CHANGE is the measure of time. The way we perceive time depends on our awareness through sensations of changes in the world about us. The regular beat of a pulse, the motions of the Sun and the stars, and seasonal rhythms all suggest that time flows regularly in an endlessly repetitive pattern. Yet, as the poet John Donne notes, we also perceive time as the ‘cruel hand’ that leads to decay, destruction, death and dissolution. This mysterious duality in the way in which things change in time is an essential feature of the Universe we inhabit.
It also points to a peculiar paradox at a more fundamental level. We accept that the Universe is made up of matter consisting of large numbers of particles, nuclei, atoms or molecules, depending on the environment. The laws of classical Newtonian physics governing the motions of the individual particles, and in many cases the laws of quantum mechanics, treat the past and future symmetrically. In other words, if we reverse the motion of a gas molecule bouncing off the walls of a container – as though we were running a film of the event backwards – the trajectory of the molecule will look identical in both directions.
Assemblies of particles, however, seem to behave in an irreversible way. A gas in an inflated balloon, if released, will expand for ever into an infinite space. It will never return to the balloon of its own accord. More complicated systems, such as ourselves, ultimately die and decay. How can the laws of physics, which treat time as symmetrical, explain the behaviour of assemblies of particles, where time is so obviously irreversible? This problem has exercised the minds of some of the greatest physicists, such as Ludwig Boltzmann and James Clerk Maxwell. I want to show through simple examples and analogies how we can reconcile these two contradictory views of time.
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Our perception of time starts with the clockwork universe of Isaac Newton. In the 17th century, Newton described mathematically how forces act on bodies or particles. He produced a set of equations relating the force to the mass of a moving body, its speed and acceleration in terms of distance and time. His laws of motion explain how the planets revolve around the Sun. They also predict the behaviour of balls colliding on a billiard table. If we reverse the motion of the balls, turning the positive quantities of time t and velocity v into negative quantities -t and -v, the equations of motion remain the same. We say that the equations are ‘time-reversal invariant’.
Victorian physicists and engineers had to reconsider the nature of time when they started relating the motions of large numbers of particles to heat energy and the work that a heat engine could do. This was the birth of thermodynamics. When a gas expands in a cylinder, converting the random motions of the gas particles (heat) into the unidirectional motion of a piston, engineers such as Sadi Carnot realised that the conversion could never be 100 per cent efficient. The overall disorderliness, or entropy, of the system had to increase. If a system, a cylinder of helium gas, for instance, were isolated from the rest of the Universe and left to itself, its entropy would continually increase until it reached a maximum value that depended on the bulk properties of the system such as pressure, volume and temperature. Entropy cannot decrease; the process is irreversible. The concept of entropy automatically endows the system with an ‘arrow of time’.
Around the same period, Maxwell and Boltzmann came up with a description of how you could derive these bulk properties of a gas by adding up and averaging the individual motions and collisions of its constituent particles – the so-called kinetic theory of gases. They used Newton’s time-symmetrical laws to define the dynamic behaviour of each gas particle. They then derived statistically an equation describing the behaviour of the whole gas. The Boltzmann equation breaks the symmetry of time reversal.
Boltzmann and other physicists were deeply perturbed by this turn of events. Although Boltzmann insisted that his derivation of the irreversible behaviour of a gas from Newton’s laws of motion was mathematically rigorous, many people, including Josef Loschmidt, a physical chemist, said that you could not deduce the irreversible Boltzmann equation logically from reversible Newtonian mechanics without making additional assumptions.
Using a theorem of Henri Poincare, a mathematician and student of Max Planck, Ernst Zermelo noted that Newton’s laws would imply the following result: no matter what the configuration of the gas was to start with, if you waited long enough, it would eventually return to its original state. In other words, you can show that, in the long term, Boltzmann’s equation is wrong. Poor Boltzmann was very depressed by all this criticism and eventually committed suicide. Ironically, with the proper interpretation, the profound contributions that Boltzmann and Maxwell made to kinetic theory have stood the test of time and experiment.
But what of Loschmidt’s and Zermelo’s objections? The answer lies in the difference between the way that systems with just a few constituents, such as the Newtonian model of the Solar System, behave in time and the way that those with very large numbers of particles, such as a gas, behave. This affects how we observe and measure their dynamical properties. It is not a trivial problem. It has repercussions in almost every scientific discipline from cosmology to psychology.
