… because mathematicians frequently make use of time,
they ought to have a distinct idea of the meaning of that word,
otherwise they are quacks …

Isaac Barrow (1630 - 1677)
Quoted in P Coveney & R Highfield, The Arrow of Time. Ballantine, NY 1990

Topics

• … not just 'time,' but 'space' too; not just mathematicians, but everyday people too!
  Despite the fact that we can not but inhabit time and space, be embedded in them, our notions about these concepts are rather primitive. Even Isaac Newton (1642 - 1727), whose laws of motion represent probably the best distillation of our everyday experience, defined time as follows:

  "Absolute, true, and mathematical time, of itself and from its own nature, flows equably without relation 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 of duration by the means of motion, which is commonly used instead of true time, such as an hour, a day, a month, a year."
  [ from Absolute and Relative Space, Time, and Motion ]

  Netwon's definition of space is in the same vein:

  "Absolute space, in its own nature, without relation 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 and which is commonly taken for immovable space; such is the dimension of a subterraneous, an aerial, or celestial space, determined by its position in respect of the earth. Absolute and relative space are the same in figure and magnitude, but they do not remain always numerically the same. For if the earth, for instance, moves, a space of our air, which relatively and in respect of the earth remains always the same, will at one time be one part of the absolute space into which the air passes; at another time it will be another part of the same, and so, absolutely understood, it will be continually changed."
  [ ibidem ]

  After defining "place [as] a part of space which a body takes up and is, according to the space, either absolute or relative," Newton then defines "absolute motion [as] the translation of a body from one absolute place into another, and relative motion the translation from one relative place into another." [ ibidem ]
  Notice that Newton's definitions imply a complete separation between space and time : you can have one without the other.
• What's wrong with this framework? Nothing, apparently! It seems to fit our experience quite adequately, and it permits the development of of the laws of motion, which are good enough to allow us to send spacecrafts to Mars and beyond. The definitions of space and time, however, smell a bit of metaphysics. In fact they assign to time and space attributes that are usually reserved in many religions for the definition of God, which, by definition, transcends our daily experience. Are such attributes needed? Can they lead to unexpected, and/or unwarranted, consequences? Can we experience 'absolute' space and 'absolute' time?
  The first crack in Newton's framework appeared quite early. It is best illustrated by a letter Newton wrote in 1692 to Richard Bentley (1662 - 1742), in which the following famous passage appears:

  "It seems to me, that if the matter of our sun and planets, and all the matter of the universe, were evenly scattered through all the heavens, and every particle had an innate gravity towards all the rest, and the whole space throughout which this matter was scattered, was finite, the matter on the outside of this would by its gravity tend towards all the matter on the inside, and by consequence fall down into the middle of the whole space, and there compose one great spherical mass. But, if the matter were evenly disposed throughout an infinite space, it could never convene into one mass, but some of it would convene into one mass and some into another, so as to make an infinite number of great masses, scattered great distances from one to another throughout all that infinite space. And thus might the sun and fixed stars be formed, supposing the matter were of a lucid nature."
  [ from Cosmlogy: Newton Cosmology ]

  As Stephen Hawking puts it in A Brief History of Time: From the Big Bang to Black Holes (Bantam, NY 1988, p. 5): "This argument is an instance of the pitfalls that you can encounter in talking about infinity." If Newton were right, all the stars in his 'infinite' universe would collapse into one fireball.
  Not only Bentley, but also Descartes (1596 - 1650) and Leibniz (1646 - 1716), among others, did not share Netwon's view. Leibniz, for example, declared: "I hold space to be something purely relative, as time is." [ ibidem ] He also persuasively argued that the universe had to be finite, though, perhaps, unbounded (how can that be?).
  Be as it may, this issue was not resolved until the beginning of the 20th century. A very interesting chapter in this story is represented by Heinrich Olbers (1758 - 1840), who in 1823 formulated what is now known as Olber’s paradox, which in plain English runs as follows:

  "Suppose that the universe is (1) static, (2) infinite, (3) eternal and (4) uniformly filled with stars (or galaxies, which are made of stars). If we look in any direction, our line of sight must eventually run into a star (galaxy), just as a frictionless arrow shot in the middle of an infinite forest will eventually stick itself into a tree. Therefore, the night sky should be as bright as the average star (galaxy) and certainly should not be dark."
  [ from Olber's Paradox: Why is the Night Sky Dark? ]

  Olber's paradox is an example of a simple but astute piece of reasoning that demands a satisfactory answer, since the might sky is definitely not as bright as the day sky. The resolution of this paradox is far from simple. In fact we are only now beginning to understand what a satisfactory answer requires. Here is a comment that shows how far an answer would take us:

  "It is interesting that in asking and answering the seemingly trivial question, 'Why is the night sky dark?' one could have inferred that the Universe was expanding and that the Universe had a finite age (or at the least the stars and galaxies had finite ages)."
  [ from Olber's Paradox ]

• Contrast now Newton's approach to time and space with Einstein's. In 1905, Albert Einstein (1879 - 1955) published his special theory of relativity. Here is how, in a later non-technical exposition, he proposed to tackle the issue of time and space (note the simplicity and clarity of his language):

  "The purpose of mechanics is to describe how bodies change their position in space with 'time.' I should load my conscience with grave sins against the sacred spirit of lucidity were I to formulate the aims of mechanics in this way, without serious reflection and detailed explanations. Let us proceed to disclose these sins.

