelibly associated with it.

Classical physics rests upon three laws of motion, summarised and explained in the Principia. The first (which Hooke pointed out to Newton) is that every object stays at rest or moves in a straight line at a constant speed unless a force acts upon it. As Hooke also pointed out, this explains the orbits of the planets around the Sun. They “want” to move in a straight line, but are tugged into curved orbits by the force of gravity, pulling them inwards (centripetally) towards the Sun. Several people realised, early in the second half of the seventeenth century, that the force needed to do the job must obey an inverse square law – that is, it is one-quarter as strong at twice the distance, one-ninth as strong at three times the distance, and so on. But it was Newton who proved that only such an inverse square law would do the job, so it became known as Newton’s law of gravity.

The second law of motion says that when a force does act on an object it causes an acceleration at a rate given by dividing the strength of the force by the mass of the object. So you need a bigger force to make a more massive object accelerate at a particular rate. But note that “acceleration” means a change in the speed of the object, or a change in its direction of motion (or both); the planets in orbit around the Sun are accelerating, both because their speeds are changing and because the direction they are moving in is constantly changing (Zeno might disagree!).

The third law says that if one object exerts a force on another object, the second object exerts an equal and opposite force on the first (action and reaction are equal and opposite). So the force of gravity from the Sun pulling on the Earth, for example, is exactly balanced by the force of gravity from the Earth pulling on the Sun, and if I am foolish enough to kick a brick, the force of the kick on the brick makes it move, but the force of the stone on my foot gives me a bruise.

Because these are absolute laws that obey precise mathematical equations, it seemed they made the Universe predictable. Before the “Newtonian” revolution in science, there was still scope to invoke a god (or gods) to explain what was going on in the Heavens. After Newton, it became clear that everything could be explained using the same laws that apply here on Earth, or across the Solar System – three laws of motion and a theory of gravity. Taking just the Sun and its family of planets, it looked as if, if you knew the exact position of every object at the same instant in time and all the forces (most notably gravity) acting on them, you could use Newton’s equations to calculate not only exactly how they would move forward into the future, but also, by running the calculations backwards, all of their movements in the past. Of course, it would be fantastically difficult for a human being to do the calculations, even with the aid of a computer, but that is not the point. The Universe “does the calculation” itself (it is its own computer, if you like) in obedience to the equations. It has no choice but to follow the rules. This is why Pierre Simon Laplace, sometimes referred to as “the French Newton”, replied to Napoleon’s enquiry about the role of God in science with the dismissive remark “I have no need of that hypothesis”.

This is what gave rise to the analogy of the Universe with a great clockwork machine, perhaps wound up by God and set running “in the beginning”, but now continuing inexorably down an undeviating and predetermined path. Not so different, when you come to think about it, from the Eleatic philosophers’ image of a fixed Universe.

But what was the stage on which all this activity took place? Newton’s idea was that it all took place against a background of “absolute space”, which was the same everywhere and always at rest, almost literally like a stage. In the Newtonian Universe, the Earth, Moon, Sun and other objects move through this absolute space, and their speeds relative to absolute space are measured in terms of absolute time, which can be imagined as the regular ticking of some ultimate cosmic clock.

It’s worth quoting Newton’s own words, from the Principia:

I.                     Absolute, true, and mathematical time, for itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure for duration by means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year.

II.                   Absolute space, in its own nature, without regard to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies. I say, a part of space; not a situation nor the external surface of a body.

So, according to Newton, time and space are independent aspects of reality, separate from each other and unaffected by the events going on within time and space. This image of the Universe dominated science until the work of Albert Einstein in the twentieth century.

Within that framework, time is divided into three regions – the past, present, and future. The past has already happened, and is fixed. The present is the boundary between past and future, where things happen. The future has yet to happen, and philosophers might argue that there is a range of possibilities in the future, although in Newton’s (and Laplace’s) world view it is predetermined. Time “passing” is a result of the present “moving” into the future, with the growing past being left behind. This, however, highlights puzzles which I shall return to, and which were already old in Newton’s day. St Augustine, whose life spanned the end of the fourth century and the beginning of the fifth century CE, was baffled by the distinction between the past, present and future. He wrote about the strangeness of the fact that neither the past nor the future seem to be real, and he worried about what is going on to produce a “flow” of time. Two hundred years after Newton’s work, however, in a development which surely would have intrigued Augustine, another insight into the nature of time came not from working with equations that describe absolute rules of motion applied to individual objects, but from statistical laws applied to very large numbers of objects.

The Pointing Arrow

The standard understanding of why we perceive a preferred direction of time – the arrow of time – stems from the industrial revolution of the nineteenth century, and in particular the development of steam engines. The development of a science of heat and motion (thermodynamics) grew out of the need to understand what was going on in steam engines, and to use that knowledge to improve the efficiency of the engines. But the advancing understanding of thermodynamics soon outstripped its practical applications, and gave insight into the workings of the Universe as a whole.

Much of the early investigation of thermodynamics was empirical, without a secure theoretical foundation. The theoretical understanding of what was going on was developed in the last quarter of the nineteenth century using statistical mechanics – an approach to thermodynamics that is based on applying the laws of statistics to the behaviour of large numbers of particles, such as the huge number of atoms or molecules present in a box of gas, each of them acting in accordance with Newton’s laws. This was cutting-edge research at the time; until the end of the nineteenth century, and even for a few years longer, there were scientists who did not accept the reality of atoms. It was the application of statistics to explain the behaviour of gases that provided some of the most compelling evidence for their existence, in the form of what was known as the kinetic theory of gases – “kinetic” because it describes the behaviour of particles in motion.

The kinetic theory describes in detail the way gases behave. We know that a gas can be compressed using a piston, for example, and we can measure the way the volume changes as pressure is applied. The kinetic theory makes it possible to predict (accurately) how much the gas will be compressed if a certain pressure is applied, in terms of the behaviour of myriad tiny, hard spheres (atoms and molecules) that bounce around inside a container, colliding with one another and with the walls of the container, while obeying “Newton’s” laws – continuing in a straight line until they hit something, bouncing off in accordance with the fact that action and reaction are equal and opposite, and so on. At any moment in time, any one molecule might be moving in any direction, but the combined effect of very many molecules bouncing off the walls each second produces a steady pressure and gives the illusion that the gas is a continuous fluid – a representative molecule of gas in the air at a temperature of 0oC and the pressure typical of sea level experiences just under four million collisions every second. At sea level, there are approximately 2.5 x 10²⁵ air molecules (that is, 25 followed by 24 zeroes) in each cubic metre of air, which is a lot of molecules, but they are each so small that there is still a lot of space between them. When the gas in a container is compressed, the average space between the molecules gets less, and the frequency of the collisions increases (in an exactly calculable way) and the pressure of the gas on the walls of the container increases (in an exactly calculable way). And when the gas is heated, the particles move faster (they have more kinetic energy), and the pressure again increases.

