Ideas about time have changed dramatically in the 20^{th} century. At the beginning of the century time was viewed as Newton's universal, absolute, mathematical time. There had been remarkable progress towards more and more accurate measurement and at the beginning of the century pendulum clocks had been perfected to the extent that they recorded time to an accuracy of less than ^{1}/_{100} of a second error in a day.

We begin our consideration of the 20^{th} century revolution in understanding time by first looking at those who began to question Newton's absolute time in the latter part of the 19^{th} century. In 1870 Carl Neumann questioned Newton's law of inertia. He considered a universe in which there was only one particle and asked what Newton's law of inertia meant in such circumstances. How did one know if the particle was moving in a straight line when there was no other reference point. He then introduced the idea of an inertial clock. If a particle is known to not be acted on by any forces then its motion can be used as an inertial clock. Equal time intervals would correspond to equal distances moved by the particle. However, how can we tell that a particle is not subject to forces?

P G Tait answered Carl Neumann's problem of the inertial clock in 1883 and in doing so he essentially showed that Newton's absolute space was an unnecessary concept, for he could create an absolute space framework. He did assume that positions of particles in different places could be measured at the same instant so he in effect used absolute time to define absolute space.

Mach published a history of mechanics in 1883. In it he argued strongly against Newton's idea of absolute space and absolute time. Newton argued that inertial motion was relative to absolute space but instead Mach argued that inertial motion was relative to the average of all the mass in the universe. As far as time was concerned Mach wrote:-

So according to Mach time is change and only relative distances are significant.It is utterly beyond our power to measure the changes of things by time. Quite the contrary, time is an abstraction, at which we arrive by means of the changes of things.

In 1898 Poincaré wrote a paper in which he asked two highly significant questions about time. Is it meaningful to say that one second today is equal to one second tomorrow? Is it meaningful to say that two events which are separated in space occurred at the same time? We should not give the impression that nobody before Poincaré had thought of these questions for these ideas had been discussed. However, it is certainly fair to say that Poincaré pointed to the problems much more clearly than anyone had before. The first of Poincaré's questions still has not received a satisfactory answer but the second of his questions was answered by Einstein only a few years after Poincaré's paper.

In 1902 Poincaré wrote another paper relevant to our topic. In this he asked what information is required to predict the future. By this he was thinking about Laplace's realisation that Newton's laws completely determined the future if the position, mass and movement of every particle were known. Laplace was, of course, right, but Newton on the other hand had based his theory on absolute space and absolute time and the positions and velocities of the particles were given with respect to this absolute coordinate system. Poincaré, however, was thinking in a relativistic manner and asked what information was needed if all that one were given were relative quantities. If, for example, the universe consisted of exactly three particles and all that were known were the relative velocities, what then?

Although Poincaré was thinking deeply about relativity before Einstein, it was the latter who made the final breakthrough. Einstein decided that time was the whole key to understanding the universe, see [14]. He wrote:-

The impact of the special theory of relativity on the understanding of time was enormous. The foundations on which the theory is based are remarkably simple. Einstein required the laws of physics to be the same for any two observers moving at a constant speed, that is not acted on by forces, and also that the speed of light is independent of the speed of its source. Looked at another way he assumed that there was no absolute space and time, but that the laws were the same in any inertial frame. Suppose we have two observersMy solution was really for the very concept of time, that is, that time is not absolutely defined but there is an inseparable connection between time and the velocity of light.

There were other remarkable effects on time with special relativity. Time was affected by velocity. A body travelling close to the velocity of light experiences time dilation. What does this mean? We must think about this statement clearly for, with no absolute space to measure velocity against, how can a body move at close to the velocity of light? Let us be more precise. If two bodies *A* and *B* are moving apart at close to the velocity of light then someone sitting on *A* would experience time normally, and someone sitting on *B* would also experience time normally. However, if someone sitting on *A* could view a clock on *B* then it would appear to run very slowly, and similarly if someone sitting on *B* could view a clock on *A* then it would appear to run very slowly.

