Whatever else may be said about time, one thing is certain. It defies definition. The best we can
say is that we all know what time is, intuitively. For it cannot be expressed in terms of other
things, though some try. The Seventh Edition of Webster's Collegiate Dictionary tells us that time
is "the measured or measurable period during which an action, process, or condition exists or
continues." Of course, what the lexicographer has done here is to tell us that time is defined by its
measurement and that that measurement is of a period during which something occurs. He has
not told us what time *really* is.

I can do no better. And so I begin this article with a description of how we measure time, without
knowing exactly what it is we measure. But in examining the measuring process, I hope we
become aware of some of the subtle properties of time. Later I digress a bit by considering "time
travel", and finally I launch into the most puzzling aspect of time, that the passage of time, (and
here I mean *physical* time, not merely our bodily sense of its passage) is *relative*.

## A. |
## Measuring Time. |

The universe is in a continual process of change and evolution. These changes are brought about
by the motion of matter, a phenomenon whose description requires the introduction of the
concept of * time*.

Many changes are periodic. The positions of bodies constituting a given system undergo changes relative to one another in such a way that the configurations of the bodies are successively repeated. Planets orbit the sun; the earth rotates on its axis; pendula swing; clockworks tick; tuning forks vibrate; hearts beat. In each of these phenomena, the system involved passes continuously through changes in its configurations that reoccur repetitively.

A comparison of the periodic changes in one system with those in others show that this
periodicity is not *capricious*. A given tuning fork executes a certain fixed number of vibrations
while a given pendulum completes one oscillation. A given pendulum completes a certain fixed
number of oscillations while the earth rotates once on its axis relative to the stars. Such periodic
phenomena may be compared. Each comparison yields fixed ratios of the number of cycles of
repetition of the phenomena. A simple pendulum whose length is one meter, for example, always
completes 43,047 oscillations while the earth rotates once on its axis.

Not all such ratios are fixed. When the number of cycles made by a beating heart is compared
with those of the pendulum, the ratio of the number of cycles varies. However, many phenomena
exist for which the ratio is constant within our ability to measure, and so these phenomena are
used for the * measure of time*.

From the set of periodic phenomena with constant ratios, one is chosen to be a standard. Time is
measured by counting complete cycles of the standard, each cycle being one unit of time. We call
the device which undergoes these cyclic changes a * clock*. Other clocks may be constructed from
periodically recurring phenomena whose cyclic rates remain always in a constant ratio to that of
the standard.

## B. |
## Philosophic Digressions. |

1. |
On the Regularity of the Flow of Time. |

There are three foundation blocks to our understanding of the physical world. The first of these is
the topic under discussion, * time*. The second is

The remarkable characteristic of time, the one that allows us to comprehend the universe in the
organized fashion in which we do, is its *regularity*. If the periodic return of a system to a
previous orientation bore no constant relation to the periodic changes in another system - that is,
if all "clocks" were no better than beating hearts, and if this *irregularity* were a fundamental
property of nature and not due merely to our inability to build good clocks, then it would not be
possible to measure time, and time as a physical entity would not be a useful concept.

In his famous *Principia*, Newton recognized the need in his physics for a *universal flow of time*,
impervious to everything. It is a concept which we take for granted and seem to perceive
intuitively. The time which flows by us, bringing the future into the present and pushing the
present into the past, flows at the same "rate" for all things. It is the uniformity of this rate which
enables us to write "equations of motion" that describe how the positions of objects change as a
function of values given to this *single* entity, time.

However it is a matter of experimentally proven fact that this seemingly universal flow of time does not exist. Lapses of time, as they are measured by the recurrence of periodic events, are not impervious to everything but rather depend upon the relative motion of the two systems whose periodicities are being compared and the positions of the systems in a gravitational field. The discovery of this dependence is recent. It is a logical prediction of the general theory of relativity, whose formulation was completed by Albert Einstein in 1916, and it has since been confirmed experimentally in laboratories and in astronomical observations.

Nevertheless, the *regularity* of the flow of time continues to hold for two "clocks" which are not
in relative motion and which are located at the same place in a gravitational field. And in
instances in which the clocks are in relative motion or are in different positions in a gravitational
field, the deviation in the regularity of their rate of time-keeping is subject to known mathematical
law. This dependence is the subject of the next section of this article.