Even before the development of the full statistical mechanical treatment, physicists had discovered, from a combination of theory and experiment, the most profound feature of thermodynamics. In spite of its importance, this became known as the second law of thermodynamics, because the first law is merely a kind of throat-clearing statement which says that in a closed system (one which has no connections with the outside world) the total energy stays the same. The second law can be expressed in different ways, but the most simple is the statement that heat cannot move from a cooler object to a hotter object without some outside influence. This is so obvious from our everyday experience that it hardly seems worth dignifying with the status of a law, but closer study shows that it really is one of the pre-eminent laws of nature. Putting it another way, things wear out, and the second law is telling us that the Universe itself will one day wear out.

Two examples show the second law at work. In the first, imagine an ice cube in a glass of water. You know what will happen – the ice will melt. You will be left with a glass of water all at the same temperature. You never see an ice cube forming spontaneously in a glass of water left alone on the table, while the water around the ice warms up. Now imagine knocking that glass of water off the table on to a stone floor. It will shatter. You never see a mess of shattered glass in a puddle of water on the floor rearrange itself into a glass of water and jump back on to the table. The second law of thermodynamics defines an (or the) arrow of time. If you made a video recording of either of the events I have just described and ran it backwards, you would know something was wrong. We all know the “right” way for the video to run.

Both these examples also highlight another feature of the second law. The natural effect of processes going on in the Universe is to move from a state of order to a state of disorder, unless there is an input of energy from outside (for example, the energy used in the factories where drinking glasses are made). Disorder is measured in terms of a quantity called entropy. The more disordered a system is, the higher its entropy. So another way of expressing the second law is to say that entropy always increases, or at best stays the same. In other words, the future towards which the arrow of time points is the direction in which entropy is greater. There is more order in a more structured system, such as [water plus ice], than there is in a more uniform system, such as [water], because it takes more information to describe an ordered system. And pretty obviously there is more order in a glass of ice water on the table than there is in the mess of broken glass in a puddle of water on the floor. In another example, there is more order in a chess board painted with black and white squares than there is in an otherwise identical board painted a uniform grey. The second law tends to smooth things out and make them more uniform.

Places where entropy seems to decrease locally always involve a greater increase in entropy elsewhere. An ordinary domestic refrigerator, for example, gets cold inside because energy is being used to, in effect, pump heat out of the fridge and release it into the air through the pipes at the back. This does indeed move heat from a colder to a hotter place; but the fridge is not an isolated system. The input of energy required involves an increase in entropy where the energy is being released and distributed. Similarly, life on Earth seems to violate the second law by growing and becoming more ordered; but it is feeding off an input of energy from the Sun, where entropy is increasing. The loss of entropy by the Earth as a whole is vastly offset by the gain in entropy of the Sun. But even the Sun will not last forever. What happens when the Sun and stars die?

There is no “outside” for the Universe itself to feed off. For the Universe as a whole, nineteenth-century physicists realised, entropy is always increasing. At present, there is a dramatic contrast (implying lots of order) between the hot stars and the cold of space. But all the stars will eventually die and cool, having poured energy out into space. This led to the idea of the “heat death” of the Universe, a far future time when the temperature everywhere is the same, there is complete disorder, and thermodynamic processes, which depend on heat moving from a hotter place to a colder place, will no longer occur. Which raises another question. If entropy is always increasing, how did the Universe get to be in the relatively low entropy state we see today? Indeed how did it get started in an even lower entropy state in the first place? What was the “heat birth” of the Universe? You don’t need to know what “the first place” was, nor how things got started, to realise that this is a major thermodynamic mystery.

This mystery was investigated by the Austrian Ludwig Boltzmann, who came up with a mind-blowing suggestion. It jumps off from another look at the classic thermodynamic example of a box full of gas, which was highlighted by the French physicist Henri Poincaré in 1890. The arrow of time becomes visible when we have a system which is not in equilibrium, like the glass of [water plus ice], and watch as it passes into an equilibrium state. The classic example is a box which is divided into two by a sliding partition. One half of the box is full of gas, the other side of the partition is empty. When the partition is slid back, the gas spreads out to fill the whole box, increasing entropy. If you now sit and watch the box, waiting for the gas to go back into one side so that you can slide the partition back across and trap it, you will have a very long wait. Common sense says that you can wait for-ever and it will never happen. But “for-ever” is a very long time. Poincaré said that it is not impossible for this to happen – for time to run backwards, in effect, while entropy decreases – it is just very unlikely. He showed that for a so-called “ideal” gas, where all the collisions take place without any loss of energy, in a closed, isolated box the particles (atoms or molecules) must eventually pass through every possible pattern that is consistent with the law of conservation of energy (the first law of thermodynamics). It is just that some patterns, the ones with high entropy, are extremely likely, and some, the ones with low entropy, are extremely unlikely. Wait long enough, and the gas in the box will eventually return to exactly the state it was in at the start of the experiment. The second law itself seems to be only a statistical law, not an absolute truth, although this is not so much time running backwards as time repeating itself.

But how long is “long enough”? The time it takes for a system such as a box of atoms (or the Universe!) to return to its starting point is known as the Poincaré cycle time. Strictly speaking it is an average – the box of atoms could return to its starting point sooner or later than this, but the cycle time is an indication of the typical time you would have to wait to see this intriguing phenomenon occur. In round terms, the Poincaré cycle time is 10N seconds, where N is the number of particles (atoms or whatever) involved. A box containing just two atoms will repeat its starting position every 100 seconds; with 10 atoms the cycle time is 10¹⁰ seconds, which is rather more than 300 years. A small box of gas might actually hold about 10²³ atoms; so you don’t need to be a whizz at maths to see that Poincaré cycle times for realistic systems are ridiculously long. But, and this is the point, they are not infinite. If it was possible to watch a glass of water for long enough you really would see ice forming in it as the water around the ice warmed up. Jumping ahead of my story slightly, that time would be much longer than what we now know to be the age of the Universe, some 13.8 billion years. The fact that we perceive a thermodynamic arrow of time is linked to the fact that the Universe is not old enough for these effects to have shown up. And the fact that the Universe is expanding away from a dense fireball (the Big Bang) gives us another indication of the direction of the arrow of time. The future is in the direction of time where the Universe is bigger, in the sense that galaxies are further apart from one another.