  It is not clear what is to be understood here by 'position' and 'space.' [ … ] In the first place we entirely shun the vague word 'space,' of which, we must honestly acknowledge, we can not form the slightest conception, and we replace it by 'motion relative to a practically rigid body of reference.' [ … ] In order to have a complete description of the motion, we must specify how the body alters its position with time; i.e. for every point on the trajectory it must be stated at what time the body is situated there. These data must be supplemented by such a definition of time that, in virtue of this definition, these time-values can be regarded essentially as magnitudes (results of measurements) capable of observation."
  [ from Albert Einstein, Relativity. H Holt & Co, NY 1920, p. 9 ]

  In other words, time is simply, but not trivially, that which a clock measures.
• In a very real sense, the difference in the two approaches lies in this: both appeal to our everyday experience, and both reflect the historical and cultural period in which they were formulated. In Newton's times, religion and metaphysics were considered as real and relevant as the natural world, and the two often mixed, quite freely. In Einstein's times, the danger of such hybrids was felt much more strongly, just as a growing trend to rely only on that which can be empirically observed and measured began to establish itself.
  These considerations are best illustrated by the two principles (or postulates) that Einstein introduced in 1905 as the foundation of his theory of special relativity :
  1. Relativity Principle : The laws of physics have the same form in all inertial frames of reference.
  2. Constancy of the Speed of Light : Light propagates through empty space at a constant rate, c = 299,792,458 m/s.
  An inertial frame of reference is defined as a frame in which non-accelerated objects move in straight lines at constant velocity (i.e. at constant speed and in a constant direction). Notice also that c is short for celeritas, the Latin word for speed.
  The assumption of the constancy of the speed of light was based on a large number of astronomical measurements, particularly in the 19th century, which left little doubt about its reality. The history of such measurements is fascinating. See for example A Case History in Astronomy and Physics: The Speed of Light .
  The amazing consequence of these two assumptions was that it was possible to derive, without any further assumptions, using only some rather elementary mathematics, a complete theory of motion. This theory differed in some fundamental and surprising ways from Newton's. Before summarizing the major features of the theory of special relativity, it is important to emphasize the basic difference between Einstein's assumptions and Newton's.
  Postulate 1 is identical in Newton and in Einstein (in fact Galileo (1564 - 1642) had already considered it). You can verify it, for example, by comparing the results of simple experiments carried out on a train moving at constant speed and in a constant direction, with similar ones carried out on land, near the train's tracks.
  Disagreement arises with regard to Postulate 2. In practice if not in theory, Newton assumed the speed of light to be infinite, while Einstein accepted the empirical evidence that it was finite. In fact it is possible to show that if one replaces the value c = 299,792,458 km/s in Einstein's theory with infinity, the special theory of relativity reduces to Newton's theory. Notice also that the speed of light was first measured in 1676 by the Danish astronomer Ole Christensen Rømer (1644 - 1710), using careful observations of the motions of Jupiter's moons [ see, for example, Ole Rømer ]. It would take more than two centuries before Einstein realized the fundamental importance of such a result.
• Here, quite concisely, are the major consequences of the special theory of relativity:
  1. Two events, simultaneous by the clock in one inertial frame of reference, in general are not simultaneous by the clocks of other inertial frames in motion relative to the first one.
  2. Given two identical, synchronized clocks in motion relative to each other, each clock will be observed (measured) to be slow by an observer moving with the other clock.
  3. The length of an object (say, a rod) at rest in an inertial frame is measured as shorter by an observer in another inertial frame moving parallel to the object (the rod).
  4. For an observer at rest in an inertial frame, the mass of a moving object increases with its speed.
  5. Mass and energy are two forms of the same substance, and can be transformed, under suitable conditions, into each other: E = mc².
  Notice that each of these statements has been verified experimentally over and over again. Why is it then that our daily life seems not to have been affected by relativity, and that the consequences described above are nowhere to be seen around us? The answer is simple and revealing: the speed of light is so great that, in the context of our daily experiences, it is practically infinite. The speeds which are experientially meaningful to us are puny in comparison. That's why we can not really say that Newton was wrong. Think: when we are told that a spaceship is traveling towards, say, Mars, at some 60,000 km/hr, we must remember that that means approximately 1.64km/s. That's only 1/18,000th the speed of light! Relativistic effects become noticeable when we have to deal with speeds which are substantially comparable to that of light.
  At the same time it is important to realize that to extrapolate from our earthly experience to the universe at large is a risky business. We may be wrong—as Einstein has shown us. Only in this sense was Newton 'wrong.'
• There is one important feature that both Newtonian mechanics and Special Relativity have in common: the mathematical equations describing motion are time-reversible. Roughly, this means that they are not sensitive to the direction of time. That's rather strange, since our everyday experience amply demonstrates that time runs only one way—towards the future. Specifically, by 'future' I mean here the tendency of all closed or isolated systems to become more and more disordered (the so-called second law of thermodynamics ). Equivalently, one can say that usable energy can only decrease in isolated systems. This situation is often claimed to apply to the universe as a whole, but that's a questionable hypothesis (why?).
  To explore this issue, however, lies well outside the the scope of this course. I will simply note that the inability of dynamical equations to account for time irreversibility has preoccupied scientists for a long time, and that no satisfactory solution has been found yet. I will mention here the work of Nobel Prize Ilya Prigogine (1917 - 2003) and his collaborators, who made important contributions that may lead to a solution. Visit the   Educational Features at the Ilya Prigogine Center for Studies in Statistical Mechanics and Complex Systems.