These results have now been verified experimentally but we should pause for a moment to think about certain problems which remain. What does the phrase "experience time normally" mean? Does a clock running slowly mean that time is running slowly? We still do not know what time is and we are identifying it with something determined by a clock, either some device or our biological clock. All we can say on this point is that all types of clocks appear to agree on time dilation and if we cannot identify time with clocks then we need a major new idea which is still totally missing.

On 21 September 1908 Minkowski began his famous lecture at the University of Cologne with these words:-

He also 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 away into mere shadows, and only a kind of union of the two will preserve an independent reality.

Weyl quickly understood the new notion that Minkowski put forward. He wrote:-Nobody has ever noticed a place except at a time, or a time except at a place.

Before we move on from special relativity, we must consider one aspect which seems particularly difficult in Minkowski's 4-dimensional space-time, and indeed in any version of relativity. Since time is only meaningful for a single observer, with different observers at different places having their own local times, what does "now" mean. Einstein believed that this was a human concept which was not meaningful in the mathematical description of the universe. Rudolf Carnap reported Einstein's views:-The scene of action of reality is ... a four-dimensional world in which space and time are linked together indissolubly. However deep the chasm that separates the intuitive nature of space from that of time in our experience, nothing of this qualitative difference enters into the objective world which physics endeavours to chrystalise out of direct experience. It is a four dimensional continuum, which is neither "space" nor "time".

In fact Einstein wrote:-Einstein said that the problem of the "now" worried him seriously. He explained that the experience of the "now" means something special for people, something essentially different from the past and 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.

In fact all relativity seems to have done is to make us realise that time is a much more difficult concept than Newton's absolute time. However it has made no contribution to answering the fundamental question "what is time?".... there is something essential about the "now" which is outside the realm of science.

General relativity incorporated gravitation into the space-time theory. This had some further remarkable implications for time. Not only was time affected by velocity, as special relativity showed, but time was also affected by a massive body. The Earth is a massive body but not massive enough to have a large effect on the passage of time. In fact a clock on the surface of the Earth will run more slowly than a clock which is not subjected to gravitational forces. The amount is very small however, and our Earth clock will loose about ^{1}/_{1000000000} of a second in an hour. In fact the difference in the rate at which clocks run at the top of a high building compared with at the bottom has now been measured. As we mentioned the gravitation of the Earth is small compared with some astronomical objects such as neutron stars. Such objects consist of atoms which have collapsed under the force of gravity. The time dilation at the surface of a neutron star is very significant, and a clock there would run 20% slower than on the surface of the Earth. The ultimate in gravitation occurs with a black hole and with such an object gravity is so strong that time effectively stops.

We have already talked about experiments capable of detecting time differences of ^{1}/_{1000000000} of a second in an hour. We began this article by noting that at the beginning of the 20^{th} century time could be measured to an accuracy of around ^{1}/_{1000} of a second in an hour. This was achieved with a nearly free pendulum clock. We should now look at the revolution in clocks that occurred.

The free pendulum is one which is completely free from mechanical tasks, such as being part of the driving mechanism of the clock, that would stop it from being completely regular. R J Rudd introduced a genuine free pendulum clock in 1898, then W H Shortt introduced a clock with two pendulums in 1921. One pendulum was a true free pendulum, the other was part of the driving mechanism of the clock. In 1928 a totally new type of clock was built by W A Marrison at Bell Laboratories, namely the quartz crystal clock. These are widely used today and are mechanical devices which utilise the fact that a quartz crystal vibrates at a standard frequency in an electric field.

In 1949 the National Bureau of Standards in the United States built the first atomic clock, using ammonia. In around 1960 the cesium atom was in use in atomic clocks. The accuracy was such that by 1967 the second was changed from its original astronomical definition as a fraction of a day, to a definition where the second was given as 9,192,631,770 oscillations of the cesium atom's resonant frequency. By 1993 the National Institute of Standards and Technology in the United States had built an atomic clock accurate to five parts in 10^{15}.