Boltzmann did not know anything about the Big Bang, of course, and the conventional wisdom at the time he was working was that the Universe is infinitely old. But as I have pointed out, the Poincaré cycle times of even complex systems are not infinitely long. In 1895, in a scientific paper with the rather prosaic title “On Certain Questions of the Theory of Gases”, Boltzmann took up the idea that the second law of thermodynamics is only a statement of probability, not really a “law” at all. After a rather technical discussion of what this means in terms of the mathematics of probability, he wound up by highlighting an idea which had actually been proposed by one of his colleagues:

I will conclude this paper with an idea of my old assistant, Dr Schuetz. We assume that the whole universe is, and rests for ever, in thermal equilibrium. The probability that one (only one) part of the universe is in a certain state, is the smaller the further this state is from thermal equilibrium; but this probability is greater, the greater is the universe itself. If we assume the universe great enough, we can make the probability of one relatively small part being in any given state (however far from the state of thermal equilibrium), as great as we please. We can also make the probability great that, though the whole universe is in thermal equilibrium, our world is in its present state. It may be said that the world is so far from thermal equilibrium that we cannot imagine the improbability of such a state. But can we imagine, on the other side, how small a part of the whole universe this world is? Assuming the universe great enough, the probability that such a small part of it as our world should be in its present state, is no longer small.

If this assumption were correct, our world would return more and more to thermal equilibrium; but because the whole universe is so great, it might be probable that at some future time some other world might deviate as far from thermal equilibrium as our world does at present.

In his book Lectures on Gas Theory, Boltzmann expressed this even more clearly:

We have the choice of two kinds of picture. Either we assume that the whole universe is at the present moment in a very improbable state. Or else we assume that the aeons during which this improbable state lasts, and the distance from here to Sirius [one of the nearest stars], are minute if compared with the age and size of the whole universe. In such a universe, which is in thermal equilibrium as a whole and therefore dead, relatively small regions of the size of our galaxy will be found here and there; regions (which we may call “worlds”) which deviate significantly from thermal equilibrium for relatively short stretches of those “aeons” of time. Among these worlds the probabilities of their state (i.e. the entropy) will increase as often as they decrease. In the universe as a whole the two directions of time are indistinguishable, just as in space there is no up or down. . . . It seems to me that this way of looking at things is the only one which allows us to understand the validity of the second law, and the heat death of each individual world, without invoking a unidirectional change of the entire universe from a definite initial state to a final state.

Boltzmann’s terminology is slightly different from mine. He says “world” where I say “Universe”, and I need another word for his “universe”, which I shall call the Metaverse. On this picture, our Universe is a bubble within the Metaverse. In Boltzmann’s scenario, in an infinite Metaverse, anything – that is, anything allowed by the laws of physics – is possible. And as physicists have been known to quip, anything which is not forbidden is compulsory. In a Metaverse that is infinite in both time and space, and which is on average in thermodynamic equilibrium, there must be bubbles (Universes) that temporarily deviate from thermodynamic equilibrium, even if the bubbles are billions of light years across (and “temporarily” here means for billions of years). Boltzmann wrote, still using “world” to mean what I have been calling our Universe and “universe” for my Metaverse:

This viewpoint seems to me to be the only way in which one can understand the validity of the Second Law and the heat death of each individual world without invoking a unidirectional change of the entire universe from a definite initial state to a final state. The objection that it is uneconomical and hence senseless to imagine such a large part of the universe as being dead in order to explain why a small part is living – this objection I consider invalid. I remember only too well a person who absolutely refused to believe that the sun could be 20 million miles from Earth, on the grounds that it is inconceivable that there could be so much space filled only with aether and so little with life.

The idea that the whole visible Universe might be no more than a kind of temporary bubble in an infinite sea of thermal equilibrium and maximum entropy became known as the “Boltzmann fluctuation” hypothesis, which seems rather tough on Dr Schuetz. Although it might seem that the Big Bang idea reduces the force of Boltzmann’s argument, an important puzzle today is where the Big Bang itself came from, and the Schuetz/Boltzmann hypothesis finds an echo in some versions of the modern idea of the Multiverse, which I discuss in my book In Search of the Multiverse, but do not have room for here.

Whatever the broader implications, though, the bottom line is that in our part of the Universe (or Multiverse) there is a definite direction in which the arrow of time points, indicated by the direction of thermodynamic processes. At the root of this statistical interpretation of the direction of the arrow of time, however, there is another mystery. The statistics apply to large numbers of particles, such as atoms. But to an individual atom bouncing around in a box of gas and obeying the three laws of mechanics, there is no arrow of time.

A Reversible Arrow?

The usual way to picture this is in terms of the break-off in a game of snooker or pool, when the neat triangular arrangement of the balls in the pack is broken up. If a video camera is placed above the pack, so that neither the players nor their cues are visible to it, and a player with more enthusiasm than skill breaks off, we can record the break, and play it back to watch a fast-moving white ball appear from off screen and smash into the pack to disturb the neat triangular pattern into a jumble of moving balls. Reversing the video shows the unlikely (but not, according to Boltzmann, impossible) scenario of a lot of red balls flying together and coming to a halt in a neat triangular pattern while transferring their momentum (in line with the laws of motion) to a single white ball that heads out of view. So far, this seems to be just another example of the second law of thermodynamics at work, like the examples described already. But now picture what happens when there is a single red ball in view and it is struck by a single white ball that stops, transferring its momentum to the red ball, which zooms off. If you run that video backwards, you get the equally plausible scenario of a red ball zooming in to hit a white ball and stop, transferring its momentum to the white ball, which heads off. An even simpler example is provided by the executive toy known as “Newton’s cradle” (which, incidentally, was invented by Robert Hooke to demonstrate what we now call conservation of momentum). A short video of the balls swinging to and fro makes just as much sense backwards or forwards (“short” for reasons I explain soon). These examples both demonstrate that the laws of physics, notably Newton’s laws, are time-reversible. They work just as well backwards in time as forwards, and there is no place in them for the second law of thermodynamics. The fundamental laws of physics do not distinguish between past and future.

Because of this, you can turn the probabilistic interpretation of thermodynamics on its head. If it is OK to say that dropping an egg breaks it – because the messy, higher-entropy, state of a broken egg is more probable than the tidy, lower-entropy, state of an intact egg – the present moment seems odd whichever way you look at it. The egg “should” (that is, was much more likely to) have been broken in the past, as well as in the future, according to probability. In general, eggs “ought” to be broken most of the time, just occasionally combining their parts (like the balls assembling themselves into a neat triangle) to make an unbroken egg for an instant before shattering again. That would be entirely consistent with the laws describing things such as the interactions between individual pool or snooker balls.

Taking this to the extreme, this approach provides another explanation for the apparent existence of a bubble of order as large as the Universe in a sea of thermodynamic equilibrium. Maybe it is just an illusion. But is it my illusion, or yours? This has led to a puzzle known as the Boltzmann’s Brain paradox, although Boltzmann did not invent it and it is not a paradox. It starts from the reasonable point that deviations from thermodynamic equilibrium are more likely for smaller deviations, and less likely for larger deviations. The smallest deviation that is consistent with everything I know (or think I know) is that an isolated human brain has appeared out of the thermodynamic sea, complete with all those sensory impressions including the impression of writing this book, (and spilling a cup of coffee after that misplaced comma) and is about to vanish.

Well, I am still here and still coffee-stained (I think).