Thus, even in the relativistic realm, the flow of time is not capricious. Its rate may not be the
same in all corners of the universe or under all circumstances, but that rate is governed by natural
law. This characteristic of time continues to be a foundation of our description of the physical
world. One wonders how a universe might behave in which the characteristic is absent.

2. |
On Time Travel. |

It is amusing and perhaps even instructive to consider here the meaning and possibility of "time travel." Could, for example, a person travel backward in time?

To say that a person travels back in time generally means that events in the past become that person's present. Both the mind and body of the person are unchanged, and the person is able to recollect the intervening life he experienced between his former present (in the "present") and his current present (in the "past"). Only the surroundings of the person have reverted to a previous time, with a loss of any awareness of its future. And yet time itself has not been displaced. For had it been, then even the person's body and mind would revert to a previous state, and he would be unaware that any displacement had occurred. Indeed, if it were claimed that such total displacements of time happen, the claim must be specious, since nothing could detect the displacement. Thus "time travel", to have any meaning, must allow the person to remain in his present, with all his memories intact, while the rest of the universe reverts to a configuration it had in the past.

In certain sense, such is possible in the small. Suppose a perfect periodic system is constructed, the perfect pendulum say, in which the configurations through which the pendulum passes in a single cycle are cyclically repeated without any variation whatsoever. A person observing the oscillation would have no way of distinguishing one oscillation from another and thus could well claim that the pendulum was in a "temporal loop," continually returning to a previous time by returning to a previous spatial configuration.

The question of time travel to the past or future is then a question of returning or advancing the
configuration of all entities in the universe to a past or future configuration without affecting the
configuration of the person doing the traveling. That such a sudden change in the configuration
of any system, much less the entire universe, seems impossible, for changes in the configuration of
a system occur in a continuous fashion, not in macroscopic jumps. (It is true, however, that
*quantum mechanics* asserts that changes in say atomic systems occur in "microscopic" jumps.)

There is a second way time travel might be accomplished. Rather than suddenly change the configuration of the universe to a past or future condition, one might change the configuration of one's own body. For example, if a person's body were to assume the state it had a year ago, he would think he had been suddenly projected forward a year when he looks at the world about him. Similarly if a person's body were to assume the state it would have in a year (along with all the memories of things yet to happen during that year), he would think he'd been projected back a year in time.

However, in both these methods, we retain the sense of a continual forward progression of time. It is not time which is altered but the state of matter. In a third conjecture, we require time itself to make leaps forward or backward. To do so, we imagine that the evolution of the universe is like the evolution of a motion picture. The events and characters in the film experience each frame as the present moment and have a memory or recording only of earlier frames. Nevertheless, all frames (i.e., the real universe) exist simultaneously in an external "super-universe" in which the real one is embedded.

If our universe were so constructed, it is conceivable that a person could, in a sense, jump forward or backward between frames using that mechanism favored by science-fiction, a spacetime warp, or a wormhole, or some other device which sunders the fabric of the spacetime continuum and allows such passage via the external super-universe.

This suggestion creates two possibilities. The first is that there is only one "film" for the universe. In this instance, if a person jumps forward or backward in time, that action must be preserved on the film to start with. Thus if a person returns to the past and appears before himself at an earlier age, then when he lived at that earlier age, he must have met his older self. He is doomed to continue the loop indefinitely. The second possibility is that the act of returning to the past generates a new film which branches from the old and allows for a different future.

Of course, there is nothing in our experience which remotely suggests that these science-fiction
conjectures have occurred or could occur. They are offered only to tickle our imaginations.

## C. |
## On the Behavior of Clocks |

1. |
Qualitative Discussion. |

We shall assume that we have a collection of perfect clocks. These are clocks which keep time at the same rate as does the standard clock, and are always in phase. In order to test for this perfection, it is necessary that the clocks be infinitely close to one another, preferably superimposed on one another, and that they be at rest with respect to one another so that the condition of superposition is maintained. For if they are separated, it is necessary that a means be found by which they can communicate with one another (or with an observer) to allow for a comparison of their rates. Any method of communication requires a lapse of time. Thus if clock A is separated from clock B, it is not possible for either clock (or an external observer) to be instantly aware of the reading of the other. Some delay is unavoidable.