I shall take one simple example of a typical Newtonian system to analyse the two different kinds of behaviour. Imagine a small bead sliding on a circular wire. For the present, we will not worry about irritating little details such as friction, gravity or the composition of the bead. Just suppose that we look down on the hoop and visualise it as a clockface or a circular race track. Take a fixed point corresponding to 12 o’clock on the hoop, and join it to the centre of the circle. At some instant, the bead will be at a certain point on the loop. We can locate the bead in two ways. First, by specifying the length of arc over which the bead has travelled from the initial starting point at 12 o’clock, or by measuring the angle that the arc makes with the centre of the loop.
The Newtonian mechanics of the bead moving frictionlessly is remarkably simple. According to Newton’s laws, force is proportional to acceleration, which is the change in velocity. Remember that velocity has both magnitude and direction. The circular motion of the bead means that it is continually changing direction, so experiencing an acceleration equivalent to a force that pulls the bead into the centre of the loop.
If we set the bead in motion at some initial time from the 12 o’clock position, the bead will continue forever moving in the same direction, clockwise or counterclockwise, and at the same speed. Obviously the bead will complete a circuit in a certain period, which equals the circumference of the loop divided by the speed of the particle. If we know the speed of the particle at any instant, we can determine its position at any future or past time. After all, this is no different from a car going round a racing circuit at a constant speed; we simply multiply the speed by the time elapsed to get the total distance travelled from the starting point. The important point is that just two quantities determine the motion of the bead, the initial position and the initial speed. We call this type of circular motion, in which speed remains the same and does not deviate from its circular direction, uniform circular motion.
Amazingly, the same formulae that work for the bead are also valid for circular motion in the Solar System, where relatively light bodies, the planets, travel around a massive central Sun. In this instance, the motion arises from the force of gravity, which pulls the planets towards the Sun. Gravity plays the same role as the hoop. Strictly speaking, planets have elliptical orbits, although purely circular motion is a possible solution of Newton’s equations and is a fair approximation. The important point here, however, is that the radius of the orbit is related to how long it takes for the planet to complete one orbit. The tighter the orbit, the faster the planet must move to avoid spiralling into the Sun. This relation is called Kepler’s third law of planetary motion. It states that the cube of the radius is proportional to the square of the period of revolution. It provides one of the best-known tests of the correctness of Newton’s theory of gravity.
So, according to Newton’s laws, the motion of the bead is uniform and, at any instant, we need two numbers to describe what the particle is doing: its position and its speed. It is as if the state of the particle is characterised by some property that is ‘conserved’, or constant in time. Like Donne’s love, the speed of the particle never decays. The position of the bead (or planet) is periodic, so time appears to repeat itself, with the period of revolution related to the conserved speed. As in the case of the billiard balls, Newton’s equations cannot tell when we reverse the motion of the bead by running the film backwards from future to past. This leads to a perception of time as a regular flow with an endlessly repeating pattern that, in the words of Donne, ‘hath no tomorrow, not yesterday’.
Things become much more complicated when we introduce another, noncentral force on the bead. What happens depends on the type of force. Some forces, such as gravity, resulting from, say, a heavy body placed at some off-central point, act so as to conserve the total energy in terms of the speed (which contributes kinetic energy proportional to the mass and the square of the speed) and the potential energy (which depends on the position of the bead). The speed can vary with time. We can visualise this by imagining gravity to act perpendicular to the plane of the hoop, and then deforming the hoop slightly to simulate ‘the hills and valleys’ along it. The motion is not strictly circular but the model is useful if the heights of the hills are small compared with the radius of the loop. The speed of the bead is lowest and its potential energy highest when it is on the crest of the hill. As the bead slides down the hill, it picks up speed and therefore kinetic energy at the expense of potential energy. Although this motion may be complicated, it is still reversible in time. There is no decay here.