Even so, the probability of a Universe like the one we see around us is much less likely than the probability that an individual human brain (mine, of course) has just popped into existence in the thermodynamic sea, complete with memories of the past, including learning about the research that led to an understanding of the Big Bang and the expanding Universe. There is no need even for a whole human body, since all a brain “knows” is the sense impressions it receives; everything “out there” could all be, if you like, a kind of virtual reality. Maybe it is your brain that has just popped into existence, with the impression of a lifetime of being you and the sense that you are reading this book. I am reminded of Alice’s reflection in Lewis Carroll’s Through the Looking Glass: “He was part of my dream of course – but then I was part of his dream too!”

Nobody takes the Boltzmann’s Brain paradox seriously. It is a reductio ad absurdum to demonstrate that there must be a flaw in the argument. The most likely flaw is that on the smallest scale, down among the atoms themselves, the laws of physics are not entirely time reversible, and that an arrow of time is built in to those laws.

In my discussion of the collisions between pool or snooker balls, I didn’t bother to include the phrase “assuming we can ignore friction”, because physicists always assume we can ignore friction. If I had been justifying this, I might have said “this is OK, because friction is a result of interactions between the atoms of the balls, the atoms of the table, and the atoms of the air; these interactions are themselves reversible in time.” The effect of these interactions is that the balls eventually roll to a halt, while their energy of motion is spread out among the atoms of the air and the table as heat, the increased kinetic energy of the faster-moving atoms (essentially like the way the energy of the incoming cue ball is shared out among the balls in the pack); similarly, the Newton’s cradle slows down and stops as its surroundings warm up slightly. Taken to extremes, the probability argument would say that even if the balls are spread around on the table in a scattered pattern, there is a possibility that the molecules in the air and in the table might conspire to jostle the balls in such a way that they all moved together to form a neat triangle, while the temperature of the table and the air decreased as a result – the equivalent of ice forming in a glass of water while the water cools down. And the balls of a Newton’s cradle sitting quietly on your desk might start swinging as the air around them cooled down. But suppose the laws of physics are not reversible in this way at a fundamental, atomic or subatomic, level.

This is a daring supposition, because experiments carried out with fundamental particles show the same kind of reversibility as experiments carried out with colliding balls. Processes like the radioactive decay of an atomic nucleus, for example, are also governed by rules which work just as well backwards in time as forwards in time. If you “run these processes backwards”, either mentally or in some cases by performing the actual experiment each way, you do have to make some adjustments, such as changing particles into antiparticles and vice-versa. A positron, the positively charged counterpart to an electron, is absolutely identical to an electron which is travelling backwards in time, an idea which intrigued no less a physicist than Richard Feynman. It was his investigation of time-symmetric electrodynamics, which he worked on as a Ph.D. student, that led him to the theory of quantum electrodynamics, for which he received the Nobel Prize. Thanks to this kind of property, the standard model of particle physics tells us that any process and its time-reversed equivalent are equally likely. So, again why does time seem to flow only in one direction?

At the level of atoms and below, there is a deeper truth than Newton’s laws. This deep truth is, of course, quantum physics. I shall have more to say about quantum physics later, but for now there are two important points to take on board. The first is that quantum physics does not undermine classical physics as it applies in the everyday world, on the scale of things like human beings, planets and stars. Newton’s laws apply perfectly well to things like calculating the trajectory required to send a spaceship to Mars, or the orbit of Mars itself around the Sun. It is only on the very small scale that things are different, and you need different laws to calculate things like the trajectory followed by an electron going from A to B, or the “orbit” of an electron in an atom. Even the notion of an orbit is not strictly accurate, but it will do for now.

The second important point is that although the equations used to describe things like the trajectories of electrons are not the same as the equations used to describe things like colliding pool balls, they are also time-reversible, or time-symmetric. They make no distinction between past and future as directions in which processes can occur. The classic example is the Schrödinger equation, named after the Austrian physicist Erwin Schrödinger, who discovered it. The Schrödinger equation can be used to describe, or predict, the trajectory of an electron from A to B. The same equation also describes the trajectory of an electron from B to A. If you could flip a switch and reverse the direction of the arrow of time, all quantum processes would happily run the other way, according to the standard version of quantum theory. It seems there is no escape from the mystery of the direction of the arrow of time. But very recently some researchers have come up with an intriguing idea which, as I shall explain shortly, ties in with some older insights into the nature of the quantum world. These ideas are radical, but clearly something radical is needed to explain why time is only seen flowing in one direction. And they are certainly less radical than the Boltzmann’s Brain idea!

Necessarily, these ideas are rather mathematical, but they can be understood in terms of one of the basic features of quantum theory, if you are willing to take the maths on trust (if not, see https://arxiv.org/abs/1307.6167 and download the PDF). The key feature of quantum theory is that things are quantised – and this applies to processes, as well as to things like the amount of energy possessed by an electron in an atom. Instead of a smooth flow of time from the past to the future, there is a discrete series of events, some of which lie in the future and some of which lie in the past. This sounds like something Zeno would have understood. At some moment in time the arrow is here, at some later moment in time it is there. How it gets from here to there is still to be resolved, but the first step towards understanding the flow of time is to find a distinction between past and future events, and that is what Marina Cortês, of the University of Edinburgh, and Lee Smolin, of Canada’s Perimeter Institute, may have done.

They have developed what they call a causal set model, in which the Universe is composed of a series of events, each of which is different from all the other events in the Universe. Processes continually manufacture new events from the set of present events, but Cortês says that events “cannot unhappen”; reversing an event does not take you back to where you started, in effect erasing the process, but gives rise to a new event. Time continues to point, and flow, in the same direction. Each set of events can only influence events in the next set, and however you juggle the equations the “next” set is always in the future.

This is as far as I can take the story of quantum time for the moment, although I shall return to it later, because as yet we have not resolved the puzzle of the flow of time. It is one thing to say that a particular set of events is in the past and another set of events is in the future, but, Zeno would ask, how do we move from the past through the present and into the future? If you had asked Albert Einstein that question, he would have told you that we don’t.

A Block of Relativity

Einstein gave us a new understanding of space and time, replacing Newton’s absolute space and absolute time. But it took someone else to point out the full significance of what he had done.

The motivation for what became Einstein’s special theory of relativity came from what he (but nobody else at the time) realised was a flaw in the classical (Newtonian) theory of dynamics, which was in conflict with a key component of the much newer theory of electromagnetism. That theory had been developed in the 1860s by the Scot James Clerk Maxwell, building on the earlier work of Michael Faraday. It provided a set of equations (now known as Maxwell’s equations) which contained everything you needed to know about the behaviour of electromagnetism, in the same way that Newton’s equations contained everything you needed to know about mechanics, and it was the greatest achievement of theoretical physics since the time of Newton. Among other things, Maxwell’s equations describe the propagation of electromagnetic waves across space, including a constant which corresponds to the speed with which those waves move. The value of that constant depends on measurable electrical and magnetic properties of matter, which Maxwell plugged in to the equations. The result astonished him; the speed was the same as the already known speed of light. In a paper he published in 1862 his excitement at the discovery shone through. “We can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena” (his italics). Two years later, he put it more calmly:

This velocity is so nearly that of light that it seems we have strong reason to conclude that light itself (including radiant heat and other radiations, if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.

