What is time? If clocks measure time, can we answer the question by a careful study of clocks?

A clock is a machine and, like all machines, subject to the laws of thermodynamics.
In popular discussions, a clock is often presented as the epitome of a reversible and
predictable dynamical system. Nothing could be further from the truth. A careful examination of the requirements for a clock show that a it cannot be reversible, and in fact requires friction, entailing heat loss, to operate. Irreversible systems generate entropy. A clock is a flow meter for entropy.

Clocks come in many forms, from bio-chemical to astronomical, from
mechanical to statistical. A dictionary definition states, “a clock is a device to
measure time”, which begs the question. Other definitions emphasise the repetitive, or periodic, nature of clock states. Yet radio carbon dating is a clock that is anything but periodic, as is the water clock used by Galileo at the birth of modern physics.

I will take the view that a clock is a physical device used to coordinate local co-
incidences of physical events. If clocks measure time, this definition captures the relational character of time: for three local events, if event C coincides with event A and also coincides with a distinct event B, then A happens at the same time as B, up to an error determined by the precision of the clock. There is no absolute simultaneity, even putting relativity aside. If clocks measure time, then now is fuzzy.

As an example, suppose you are in a kitchen with a dripping tap. A digital clock/timer on the wall is counting up in unit steps. When a drip hits the basin, a sound pulse is generated. At each repetition of the sound, take a note of the count on the clock and tabulate the results. Place a 0 if no sound occurs at a particular count and a 1 if a sound does.

Event (x)	Count (n)
1	1
0	2
1	3
0	4
1	5
1	6
0	7
1	8
0	9
1	10
0	11
1	12

It is clear that the event is almost periodic according to this clock but this is entirely relational. There appears to be an error at n=6, but we can equally well claim that the event is truely periodic but that the clock is imprecise. I have not said what the units of the clock are … they are irrelevant. The phenomenon is already captured by the table.

The table represents an integer valued function of a single binary variable. Where is the continuous time of physics, the little t ? A little known theorem of integer valued functions tell us that they can always be parameterised in terms of a real variable t. As far as clocks are concerned this is a mathematical trick, but it makes the job of theoretical physicists much easier. But keep in mind that little t is not time. It is simply a mathematical device. All real parameterisations are equivalent. We chose the one that makes the calculations easier. This was the key insight that Einstein injected into general relativity.

The relational nature of time as measured by clocks is critical to understanding the nature of time. Ernest Mach, the famous 19th century physicist that inspired Einstein, saw this very clearly.

Physics sets out to represent every phenomenon as a function of time. The motion of a pendulum serves as the measure of time. Thus, physics really expresses every phenomenon as a function of the length of the pendulum. . . If one were to succeed in expressing every phenomenon. . . as a function of the phenomenon of pendulum motion, this would only prove that all phenomena are so connected that any one of them can be represented as a function of any other. Physically, then, time is the representability of any phenomenon as a function of any other one

Returning to clocks as machines. The pendulum clock invented by Huygens in the 17th century is typical. Friction causes it to run down unless we give it some kicks. Huygens figured out a way to do this using a falling mass and an escapement wheel coupled to the pendulum. The escapement gives the pendulum two kicks every cycle. That is the sound you hear as tik-tok.

The escapement must be heavily damped to prevent it from bouncing. Most of the energy per cycle is lost here. The combined system is no longer a simple oscillator but a special kind of non liner oscillator, a relaxation oscillation. It remains to add a counter…the dial.

There is a price to pay for stable, continuous operation. The clock is no longer reversible and necessarily dissipates heat. The laws of thermodynamics mean that the period of such a system is not fixed but suffers small fluctuations from one cycle to the next. The average period can be fixed but the uncertainty cannot be reduced to zero. Engineering a good clock means finding a way to make the uncertainty in the period as small as possible.

It turns out that the greater the rate of heat and entropy produced by the clock the better it is. Good clocks are pumping disorder into the world and the faster the better. A clock is a flow meter for entropy. If it measures anything it measures the rate at which disorder is increasing in the world.

You may object that astronomical time, based on the motion of the earth and planets, looks pretty regular, quite ‘clockwork’. Where is the heat being generated? The answer is surprising: heat is generated in the planets due to non uniform gravitational fields stretching and squeezing the matter and generating heat. This is called tidal heating. It leads to a loss of precision over vast numbers of orbits. The phase of the orbits becomes scrambled on the order of 100 million years. This is much longer than our quotidian concerns, and so astronomical time is good enough for us.

National governments spend billions of dollars annually to support laboratories to make better clocks. Our sophisticated technologies for computing and communication require better and better clocks. Our global navigation system is limited by the precision of clocks in satellites. Currently the best clocks under development are based on a special kind of relaxation oscillation, the laser, locked to a quantum oscillation of the charge distribution in an atom that the light illuminates. Engineering better clocks requires new quantum technologies and this is big business.

Clocks are irreversible, entropy producing devices. What does this tell us about the nature of time? Firstly, time is not the reversible, perfect little-t of physics. The equations of physics are reversible in little-t. The world is not, and neither are clocks. The equations of physics provide an idealisation of the changing world. Secondly, we are only too aware that time is irreversible for that is what aging is. Time is personal for us. We change and decay in our little patch of the universe. All time is unredeemable and local. Here and now is all there is.