What about other types of noncentral forces, such as friction? These come in many forms but they all have one thing in common: the force always acts opposite to the direction of the particle’s motion, tending to slow it down. Friction, in the simplest cases, works like this: if the initial speed is, say, 100 metres per second, then after a certain time, say, five minutes, the speed reduces to 50 metres per second. After another five minutes, the speed drops to 25 metres per second and so on. We represent the time for the speed to halve as t(half). Friction tends to reduce the speed or, more precisely, the kinetic energy, in a geometrical progression.
The situation is a bit like that of someone with a certain amount of money in the bank, who each year spends half of the balance but does not receive any income. The result is totally predictable. The person’s wealth will slowly but surely decay in geometrical progression, just like the speed of the bead. The main difference is that the speed reduces continuously, while wealth is measured in ‘quanta’ of the smallest units of currency and, therefore, eventually disappears entirely. Incidentally, the laws of radioactive decay are exactly the same. The time t(half) is the so-called half-life. It is precisely this difference between past and present implied by the decay that allows archaeologists to use the radioactivity of carbon-14 to date ancient objects.
Going back to the circulating bead, note that friction introduces two distinct timescales: the period of revolution (which, in the absence of friction, depends on the initial conditions) and the decay time, t(half). If the decay time is very long compared with the period of revolution, frictional forces are weak. The motion is analogous to a pedalled piano note. If, on the other hand, friction is strong, and the decay time is short compared with the period of revolution, we have something like a pendulum moving through thick syrup.
What is the origin of these mysterious forces that violate the pristine symmetry of time reversal in Newton’s equations? I am going to argue that the answer lies in the introduction into the mechanics of very large numbers, and averages over these numbers. It also has very much to do with the kinds of quantities we measure and the nature of the averaging processes inherent in such measurements. This is extremely important. We can obtain a surprising amount of insight by looking at quite elementary cases.
The first example of a frictional force comes from ‘wave damping’. Remember that when the bead slides round the hoop it is actually accelerating because it is continually changing direction. The same thing applies to the motion of the planets: they are continuously accelerating under the force of gravity. In fact, Albert Einstein made an important correction to Newton’s theory of motion and gravity. His general theory of relativity showed that acceleration and gravity were equivalent. Consequently, we would expect our accelerating bead to emit gravitational waves, albeit at an extremely low rate. So, really, the kinetic energy of the bead steadily decreases as these waves go off into space.
The same thing applies to the planets. It turns out that for the Earth moving around the Sun, the decay time t(half) due to the loss of kinetic energy is about 10 million million million million years. This is far longer than the estimated age of the Universe of between 10 and 15 billion years, not to mention that of the Solar System, so the damping of motion due to gravitational waves is minuscule. If, however, the bead were a charged particle, such as an electron, then according to classical electromagnetism it would emit light waves, or synchrotron radiation. This kind of wave damping can be large, the electron can lose a lot of kinetic energy, so the decay time can be much shorter than the period of revolution.
We can explain wave damping with the following analogy. Consider the still surface of a lake, then put a cork in the water and make it bob up and down. If the water had no friction, in other words, no viscosity, you might expect the cork to bob up and down forever. In fact, it always produces surface waves which travel outwards, carrying energy away from the cork. If the lake is a small pond, the waves would reflect against the sides of the pond and eventually return to the cork. If the reflections were ‘perfect’, the energy of the cork would be entirely conserved.
If the lake is huge so as to seem ‘infinite’, the energy carried by the waves would never return. The cork would steadily lose energy and settle down to rest. This is despite the fact that Newton’s equations of fluid motion without viscosity do not contain any terms related to the irreversibility of time. We learn from this example that dynamic systems like the cork in the lake depend not only on Newton’s equations but also on what happens at the ‘boundaries’ of the system and where the boundaries are. Going back to the bead emitting gravity waves, if we put the system in a smallish box with walls that perfectly reflected the waves, the system would not decay. The symmetry of time reversal would not have been broken.
What about ordinary friction between the bead and the hoop – the kind due to the rough surfaces of the bead and hoop rubbing together? This force is very complicated. It is due to the myriad little molecules on the surface of the hoop obstructing and pushing against those on the surface of the bead. Although it is not easy to see the effect of infinity in this case, the energy given up by the bead is shared rapidly by the billion of molecules of the hoop, so it never has a chance to return undiminished.