But there was something odd about this. The number which matched the speed of light was indeed ‘a’ constant, which became represented by the letter c. The equations seemed to be saying that the speed of light is always the same, wherever you measure it from. And this conflicts with classical mechanics. If you were moving towards a light source (a lamp, or a star, or any source of light) at a speed of 50 km per hour you would expect to measure the speed of the light coming towards you as c + 50 km/hr. If the light was coming from behind you, overtaking you as you moved away from the lamp, you would expect to measure its speed as c - 50 km/hr. But Maxwell’s equations make no allowance for this. They just say that if you measure the speed of light the answer you get will be c, regardless of where the light is coming from, or how you are moving relative to the light source.

The received wisdom in the late nineteenth century was that there must be some invisible medium, the “aether” filling “empty” space, and that the speed implied by Maxwell’s equations is the speed of light through that aether. Many experiments were carried out to try to measure the motion of the Earth through the aether by detecting variations in the measured speed of light, but without success.

On the theoretical side, there were two possibilities. Either Maxwell was wrong in some subtle way, and the speed of light was not an absolute constant; or Newton was wrong in some subtle way, and adding up two velocities v and u did not give the simple answer (v + u). In particular, it raised the ludicrous possibility that v + c = c.

Maxwell was, to most physicists, the obvious fall guy. By the beginning of the twentieth century, Newton’s god-like status was such that only a genius or a fool would say that he was wrong and Maxwell was right. But that is exactly what Einstein did, constructing a whole new description of the physical world and the dynamics of moving objects which started out from the basic postulate that the speed of light is an absolute constant and that all observers, however they are moving through space, will always measure the same value for the speed of light, no matter how the source of the light is moving. This was the special theory of relativity, published in 1905.

Part of the beauty of the special theory is that for speeds that are much less than the speed of light, everything in Einstein’s world is the same as in Newton’s world. His equations “reduce” to Newton’s equations, a physicist would say, in the low-speed limit. Velocities add up in the familiar way, and so on. It is only when we are dealing with things that are moving at a sizeable fraction of the speed of light (which is just under 300,000 km per second) that things get interesting. But then they get very interesting indeed.

Einstein’s equations tell us that any “observer” can be regarded as being at rest, with things moving relative to them. From the point of view of the observer, things moving past are shrunk in the direction in which they are moving (their length contracts), and any clocks carried by those moving things run slow (time dilates). But an observer being carried along on one of those moving things, maybe in a spaceship, is also entitled to say that they are at rest, and that it is the first observer who is moving, has shrunk, and whose clocks are running slow.

All of this is real, has been tested and is built in to such mundane things as Satnav systems, which use data from satellites which are moving fast enough that the effects of the special theory have to be taken into account. But Einstein himself did not at first appreciate the most profound feature of the special theory, which had to be pointed out by one of his former teachers, Hermann Minkowski, who, deliciously, at one time referred to the undergraduate Einstein as a “lazy dog” who “never bothered about mathematics at all”. With hindsight, the insight Minkowski gave into the special theory can be understood in terms of one of the basic things we learned about in school, Pythagoras’ theorem.

The famous theorem says the length of the hypotenuse (h) of a right-angled triangle is related to the length of its other two sides (x and y) by the simple equation h² = x² + y². This, of course, applies to an object in two dimensions, a flat triangle. It is a way to measure the shortest distance between two points, because the hypotenuse links the ends of the other two sides of the triangle, two points which can be represented by coordinates, like a map reference, even if we do not draw the other sides in. There is an equivalent equation for three dimensions, which gives the shortest distance between two points in terms of three coordinates, and can be written as h² = x² + y2 + z². An important point is that this works wherever we make the zero for our coordinate system, wherever we measure from, because all we are calculating is the relative positions of points in space. Minkowski realised that Einstein’s equations were telling us that we can do very much the same in four dimensions, specifying some kind of four-dimensional length a by the equation a² = x² + y2 + z² - (ct)², where t denotes a time interval and c is the speed of light, so the two of them combined together make a distance. Time, or rather ct, appears as a fourth dimension (not like a fourth dimension; it is a fourth dimension), with two special features. The first is that minus sign, which makes time a kind of negative dimension. When space expands, the time shrinks, and vice-versa. The second is that c. It means that one second of time is equivalent to 300,000 kilometres of space, and this is why relativistic effects only become important if things are moving at a large fraction of the speed of light. The equation tells us that the four-dimensional length of an object, usually known as extension, stays the same however the object moves. If the length contracts, the time stretches, by an equivalent amount, and vice-versa.

Minkowski had found a way to simplify the special theory by recasting it into geometry, and presented his dramatic discovery at a meeting in Cologne in 1908, where he said:

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade into mere shadows, and only a kind of union of the two will preserve an independent reality.

This was the death-knell of Newton’s absolute space and absolute time, and the birth of the modern understanding of a combined entity, “spacetime”. Although Einstein was initially reluctant to accept this geometrization of his theory, it became a cornerstone of his masterwork, the general theory of relativity, which relates gravity and matter through the curvature of spacetime. But that is another story. My focus here is on the fact that “space by itself, and time by itself, are doomed to fade into mere shadows, and only a kind of union of the two will preserve an independent reality”, which applies in both the special and the general theories.

To see the full significance of this, go back to the idea of describing positions in space in terms of coordinates. This was one of those flashes of insight that great minds are prone to. In this case, the great mind was that of René Descartes, and the insight came to him, he later recounted, while lying snug in his bed on 10 November 1619, watching a fly buzzing around his room. He realised that at any instant the position of the fly could be uniquely described by its distance from each of the three surfaces (two walls and a ceiling) that met in the corner of the room. He immediately saw this in three-dimensional terms, although we are more used to the idea in two dimensions, like triangles drawn on a piece of paper, or grid coordinates on a map. The surprising thing to us, who have grown up with the idea, is that someone had to come up with the idea at all. But Descartes did, and so anyone using such methods of recording the position of an object, or a point on a map, is said to be using Cartesian coordinates.

But Minkowski’s geometrization of the special theory goes one better (exactly one better) than Descartes. Descartes realised that he could specify the position of the fly at any instant using three coordinates. But he did not appreciate (how could he, in 1619?) the significance of the fact that by specifying the time he was adding a fourth coordinate.

We use four-dimensional Cartesian coordinates in everyday life, without pondering the deeper implications. If I say I will meet you on the corner of Second Street and Fifth Avenue, I am specifying the location in two-dimensional Cartesian coordinates. If I say I will meet you in the shop on the corner of Second and Fourth, by the elevator on the third floor, I am specifying the location in three dimensions. And if I say I will meet you outside that elevator at two o’clock, I am specifying a location in four-dimensional spacetime, using three spatial coordinates and one time coordinate.