  "Until the second half of the 19th century the ideal of the natural sciences were the laws of dynamics, which are symmetrical in respect of time (for a dynamic system, the past and the future are absolutely equivalent). The Second Law of Thermodynamics, discovered more than a hundred years ago, created some confusion in the minds of scholars by proclaiming that, left to itself, a system will tend from order towards chaos, this process being irreversible, i.e. 'time-oriented.' But the 'arrow of time' and the concept of chaos did not become fully incorporated in the natural and social sciences until the second half of the century that has just ended. And their incorporation was wrought first and foremost by the works of Ilya Prigogine, who showed that time-orientation is a fundamental property of all natural systems (physical, chemical, biological and social) and that the 'natural tendency' towards chaos by no means entails loss of harmony. Prigogine succeeded in explaining in the language of mathematics that chaos can be constructive—that it is precisely chaos that gives birth to the new order."
  [ from The Rediscovery of Time ]

  We will deal with chaos in another lecture. What is important here is to note that Prigogine showed that time may be, not a general property of the universe, but a property of each individual system. In other words, each system is the origin of its own time, and that the basic behavior of each system is governed by dynamical laws which are not time-irreversible.
• What about time in quantum mechanics? We can barely scratch the surface in this course. Here is a few important points:

  Time measurements are defined by the type of clocks used:

  • There is always some uncertainty due to the very act of measurement, but in the case of quantum clocks there is additional uncertainty.
  • The time measured by clocks may depend on the physical situation, and there is no guarantee that the clock is not influenced by some nearby field.
  • Measurements may depend on the path of the clock, not only on the start and end points.

  Notice that, despite the above qualifications, the equations of the standard theory of quantum mechanics are also time-reversible, and in that sense they too are 'unrealistic,' just like Newton's and Einstein's equations. We are still awaiting a theory that embodies the irreversibility we so easily observe in our everyday world.
The success of Einstein's theory of special relativity and of the general theory of relativity (which lies outside the scope of this course) points to a new conception of space and time. What Einstein showed is that the really meaningful concept is that of spacetime (also spelled space-time or space time). Space and time are inextricably woven together. We can not speak of one without using the other. For example, within the framework of special relativity, the measurement of speed (the rate of displacement in space as a function of time) requires both clocks and rulers. But the behavior of these depends on the state of motion of the observer relative to what he is measuring. For a very clear, fairly elementary, and nicely illustrated explanation, visit
  Spacetime 101

 
Readings, Resources and Questions

• A good resource for horology, the science of time measurement, can be found in the Internet Public Library.
• A very helpful and entertaining illustration (also in a literal sense) of the special theory of relativity was published sometime before 1966 in an issue of Time Magazine, under the title The Great Relativity Bomb Plot. It was reprinted in 1966 as part of a Time-Life book by S A Goudsmit entitled Time. (Toronto Public Libraries have copies). You can find a scanned copy at   The Great Relativity Bomb Plot (click on each section to see the corresponding scan of the original, illustrated pages).
  Visit Han Erim's Alice Law, where you can download a free program that will teach you, in a very original and interesting way, the basics of Special and General Relativity.
• The number of books on the nature of space and time, on the evolution of our ideas about space and time, and current research trends, is enormous. I will mention here only a few, accessible ones. Notice thay they all seem to be about 'time' only. In fact, because time is really part of 'spacetime,' all these books also deal with 'space.'
  • A Einstein and L Infeld, The Evolution of Physics: From Early Concepts to Relativity and Quanta. Simon & Schuster, NY (1938), 1966.
  • A Einstein, Relativity. H Holt & Co, NY 1920.
  • S Hawking, A Brief History of Time: From the Big Bang to Black Holes. Bantam, NY 1988.
  • C A Pickover, Time: A Traveler's Guide. Oxford UP, NY 1998.
  • P Coveney and R Highfield, The Arrow of Time: A Voyage Through Science to Solve Time's Greatest Mystery. Fawcett, NY 1990.
  • B Russell, The ABC of Relativity. Unwin, London 1958.
  • G Smoot and K Davidson, Wrinkles in Time. W Morrow & Co., NY 1993.