Let us now look at another revolution in time which took place in the 20^{th} century with the discovery of quantum mechanics, see [10]. It is really impossible in an article such as this to cover all aspects of time in relation to quantum theory but we will look at one or two issues to gain a feeling for their relation. The first point to note is that quantum theory was developed within the absolute time scenario of Newton.

Heisenberg discovered the Uncertainty Principle in 1927. This states, in its best known form, that there is a lower limit to the product of the uncertainty in a particle's position and the uncertainty in its momentum so that the more accurately one is able to measure the position of a particle, the more uncertainty there is in the knowledge of its momentum. Even in this form it has a direct consequence for aspects of time we have already discussed, for it means that Laplace's realisation that Newton's laws meant that the future was completely determined by the present would not extend to quantum theory. In practice, predicting the future from Newton's laws was impossible but theoretically it could be done. However the Uncertainty Principle meant that it was not theoretically possible to know the present with the arbitrary degree of accuracy needed to predict the future.

The Uncertainty Principle also connects other pairs of quantities in the same way. For example the uncertainty in the energy of a particle and the time at which this energy is measured cannot both be determined to an arbitrary degree of accuracy. The more precisely one determines the time at which the energy is measured, the less accurately one can know that energy. Einstein was deeply unhappy about the Uncertainty Principle for it meant that the world could never be described with complete accuracy and he felt that it should not be so. He devised a number of thought experiments to try to disprove the Uncertainty Principle and posed them as challenges to Niels Bohr. The most famous one was the clock in the box which Einstein presented at the 1930 Solvay Conference in Brussels. Before we describe it, however, we should stress that the Uncertainty Principle is not about practical problems of measurement but about theoretical uncertainty. The point of Einstein's "clock in the box" thought experiment was to argue a theoretical case. The fact that it would be impossible in practice to carry out the experiment is not relevant.

Einstein's "clock in the box" consists of a box suspended from a spring. The box contains a clock which operates a shutter. There is a scale beside the box and a pointer attached to the box to measure its height. Clearly if the box has a weight added the spring stretches and the pointer comes down the scale. Similarly if the box becomes lighter then the spring will lift the box up and the pointer will move up the scale. The experiment as proposed by Einstein was to open the shutter for a very brief period and allow one particle to escape. We can fix the time of the escape as accurately as we want by having the shutter open for as short a period as we want. But, claimed Einstein, we can measure the energy of the particle as accurately as we want for its energy is determined by its mass and so we measure the mass by attaching a weight to the bottom of the box to bring the pointer back to its original position. The clever feature was that the time and the energy were calculated independently.

At first the "clock in the box" worried Bohr, but he soon realised how the Uncertainty Principle operated in this case. To weigh the particle one must measure the position of the pointer at rest on the scale. But deciding the pointer is at rest and measuring its position are subject to the Uncertainty Principle. The more accurately we determine that the pointer is at rest, the less accurate will be our determination of its position. There is a second uncertainty in this experiment. If we cannot measure the height of the box to arbitrary precision, we cannot measure the height of the clock inside the box with arbitrary precision, so we do not know the rate of the clock with arbitrary precision (by Einstein's own general relativity results).

In the "clock in the box" thought experiment we have seen how relativity and quantum theory begin to interact. Several early attempts to bring the two theories together revolved round the problem of time. Milne developed a complex theory of cosmology, attempting to unify relativity and quantum theory, that included a non-constant value for *G*, which we know as the gravitational constant. In order to account for this, Milne actually developed two separate time scales: kinematic time *t* and Newtonian time *τ*. The two time scales were related by the following relation:

Dirac tackled a similar problem in his development of his Large Numbers Hypothesis. He was forced to initially create two time scales, much like Milne, one being atomic and the other being global (Newtonian). Atomic time was supposed to describe radioactive decay while global time would be applied to large-scale phenomena. Dirac was forced into this conclusion based on results of the Large Numbers Hypothesis that threw off age calculations of the Moon and Sun. He also had a changing value for *G*, though his decreased while Milne's increased. Dirac later abandoned the dual time scale idea.