To get back to the fly buzzing around in Descartes’ room, if we imagine time passing like an ever-rolling stream, the path of the fly will trace out a wiggly line around the room as time passes. In four dimensions, this is called the “world line” of the fly, and it can in principle be extended back in time to the birth of the fly, and forwards in time to the death of the fly. Everybody, and everything (including the Universe), has a world line through four-dimensional spacetime, a world line which traces their entire history. Zeno’s image of the arrow that is stationary at any instant is like a series of snapshots of the world line of the arrow. But is the arrow moving? Do flies, and people, and universes, move along their world lines, or are the world lines fixed in four-dimensional spacetime with only our perception of “now” moving?

This is the heart of what is known as the “block universe” model of reality. The name comes from the idea of spacetime laid out like a “block” – a block of ice cream, or a slab of concrete are the images that come to my mind. The time dimension is represented by one direction along the side of the block, and space is represented by slices across the block, at an angle to the time line. World lines are fixed in the block, and unchanging, but observers that are moving in different ways (that is, following different world lines) see things differently, because the effect of their motion is to change the angle at which they view the slices through spacetime. The exact angle of the slice depends on the exact speed, and all of this can be described perfectly in accordance with Einstein’s equations. Indeed, it is a requirement of Einstein’s equations. It is not “merely philosophical” as some critics with a jaundiced view of philosophy have claimed.

This is the logical development of the idea that given enough information at a specific time, everything that occurs at a later time, and everything that has occurred at an earlier time, can be determined. There is nothing special about the present moment, the Now, or indeed about any other moment. Past, present and future are all on an equal footing, because there is no slice through spacetime which can be uniquely identified as “the present”. The Universe does not change, but it exists, as a fixed block of spacetime that contains all the things that have ever happened, and all the things that ever will happen. The flow of time is an illusion. Philosophers call this idea (or something so similar to this idea that as a physicist I cannot spot the differences) “eternalism”. The two key assumptions of eternalism are the same as the basic features of the block Universe – time is a real dimension, and nothing changes.

The block Universe model has been discussed by, among others, David Deutsch, of the University of Oxford, and another Oxford physicist, Julian Barbour. But if you are looking for a place to start finding out more about time, with a good summary of the block Universe idea, I recommend the book Time’s Arrow and Archimedes’ Point, by Huw Price, who is a philosopher, but no “mere philosopher”, at the University of Sydney. An even more entertaining read, entirely based on the block Universe/eternalism idea, is Kurt Vonnegut’s novel Slaughterhouse Five, which I recommend even more highly (I’m sure Price will forgive me). The fundamental feature of the block Universe is that nothing changes. The world lines are fixed.

In his book The Fabric of Reality, Deutsch says:

We think of causes as preceding their effects; we imagine the moving present arriving at causes before it arrives at their effects, and we imagine the effects flowing forwards with the present moment. Philosophically, the most important cause-and-effect processes are our conscious decisions and the consequent actions. The common-sense view is that we have free will; that we are sometimes in a position to affect future events . . .

But:

according to spacetime physics, the openness of the future is an illusion, and therefore causation and free will can be no more than illusions as well . . . In reality, we make no choices. Even as we think we are considering a choice, its outcome is already there, in the appropriate slice of spacetime, unchangeable like everything else in spacetime, and impervious to our deliberations.

This worried Einstein. Barbour, in his book The End of Time, reports a conversation Einstein had with the philosopher Rodolf Carnap, who tells us:

Einstein said that the problem of the Now worried him seriously. He explained that the experience of the Now means something special for man, something essentially different from the past and the future, but that this important difference does not and cannot occur within physics. That this experience cannot be grasped by science seemed to him a matter of painful but inevitable resignation.

It is with that, and the block Universe, in mind that we can understand why, when his old friend Michele Besso died in 1955, Einstein wrote in a letter of condolences to Besso’s family:

Now he has departed from this strange world a little ahead of me. That means nothing. People like us, who believe in physics, know that the distinction between past, present and future is only a stubbornly persistent illusion.

Einstein died on 18 April 1955, a few weeks after he wrote those words:“But our language, which has built in to it our subjective feeling of time passing, struggles to deal with these concepts.”

I said earlier that the block Universe “contains all the things that have ever happened, and all the things that ever will happen”; but that is itself implying a flow of time, and there is no flow of time in the block Universe! Time is as much a part of the block that makes up the Universe as space is, and it cannot be “an entity in time”, as Price puts it:

Defenders of the block universe view deny that there is an objective present, and usually also deny that there is any objective flow of time. Indeed, perhaps the strongest reason for denying the objectivity of the present is that it is so difficult to make sense of the notion of an objective flow or passage of time. Why? Well, the stock objection is that if it made sense to say that time flows then it would make sense to ask how fast it flows, which doesn’t seem to be a sensible question.

But there is an even more compelling reason to reject the idea of time flowing in the block Universe, which Price also spells out:

If time flowed, then – as with any flow – it would only make sense to assign that flow a direction . . . we don’t have an objective sense in which time is flowing one way rather than the other [in the block Universe]. In other words, not only does it not seem to make sense to speak of an objective rate of flow of time; it also doesn’t make sense to speak of an objective direction of flow of time.

You might think that this implies that time’s arrow can point in either direction, “backwards” or “forwards” relative to our everyday “common sense” understanding of things. But what it really means is that the arrow does not point in any direction at all. Time just is.

Which brings me back to quantum physics, which is missing from Einstein’s description of time. I should warn you, though, that this will not give us back the common sense notion of time flowing like an ever-rolling stream. And since the conclusions will be so counter-intuitive, I had better fill in some background to the quantum story.

Quantum Realities

At the end of the nineteenth century it was firmly established that light is an electromagnetic wave described by Maxwell’s equations. About the same time, the electron, a particle smaller than an atom, was identified. Then Max Planck showed that light is only emitted or absorbed by atoms in discrete chunks. It seemed at first that this was an oddity about the behaviour of atoms, nothing to do with light itself. But in 1905 Einstein showed that the way electrons are ejected by atoms when light shines on a metal surface could only be explained if the light arrived at the surface as little particles – particles of light which became known as photons. This was puzzling enough, but in the 1920s, following a prediction by the theorist Louis de Broglie, experimenters discovered that electrons, which they had thought were particles, also behaved as waves. In the sub-atomic world, it became clear, entities are not simply waves or particles, but exhibit “wave-particle duality”.