For example, if we look at a clock across the room, we do not see the reading on its face as it is at the instant we perceive the face. The light that is reflected from the clock and into our eyes takes time to travel the distance from clock to eye, and so we see the clock as it appeared earlier in time. The effect is much more evident (at least intellectually so) when we observe the night sky. Astronomical measures indicate that the stars are so far away that many years elapse in the time the star emits light and time moment at which that light enters our eye. The more distant the star, the earlier in time we see it. Yet our minds, unaware of the lapse, are fooled into thinking we see the starry heavens as it exists at this instant. But we can see nothing as it exists at this instant. Indeed, as will be shown later, "at this instant" is dependent upon the "observer" and is not an absolute moment in time.

As a further condition, we require that the geometry of the space in which these clocks lie huddled together in relative rest be Euclidean. Exactly what we mean by a Euclidean space is discussed in another article, one devoted to the properties of space. It is assumed here that the reader understands generally what is meant, namely the space in which he or she lives, or at least a close approximation to it.

Out of this collection of clocks, one, say clock A, is moved away from the group. It is caused to
travel in an arbitrarily circuitous path, returning to the group at a later time. What is remarkable
about this motion is that on its return clock A no longer reads what the collection reads. Rather it
has recorded a lapse of time which is *less* than the time universally recorded by the others.
Admittedly, this loss in time is minuscule, so minuscule that only an extremely sensitive and
accurate atomic clock is precise enough to detect it. Nevertheless, it occurs and has been
measured to occur.

Although the experiment has been performed only with atomic clocks, theory states that this
temporal retardation has nothing to do with the clock itself but rather with the nature of time.
Any object, say even a person, traveling with the clock, would experience the same retardation.
That is, this person would not have aged quite as much as he would have had he remained at rest
in the collection, and his awareness of the lapse of time would be less than it is for a person who
does remain with the collection. Again, for ordinary travels, the retardation is far too small to be
perceived by ordinary senses. It is only if the circuitous journey be quite extensive and completed
at nearly the speed of light would a discernable slowing of physical processes be evident to human
senses.

2. |
Quantitative Discussion. |

a. |
On Mathematical Physics. |

Acceptable physical theories must be fashioned around measurement. Quack theories have as a
common character the introduction of strange entities, undefined in terms of measurable
quantities, and without the ability to predict the outcome of events in measurable terms.
Measurement itself involves *number*, and a theory involving measurement must necessarily be
concerned with relationships among numbers, that is, with *mathematics*.

It is unfortunate that the mathematical nature of the physical world generates such apprehension
in many of those who would hope to understand it. There is really nothing to fear, for most of us
who are reasonably rational have no trouble with *counting*, and counting is all that mathematics is
about. However, the *language* of mathematical expression is apt to be intimidating. Disciplines
like algebra and trigonometry, calculus and differential geometry, can be terrifying not because of
our fear of numbers but because of the arcane symbolism in which these disciplines are expressed.

While I have refrained from the use of numbers so far, I now must resort to them, for without them, I cannot describe how we arrive at the theoretical prediction of the outcome of experiments. I shall not shrink from standard mathematical symbolism, advanced though it might be. If it is mysterious to the reader, I trust he or she will skip over it, accept its deductions, and continue on with the narrative. At the end of each mathematical discussion, I shall illustrate its conclusions with a numerical thought experiment to illustrate the point of the mathematical development.

Finally, because of the limitations of hypertext, I shall express the square root of *x* by sqrt(*x*).

b. |
Proper Speed. |

It is necessary, in developing formulas which describe the lapse of time on clocks which move, to
express that motion in precise terms. Such is accomplished by first introducing a *spatial frame of
reference*, by which we mean a space in which the location of fixed points (i.e. positions or
locations) can be expressed numerically, and in which the *shortest distance* between these points
can be measured. We symbolize that distance by *s*. If the distance between two points is
infinitesimally small, we symbolize it by *ds*.