There are ways that quantum theory time appears to contradict relativity time, and this is worrying. The idea was first put forward by Einstein, together with Nathan Rosen and Boris Podolsky in 1935 and it is known by the initials of its proposers as the EPR experiment. It relies on the fact that a quantum event sometimes creates a pair of particles with complementary properties - for example they must have opposite spins. In quantum theory the particle will have the properties of both possible states until we measure it when it collapses into one of the two states. However, when we measure one particle and it collapses into one state, the other particle must instantly have the complementary property. Einstein firmly believed that no information could be transmitted faster than the speed of light, and saw this as an objection to quantum theory. The EPR experiment in this form, however, did not seem possible to test.

John Bell sharpened the EPR experiment in the 1960s by devising a way to check that the particles had all possible states until tested. The classical theory (or a common sense theory) would say that the two particles had definite states when created, it was just that we do not know what they are until we test one of them. Bell discovered "Bell's inequalities" which would hold in the classical case. If such an experiment could be carried out it would verify whether the particles only chose their state when tested but at the time Bell proposed his version the experiment was beyond existing experimental techniques. In the early 1980s Alain Aspect successfully carried out the experiment at Orsay in Paris. He showed that Bell's inequalities were violated and so the quantum interpretation held rather than the classical one. The implications are, however, that when one particle is tested and chooses a particular state, its partner must chose the complementary state at the same instant. This violates the basic principle of relativity that no information can be transmitted faster than the speed of light. The implications for "time" are still not fully understood.

An interpretation of quantum theory put forward by Hugh Everett in 1957 is the many worlds interpretation. In this the universe splits into two every time a quantum event is forced to choose between two states. What is the effect of this theory on time? In [3] Deutsch supports the many worlds interpretation and argues against the idea that time is flowing from past to present to future. His argument against this is that to measure how the present moves forward one would need a second "time" against which to measure the progress our standard time. Again to measure this second time's flow one would need a third time and so on. Deutsch presents a universe consisting of snapshots, rather than a continuous progression with the flowing of time.

Other ideas consider that instead of space-time being four dimensional, there are many more dimensions. Such theories attempt to unify all the theories of physics under a single mathematical framework. In such high dimensional spaces time travel seems a possibility but we will not look at this possibility in this essay. It is, however, reasonable to ask if there is more than one time dimension. What would it mean if we lived in a universe where time was two dimensional?

Hawking has presented some ideas concerning imaginary time; see for example [7] and [4]. He writes in [7] about a model for time/size of universe as a sphere:-

After describing the real time appearance as beginning in a singularity he then wonders which is the "real" time:-... with the distance from the North Pole representing imaginary time and the size of the circle of constant distance from the North Pole representing the spatial size of the universe. The universe starts at the North Pole as a single point. As one moves south, the circles of latitude at constant distance from the North Pole get bigger, corresponding to the universe expanding with imaginary time. ... Even though the universe would have zero size at the North and South Poles, these points would not be singularities. ... The laws of science will hold at them ... The history in real time, however, will look very different...

Penrose, in [6], takes a different approach but reaches similar conclusions about our perception of time:-This might suggest that the so-called imaginary time is really the real time, and that what we call real time is just a figment of our imaginations. In real time, the universe has a beginning and an end in singularities that form a boundary to space-time and at which the laws of science break down. But in imaginary time, there are no singularities or boundaries. so maybe what we call imaginary time is more basic, and what we call real time is just an idea we invent to help us describe what we think the universe is really like.

Time is a fascinating topic and new ideas are continually being put forward. It is still perhaps the most mysterious property of the universe.The temporal ordering that we 'appear' to perceive is, I am claiming, something that we impose upon our perceptions in order to make sense of them in relation to the uniform forward time-progression of an external physical reality.

**Article by:** *J J O'Connor* and *E F Robertson*

**MacTutor History of Mathematics**

[http://www-history.mcs.st-andrews.ac.uk/HistTopics/Time_2.html]