These discoveries were applied to try to develop an understanding of what is going on in sub-atomic processes, such as the emission of light by an atom or the way an electron gets from A to B. In the late 1920s, Werner Heisenberg and his colleagues developed a mathematical description of what is going on, in terms of equations incorporating the idea of “jumping” between energy levels. The equations said that the energy of an atom (or anything else) can only have certain values, related to the states of the electrons in the atom. An electron can disappear from one energy level and instantaneously reappear in a level with lower energy, provided that a quantum of light (a photon) carrying energy equal to the energy difference between the two levels is released. The same thing happens in reverse when light is absorbed. (The “quantum leap” of popular mythology is actually a very small change made entirely at random.) At about the same time as Heisenberg came up with this idea, Erwin Schrödinger found an equation which he thought could describe the electron as a wave, and do away with this notion of quantum jumping. He thought of the energy levels as like different harmonic notes played on a single guitar or violin string. But it turned out that the best way to interpret his wave equation is that that it describes a “wave of probability” (whatever that is) which does not describe an electron, but rather the probability of finding an electron in one place or another, or in one energy level or another. Schrödinger himself was disgusted, and said that if he was stuck with quantum jumping he wished he’d never got involved in the subject. What he abhorred is the idea that an electron (or other quantum entity) disappears from energy level A and appears at energy level B without passing through any intermediate stages, and without taking any time at all to make the transition. But that is what the equations tell us. And those equations work – they are the basis of the design of such practical things as lasers and computer chips, and in explaining the workings of DNA and heredity.

As the dust settled after the 1920s, and physics had a working description of what is going on at the quantum level (a description, not an understanding; as Richard Feynman was fond of pointing out, nobody understands quantum physics), the now-standard image emerged. Sticking with the example I have already used, this says that an atom with an excess of energy (in an “excited state”) can release a photon and fall down into a lower-energy state. The image is of a process which takes place in time, with the photon being released like an arrow being released by a bow. But although that image is popular, it is not what the equations are telling us. The equations say that you can have a mathematical description of an excited atom, and you can have a mathematical description of a system with exactly the same energy, shared between an unexcited atom and a photon. The equations also tell us what probability there is of finding the system in either state. But there is no mathematical description of a process which changes the excited atom into an unexcited atom plus a photon. We perceive one state as coming before the other in time, but there is nothing in the laws of physics which distinguishes a direction of time. The equations equally happily describe an unexcited atom and a photon as what we would call the initial state, with an excited atom as the final state. Either way, they say nothing about any process of transition from one state to another. And in spite of some very careful experiments having been carried out to try to catch such quantum processes in action, nobody has ever seen an atom (or any other quantum system) in the act of changing from one state to another.

The first person to appreciate the significance of this was Alan Turing, best remembered today as the founding father of computing. In 1954, he pointed out that a quantum system that is constantly “watched” (that is, monitored in some way) will never change. This is the ultimate example of the adage “a watched pot never boils”; although originally known as the Turing paradox, it has since become known, for obvious reasons, as the quantum Zeno effect. Turing actually wrote (quoted by Alan Hodges in Alan Turing: Life and Legacy of a Great Thinker):

It is easy to show using standard theory that if a system starts in an eigenstate of some observable, and measurements are made of that observable N times a second, then, even if the state is not a stationary one, the probability that the system will be in the same state after, say, one second, tends to one as N tends to infinity; that is, that continual observations will prevent motion.

This is the kind of thing that is easy for someone like Turing to show, but a bit less easy for us to understand in mathematical terms. The physicists try to explain what is going on (or not going on!) in a variety of ways. One way of looking at it is to imagine the wave of probability that Schrödinger hated spreading out from the starting position of an electron, or some more complicated system, and gradually increasing the probability of finding the electron somewhere else, or the system in a different state. If you wait a long time, then look, you probably observe a different state. But if you look very quickly, the probability has not had time to change, and the system will still be in the same state. It cannot be in an intermediate state, because there are no intermediate states. There are more complicated “explanations” based on energy considerations, and Heisenberg’s famous uncertainty principle, which, happily, need not bother us here, thanks to the experimenters. What matters for our peace of mind is that Turing’s prediction has been borne out by experiments. Turing’s idea was rediscovered in the 1970s. The experiments were first carried out in the following decade, and more recently with higher precision.

The earlier experiments, undertaken by David Wineland and his colleagues at the US National Institute of Standards and Technology (NIST), give you a feel for how this kind of work is done. The “pot” which they watched contained a few thousand ions of beryllium, trapped by electric and magnetic fields. An ion is an atom from which one or more electrons have been stripped, leaving it with a positive charge which makes it easy to manipulate with such fields. The ions were held in this way in what is called a Penning Trap, at a temperature below 250 milliKelvin, almost at the absolute zero of temperature. At the start of the experiment, the ions were all in the same energy state, which the team referred to as Level 1. The ions could all be moved up in energy to a another state, called Level 2, by applying radio waves with a particular wavelength for exactly 256 milliseconds. This was the equivalent of making the pot boil. But what happens if you look to see what is going on during that 256 milliseconds? That time corresponds to there being an almost 100 per cent probability of the transition. After 128 milliseconds, the equations tell us, there is a 50:50 chance of the transition having taken place, so half the ions should be in Level 1 and half in Level 2, and so on for shorter time intervals.

To test this, the team fired a brief flicker of carefully tuned laser light at the quantum pot. This had the effect of jumping ions in Level 1 up to another state, Level 3, from which they instantly fall back, emitting a characteristic photon. But ions in Level 2 are unaffected by the laser, so the number of photons detected revealed how many ions were still in Level 1. As expected, after 128 milliseconds half the ions were in Level 1, so the other half must have been in Level 2. After 64 milliseconds, just a quarter of the ions were in Level 2. But if the laser peeked 64 times (once every 4 milliseconds), almost all of the ions stayed in Level 1, throughout the duration of the experiment. Even after 256 milliseconds almost all the ions were in Level 1. This is because after 4 milliseconds the probability of an ion making the transition was only 0.001 per cent, so 99.99 per cent of the ions had to still be in Level 1. And this holds for every 4 millisecond interval. Even though the radio waves were trying to warm the pot, even after 256 milliseconds it still had not boiled – there were hardly any ions in Level 2. Which is compelling evidence that if you could watch the ions continuously, none of them would ever make the transition.

Several similar experiments have been carried out since, confirming and refining these conclusions. These show that there is no limit to the effect – the shorter the time interval between observations, the stronger the quantum Zeno effect. But one of the most recent experiments involves a more obviously physical system, which reaches the same conclusions in a significantly different way (and which was, in fact, the inspiration for me to write this essay, if “inspiration” is the right word). This experiment was carried out at Cornell University, by Yogesh Patil and Srivatsan Chakram, and published in 2015. It was based on the way the position and velocity of a particle interact with one another, in line with the uncertainty principle. Simply put, the more slowly a particle moves the less certainty there is in its position.