We next allow perfect clock A to move from one point *P* to another point *Q* infinitesimally close
to the first point. During this motion A travels through the distance *ds, *a* *quantity that may be
leisurely measured using a measuring rod. The duration of this motion, *as measured by A*, is *dt*_{a}
The proper speed of A is called *u* and is defined by

(1) | u = ds/dt_{a}. |

For example, if the distance between *P* and *Q* is 10^{-9} m, and the time in which clock A traverses
this distance is 10^{-10} s, as measured by itself, then the proper speed of A is 10 m/s.

c. |
The Retardation of Time |

Let clock B be one of the clocks at rest in the collection of clocks, at rest in the Euclidean frame,
and let clock A start moving at a constant speed *u* relative to that frame and hence relative to the
collection. Of course, since A is initially at rest, it must accelerate from a speed of zero to the
speed u, and so its speed is technically not constant during its motion. However, we assume that
the acceleration is nearly instantaneous so that the time spent in accelerating is virtually zero.

Clock A travels away from the collection along some arbitrary path, say a very large loop, that
eventually brings it back to the collection. During its motion, it has recorded a lapse of time of *t*_{a}.
Correspondingly clock B has recorded a lapse of time of *t*_{b}. That is, the hands of clock A have
advanced through a distance indicating a temporal measure of *t*_{a} and the hands of clock B have
advanced through a distance indicating a temporal measure of *t*_{b}. These two measures of a lapse
of time are not the same. Rather *t*_{b} is greater than *t*_{a} by a factor which depends upon the constant
speed *u*. The relationship as deduced from theory is that

(2) | t_{b} = t_{a}sqrt(1+u^{2}/c^{2}). |

Here *c* is a speed very nearly equal to 3x10^{8} m/s and represents the speed of light.

In order to give an example of the use of this equation under circumstances which are amenable to
imagination, we reset the value of *c* to 10 m/s. With this new value, we consider a runner who
runs around a 400-meter track at a steady space, completing one circuit in 100 seconds measured
by the watch he carries with him. That is, the time *t*_{a} in Eq. (2) is 100 seconds and his proper
speed *u* is 4 m/s. His coach stands beside the track, at rest relative to the track at the runner's
starting position. The coach starts his watch at the moment the runner leaves and measures the
lapse of time to be *t*_{b}, which, by Eq. (2) is 107.70 s.

Specifically, both watches read zero when the runner starts out next to the coach. When the runner returns to the coach, his watch reads 100 s and the coach's watch reads 107.70 seconds. The runner has aged 100 seconds while running, and in the same interval, the coach has aged 107.70 seconds. The lapse of physical time on the runner is 100 seconds and the corresponding lapse of physical time on the coach is 107.70 seconds.

Which of these times, if either, is the "true" lapse of time? Both are "true." Neither is to be preferred over the other.

Why is the lapse of physical time at the coach greater than that at the runner? That is, why is Eq. (2) true? We don't know. However, as we shall see later, Eq. (2) arises because,.loosely speaking, the speed of light is the same for all observers. One could just as well ask, why is the speed of light the same for all observers? The answer remains: We don't know.

Actually, the speed of the runner is not steady and so Eq. (2) must be adjusted to compensate for
this variation. Such is done through the integral calculus. Suffice it to say that the calculus
enables us to enter an *average *proper speed *u* in Eq. (2) which yields the value of *t*_{b} when *t*_{a} is
given.

d. |
Coordinate Speed. |

Using the lapse of time measured by the coach for the run, namely 107.70 seconds, and the
distance covered by the runner, namely 400 meters, we evaluate the average speed of the runner
from the point of the coach to be (400 m/107.7 s), or 3.71 m/s. This speed is called the
*coordinate speed* of the runner. Its general definition is that *coordinate speed**v* is the ratio of the
distance *ds* traveled by an object to the corresponding lapse of time *dt _{}*b as measured by a clock at
rest in the frame in which the distance is measured. In equation form, this definition is written

(3) | v = ds/dt_{b}. |

## D. |
## Setting Separated Clocks. |

We here consider the problem of setting clocks which are separated from one another but which are at rest relative to one another, both of them being at rest in the same spatial Euclidean frame. We want the clocks to "read the same time," an expression which is fraught with difficulty, as we shall soon see. For specificity we call one of these clocks B and the other C.