At this point, I cannot resist posing a problem: what if we deliberately drive our bead against the frictional forces of whatever origin, with another, constant, noncentral ‘pushing’ force? This is similar to cycling down a not-too-steep hill, where we accelerate under gravity but ultimately reach a constant speed as wind resistance and friction from the bicycle brakes balances the effect of gravity. In the case of the bead, the final constant speed has nothing to do with the initial speed, which has been ‘forgotten’. This loss of memory happens on the timescale of the decay time t(half), which is a measure of the strength of the friction. Thus, amnesia is a characteristic of irreversible systems, which must involve frictional forces.
An extreme example of this is driving a car around a race track at 100 kilometres an hour with the handbrake on. Yet, despite this apparently eternal periodic motion, the system composed of the car engine and handbrake does not have time reversal invariance because there is an outside agency, the petrol tank, which supplies energy to offset that lost through the frictional force of the handbrake. The system is therefore ‘nonconservative’. Time-reversal symmetry applies only to conservative systems in a bounded domain, such as the circling bead in a perfectly reflecting box, or the cork floating on the small pond. Such systems are like a Newtonian miser who has no net income and no net expenditure and, indeed, no communication whatever with the outside world but simply moves his wealth from room to room in his house forever.
Forces like friction break the symmetry between past and future. They introduce an arrow of time. I have suggested that this has something to do with introducing large numbers. We can show this by enlarging the bead model. Consider two identical hoops with identical beads. Imagine that these hoops are fixed a metre apart, one above the other. The beads do not ‘know’ about each other. If we know their starting positions and speeds, we can predict what each bead will be doing at any time because the speeds are conserved. Unlike the case of the single bead, we have four numbers to describe the system. If we have N beads and hoops arranged in the same way, we need 2N numbers to describe the system. Just by increasing the number of beads we have not changed the time reversal symmetry of the system. We should have no trouble verifying this if we can simultaneously watch all the beads.
If we start all the beads from the same initial position on their respective hoops, they will all go round as if they were a single bead. The system with 2N numbers is equivalent to the system with two numbers. Suppose, instead, we start them all off at the same position but give them different speeds. An amusing alternative model is a single satellite moving around Saturn in a circular orbit. Suppose the satellite suddenly breaks into N pieces. We assume that these fragments are sufficiently far apart to avoid collisions, and that they start moving in circular orbits with nearly the same radii and a spread of speeds. If N is large – say, a million – we would, in time, see the formation of a ring composed of a million pieces that persists forever.
Getting back to the hoops and beads, if we look down on the hoops, we will see the initial clump of beads separate and steadily spread as it moves around the circular track. After a longish time, there will be, on average, as many beads per degree in any sector of the circle as in any other. If there are just two beads on two respective hoops with bead 1 going round twice as fast as bead 2, the whole system will return to its original configuration each time the slowest, bead 2, completes one orbit.
You can work out this ‘recurrence time’ for any number of beads. It is simply the least common multiple of the individual periods of revolution. Now, see what happens with seven beads that have periods of revolution in seconds equal to the prime numbers between 1 and 13 – 1,2,3,5,7,11 and 13. The recurrence time is 1X2X3X5X7X11X13 equals 30 030 seconds. This is far larger than the period of the slowest bead (13 seconds). You can carry out this ‘thought’ experiment on a home computer with different colours for the beads.
You can see that the time-reversal properties of the system are completely lost on the timescale of the periods of the individual beads. To be sure, for any finite number of beads, however large, the system will eventually revert to its initial configuration of all the beads starting at the same place on their respective hoops – if we wait long enough. When all the beads start in the same position but with a wide range of speeds, they quickly become spread out. The timescale over which this happens depends on the difference in speed between the fastest and slowest beads. The degree of spread eventually reaches an equilibrium value – an average value for the density of beads in each sector of the ring. For large numbers of beads, we quickly lose information about individual beads and have to be satisfied with measuring averages, such as the density of beads, which appears to behave irreversibly in time.