The team observed the quantum Zeno effect by measuring the behaviour of roughly a billion atoms of rubidium, cooled almost to absolute zero in a vacuum chamber. Under such conditions the atoms have hardly any motion at all and settle down into an orderly lattice, like a crystalline lattice in a solid at room temperature. But the uncertainty in their position means that they ought to be able to “tunnel” from one part of the “crystal” to another, making a changing pattern. An individual atom is too cold to move through space, but it can disappear from one place and reappear in another without moving at all, in the everyday meaning of the term. The rubidium lattice was monitored through a microscope, and “looked at” with a laser. The laser caused individual atoms to fluoresce, in much the same way as the ions jumping from Level 1 to Level 3 and back again in the NIST experiment, revealing their position. The microscope was able to see the flashes and show how the pattern changed because of tunnelling. When the laser was turned on occasionally, the atoms tunnelled away happily, but when the laser flashed more frequently the amount of tunnelling was reduced dramatically. By watching the atoms all the time it would be possible to freeze the lattice in one pattern. The analogy with the “weeping angels” of Dr Who has been too tempting for many commentators to resist – the “angels” look like motionless statues, as long as you are watching them; they only move when you look away. But the quantum world is not quite the same. The message you should take away from these experiments, and the theory that underpins them, is that quantum systems do not move at all, as I shall explain.

As Minkowski said about the special theory, these ideas “have sprung from the soil of experimental physics, and therein lies their strength”. The quantum Zeno effect is real; a watched quantum pot never boils. Many physicists don’t worry about the deeper implications, but are excited by the success of experiments involving the quantum Zeno effect because it opens up the possibility of freezing systems in single quantum states and studying them, or manipulating quantum systems in practical ways, for things like quantum computing. But others worry about what is going on. Does time flow in the quantum world? Regardless of any practical applications, the quantum Zeno effect is pointing to a Deep Truth about the nature of time, by raising a Big Question. Do quantum systems ever actually make a transition from one state to another? To answer that question, I will first take you back some fifty years, to my first encounter with the puzzle.

In 1966 I read a science fiction story, October the First is Too Late, written by the eminent astrophysicist Fred Hoyle. It is an entertaining tale involving an unusual kind of time travel – time on Earth is jumbled up so that, for example, Britain was “in” 1966 while the North American continent was “in” 1750, and a traveller could pass from one time to another by moving around the globe. But what particularly struck me was that instead of the usual disclaimer about everything in the book being fiction, Hoyle included a note which said “the discussions of the significance of time and the meaning of consciousness are intended to be quite serious”. Just over a year later I was a student at the Institute of Astronomy in Cambridge, where Hoyle was the Director. As the most junior member of the Institute, I diffidently asked him one day, during coffee time, if he really meant that. I received the one-word answer “Yes”, and (to my regret) I left it at that and shuffled off to chat to the other students. But Hoyle’s ideas about time, put forward in that novel, stuck in my mind.

Hoyle’s idea was a development of the block universe model, in which 1750 is just as real as 1966 even if you perceive yourself living in 1966, in the same way that Washington is just as real as London, even if you perceive yourself living in London. His image was not, however, of a continuous block of time, but of an infinite stack of separate moments. He made an analogy with a stack of pigeon holes, the little compartments used to sort letters in the days before automation. Information is stored in the infinite array of little boxes, which you can imagine being in a numbered sequence. Each box has information about what is in boxes with lower numbers, but not about the contents of boxes with higher numbers. Consciousness, Hoyle says, is like a light dancing around the stack, illuminating one little box at a time, but, crucially, not in any particular order. The box that is illuminated is the present, as far as you are concerned. Pigeon holes with lower numbers, which are described by the contents of the illuminated box, are the past. The higher-numbered pigeon holes, of course, are the future. The spot of light – consciousness – does not have to visit each of the pigeon holes in numerical order, starting with number 1. It jumps about at random. Whichever hole it visits it will “remember” the past relative to that pigeon hole, and “experience” the present described in that pigeon hole. “The actual world is very much like this”, one of Hoyle’s scientist characters says, but “instead of pigeon holes we talk about states”. In other words, although it is not spelled out in the book, Hoyle is talking about quantum states.

A third of a century later, in 1999, I read Julian Barbour’s superb book The End of Time, which was the culmination of years of his thinking about the nature of time. The central feature of the model he developed in the book is the idea of what he calls “time capsules”, moments in time which we can identify with quantum states, which contain information about the past, but not the future. This immediately brought Hoyle’s ideas back to mind, and I mentioned this to Barbour, who had been unaware of them. I was not alone. In the paperback edition of the book, published in 2000, he added a note that “it has also been pointed out to me by several email correspondents that there is a clear anticipation of some of my ideas about time in Fred Hoyle’s 1966 novel October the First is Too Late . . . Sir Fred’s ‘pigeonholes’ are essentially my time capsules.” The difference is that Barbour provides a full quantum-mechanical discussion of the idea, putting it on a secure scientific footing. If you want the nitty-gritty, you can find it in his book. But the essence is that just as in the quantum pot experiments, different quantum states are real, but nothing is ever seen moving from one quantum state to another. There is still a sequence of events, not unlike Zeno’s image of the arrow that is stationary at any moment. But one state does not change into another. We regard some states as being in the future and others as being in the past, but there is no movement from the past into the future. This is entirely consistent with the ideas of Marina Cortês and Lee Smolin, which identify a direction of time at the quantum level, but not a flow of time.

My favourite analogy is with a cosmic library full of books which contain every detail of the history of the Universe. The volumes are in a numbered sequence which defines their place in history. Each volume starts with a summary of the information contained in earlier volumes, then goes into detail about “present” events, but says little or nothing about “future” events. The books do not even have to be arranged in any order on the shelves in the library. They could be in a jumbled heap on the floor, and if you picked one up at random you would be reading about, or experiencing, a particular “present”, with a reasonably clearly understood history, and maybe some speculation, but no clear facts, about the future. Each volume is a quantum state, one of Hoyle’s pigeon holes or Barbour’s time capsules, corresponding to what we think of as a moment of time. This exactly matches our everyday experience of time.

In terms of quantum realities, David Deutsch summed it up neatly in his book The Fabric of Reality. Using the term “snapshots” rather than “time capsules”, he says that:

Any one of the snapshots, together with the laws of physics, not only determines what all the others are, it determines their order, and it determines its own place in the sequence. In other words, each snapshot has a “time stamp” encoded in its physical contents . . . That is how it must be if the concept of time is to be freed of the error of invoking an overarching framework of time that is external to physical reality.

That “overarching framework of time” is what is represented by the popular image of a river of time flowing along steadily. It is Newton’s “absolute, true, and mathematical time”, which “for itself, and from its own nature flows equably without regard to anything external”. And it does not exist. The image that emerges from quantum physics is similar in some ways to the way that the illusion that air, or water is a continuous fluid emerges. Myriad tiny particles separated by tiny gaps feels to you like a smooth fluid. Myriad quantum states separated by tiny gaps feels to you like a smooth flow of time. Zeno was right. The arrow of time points, but it does not move.

THE END