First we establish that B and C are good clocks. They remain in phase with the standard clock
when compared to the standard in close proximity to it. Next we place B and C at a point
halfway between the positions at which they are to be located. With their readings the same, we
then move them apart with exactly the same proper speed, and they travel over the same distance.
When they arrive at their destinations, each makes a record of the lapse of time each has taken for
the journey. These records are later compared, and it is found (not surprisingly in this thought
experiment) that each has taken the same lapse of time to make the journey. That is, at the
moment B and C arrive at their destinations, they read the same time according to the record.
Next C sends an intermittent signal to B, say a flash of light or a burst of sound every second as
measured by C. It is found that B receives the signal at the same rate, namely one pulse per
second. By this criterion, clocks B and C keep time at the same rate and may be said to be
*synchronized*.

The stage is now set to test the applicability of Eq. (2) when clock A moves from B to C. Again, to avoid the mathematical complication of a variable proper speed, we require A to move at a constant proper speed from B to C. It is found that Eq. (2) applies provided it is written as

(4) | t_{c}-t_{b} = Dt_{a}sqrt(1+u^{2}/c^{2}), |

where *t _{}*b is the reading of B when A leaves B,

It is important here to recognize that the symbol *t* for time (like the word *time* itself) takes on two
different roles here. When referring to A, it symbolizes a *lapse *of time. When referring to B and
C, it symbolizes a moment in time. An attempt is made here to symbolize this difference by letting
*t* alone represent a moment in time and *Dt* a lapse of physical time

The difference in times, *t*_{c}-*t _{}*b, is not a lapse of physical time, since it is merely the difference in the
readings of two separated clocks. Physical time elapses only on objects themselves. It is a

## E. |
## The Speed of Light. |

There is a circumstance in which the difference in synchronized clock readings plays a physically significant role. This circumstance occurs in the measurement of the speed of light.

We imagine a pulse of light passes by B at the moment B reads zero. It arrives at C (which is
synchronized with B) at the moment C reads* t*. This *t* is also the difference in the readings of
clocks B and C since the reading of B is zero when the light passes by it. Letting s be the distance
between B and C, we can write that the "speed" of light is *s/t*. Since *t* is not actually a lapse of
physical time but is merely the difference in the reading of clocks B and C, it is not clear that this
speed is of any physical significance. Nevertheless, when this experiment is performed, it is found
that the ratio *s/t*, always has the same value, namely *c*. That is

(5) | s/t = c, |
or | s = ct. |

The value of this ratio *c* is independent of anything external to the experiment. It is independent
of the motion of the earth around the sun. It is independent of the motion of the laboratory
relative to the earth. It is independent of the source of the light, whether it be fixed on B, or fixed
on some object moving relative to B, or very remote from B. The only experimental condition
imposed is that the light, whatever its source, pass by B on its way to C and that it travel over the
distance *s*.

This character of light is one of the foundations of the theory of relativity, but we shall leave it here since its further examination requires that we examine first the properties of space, something we shall do in the next article.

There is, however, one important conclusion we can offer regarding the character of light. To arrive at this conclusion, we return to Eqs. (1) and (4), which we rewrite here:

(1) | u = ds/dt_{a}. |

(4) | t_{c}-t_{b} = Dt_{a}sqrt(1+u^{2}/c^{2}), |

In the thought experiment described above, we first imagine light to take on a corpuscular form,
which we call the photon.. The proper speed of light *u* would be the ratio of the distance *s* to the
lapse of time *Dt*_{a} on the photon as it travels from B to C over the distance *s*. In the thought
experiment *t*_{b} = 0 and *t*_{c} = *t*. Thus we write Eq. (1) as

(6) | u = s/Dt_{a} |

and Eq. (4) as

(7) | t^{2} = Dt_{a}^{2}(1+(s^{2}/Dt_{a}^{2})/c^{2}). |

Multiplying Eq. (7) by *c ^{}*2 and expanding the parenthesized terms, we see that Eq. (7) becomes

(8) | c^{2}t^{2} = Dt_{a}^{2}+s^{2}. |

However, because we have discovered that *ct = s* in the thought experiment described above, we
see that Eq. (8) becomes

(9) | s2 = ^{}Dt_{a}^{2}+s2.^{} |

That is,* Dt*_{a} must be zero! There can be no lapse of physical time on a photon. The proper speed
of light is infinite! Light created in a distant galaxy millions of light years from us and striking the
earth, travels those millions of light years instantly in its own time. To a photon, the universe and
the photon exist, but only for a moment of no duration.