If all the beads could act as one, then the recurrence time would be the same as the period of revolution and the system would stay reversible. If, for example, you dropped an ice cube from a glass, it would be easy to restore the ice cube to its original state, simply by picking it up and putting it back in the glass. The molecules of frozen water are not free to move so they act as one. Repeat this after the ice cube has melted and you’ll find it a good deal more difficult to reverse the process because the water molecules are much freer to move so their relative positions spread very quickly. This rapid spreading of possible positions happens even more quickly when you release a fully inflated balloon. The air rushes out and the balloon zips about all over the place. No one has ever seen an empty balloon move around, suck in air and inflate itself, although this is a possible solution of Newton’s equations.
To sum up, even if the dynamics of a system obey Newton’s laws and are reversible on the microscopic scale of individual particles, large numbers or an infinite environment can intervene to produce an arrow of time. This happens in ‘macroscopic’ systems such as a gas. Irreversibility need not be an intrinsic property of the system itself. It arises when we have to make measurements based on averages because there are too many particles to observe them individually. This is what happened when Boltzmann derived his equation. By measuring average properties, we automatically lose information about the behaviour of the constituent particles. When dealing with a macroscopic system, computer scientists like to relate the amount of information lost directly to the concept of entropy – the apparent randomness within a set of data.
Our simple example of circulating beads shows that although collisions between gas molecules help to make the behaviour of a gas irreversible, as Boltzmann showed in his kinetic theory, irreversibility creeps in, even when we have an assembly of particles that do not collide. The arrow of time seems to depend on how we observe a system and what kind of measurements we can make. For the ‘all-seeing’, there is no past or future in a bounded Newtonian universe. For the rest, as Jaques remarks in As You Like It, ‘Time travels in divers paces with divers persons’.
All kings and all their favourites, All glory of honours, beauties, wits, The sun itself, which makes times, as they pass, Is elder by a year, now, than it was When thou and I first one another saw: All things to their destruction draw, Only our love hath no decay; This, no tomorrow hath, not yesterday; Running it never runs from us away, But truly keeps his first, last, everlasting day.
from ‘The Anniversary’ by John Donne
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Many dances to the music of time
ALTHOUGH I have described how the symmetry of time breaks in classical systems when we introduce large numbers, Nature is, of course, much more complicated.
Even in classical physics, there are situations in which you might expect large numbers to lead to irreversible behaviour but yet something like periodic regularity applies. Despite the fact that a gas is made up of very large numbers of molecules, sound waves are examples of highly ‘coherent’ motions that are nearly periodic.
Quantum theory introduces fascinating new perspectives where quantum laws suspend ‘decay’. In a hydrogen atom, an electron in its lowest state of energy (ground state) is stable. It does not emit light, even in an infinite universe, and cannot ‘decay’ (unlike a charged bead moving according to classical physics), thereby, exemplifying John Bunyan’s observation: ‘He that is down needs fear no fall.’
A superconducting current continues indefinitely without losing any energy. It is an intriguing example of how quantum laws manifest themselves on the macroscale. The electrons responsible for superconductivity appear to ‘cooperate’ throughout the bulk of the material. In this case, large numbers are necessary but not sufficient to break the symmetry of time and cause attendant ‘disordering’ of energy. This fascinating phenomenon where dynamical complexity is reduced in apparently ‘large’ systems is difficult to understand, and many theorists are interested in trying to interpret the physics of such processes.
As if this were not confusing enough, time-reversal invariance is actually violated in particle physics in the well-known case of the decay of an exotic particle called the neutral K-meson, which is produced in some collisions between high-energy particles. The neutral K-meson decays into two other particles called pions. In decaying, it breaks one of the fundamental tenets of quantum mechanics, that particles and their antiparticles show complete symmetry in charge (C), position (P) and time (T). The decay of the neutral K-meson is complicated but the CPT symmetry of the process is ‘broken’ although the product of the symmetry still holds. Measurement in quantum mechanics usually involves the presence of an ‘apparatus’ that often behaves in a classical, irreversible manner due to its large number of dynamical degrees of freedom. The results of ‘observing’ quantum systems are, therefore, usually irreversible.
Einstein’s theory of special relativity also introduces another perspective – that simultaneity is relative. There is no single universal Newtonian time inexorably marching on but many ‘times’, to different beats, with diverse persons. The nature of time is truly mysterious and perhaps grasped better by poetic insight than dry discourse.
Anantanarayanan Thyagaraja is interested in plasm physics and fluid mechanics. He works on these subjects at the Culham Laboratory near Oxford
