Clock Motion and Time Dilation

9 Apr 2008

When time is named in natural science, particularly physics, what is meant is a standard motion or a probabilistic tendency, and most physicists have the grace to eschew definitions of time. . . . when it is spoken of in anthropology, for example, when the Mayan "concept of time" is presented, the object of study is an intricate and significance-bearing calendrical system, and I have not come across the claim that the Mayans even had a separate word for the notion of time . . . they seem to have used the word for "sun".

Eva Brann, What, Then, Is Time?


I will describe a phenomenon involving imaginary motions of two ordinary circular mechanical clocks along a designated imaginary path, and a way of measuring angles between hands of the clocks as they travel between two positions along the path. Measuring the angles will involve angular units which will be renamed in order to make definitions of the term time and some associated terms, using only the angular units and units of distance. In this way, I will show how, using these derived names, a phenomenon can be predicted which is similar to one predicted by special relativity theory, known as time dilation, involving slowing down of time as measured by clocks with speeds which get closer and closer to the speed of light in a vacuum. My point is not to show something about theories of relativity, but to illustrate by way of an example that the term time, like many other terms, can be used in different useful ways. It appears to me that instead of trying to answer the question "What is time?," it would be more useful and rewarding to try to answer the question "What are times?," so that grammar wonít misguide a questioner into trying to find some fundamental and useful time from which all useful times can be consistently derived.


Two clocks moving the same distance along the same path.

Start with a standard circular clock run by a mechanism intended to make its long hand track movements of the sun or perhaps other celestial objects as observed and measured from some position on the earth. Such clocks may be considered to be descendants of sundials, with their gnomons and shadows. The circumference of the dial of such clock will be marked in such a way that it is divided into 60 arcs of equal length. It will also be marked in such a way that each successive 5 of these arcs are labeled with the integers 1 through 12, in the usual way of circular clocks. This divides the circumference into 12 arcs of equal length, the endpoints of which are labeled with these integers.

Suppose we have two such clocks C1 and C2 which are next to each other at this position on earth, and are synchronized in such a way that in each clock the long and short hands are pointing to the position labeled 12 on their faces. Suppose further that C1 is projected in a vacuum along a path Q1 from a position P1 on the path in such a way that it travels s1 = 1.5*107 meters from P1 to a position P2 on the path during some one revolution of its long hand. At the end of its journey, its long hand points to the 12 and its short hand points to the position on its face labeled 1. Suppose also that C2 is projected from P1 to a position P3 on a path Q2 close to Q1 in such a way that it travels

s2 = (2/3)s1 = 1.0 * 107 meters for some one revolution of its long hand, so that its short hand then points to the position marked 1 on its face. Then C2 passes a position P3 on the path which is as far from P1 as 2/3 of the distance C1 traveled from P1 to get to P2.

Suppose C2, without pausing, travels an additional (1/2)[(1.0) * 107 = (0.5) * 107 meters. Then, since (1.0 + 0.5) * 107 meters = 1.5 * 107 meters, so that it has reached the position P2 on the path that C1 got to after one revolution of its long hand. During the second part of its journey C2ís long hand makes an additional 1/2 of a revolution, so that at the end of its journey, C2ís long hand has rotated 1.5 revolutions (or 3π radians, or 540˚) in traveling the same distance as C1 did while its long hand rotated 1 revolution (or 2π radians, or 360˚). At position P2, the long hand of C2 again points to 12, but its short hand points to the 6 on its face.

Suppose that when the clocks reach the position P2 from P1, there is a device that reports the positions of the long and short hands of each clock, and also the number of revolutions each clock has made in reaching P2. The report will show that C1 made 1 revolution, its long hand was pointing to the 0 position, and the short hand to 1. It will also show that C2 made 1.5 revolutions, its long hand was pointing to the position on the circumference marked 6, and its short hand was pointing to a position halfway along the piece of arc of the circumference of its dial between 1 and 2.

Now introduce some other names for the units used in measuring these distances and rotations, the rotations having been measured, say, with units of angular change between the hands of the clocks, using radians. Call each revolution of a long hand an hour, and each of the 60 arcs marked off on a circumference a minute. Also define a second to be 1/60 of a minute. Then the report can be worded this way: In traveling a distance of s1 = 1.5*107 meters in one hour, or 250,000 meters in one second, clock C1 shows that 1 hour has elapsed, while in traveling this distance clock C2 shows that 1.5 hours have elapsed.

Also introduce the term average speed to stand for "distance traveled per revolution", i.e., using our terms hours and seconds, speed is "distance per hour" or "distance per second". Then we can say that the average speed of clock C1 was v1 = 250,000 meters/second, while the average speed of C2 has been slightly more than v2 = (2/3)v1 = 166,666 meters/second. We can say that it took C2 1.5 hours to travel the same distance that C1 traveled in 1 hour. I am italicizing the word "say" in order to emphasize that if we use the term "hour" in the way I have specified, we will be using, under another name, a measure of angles determined in effect by rotations of radii of circles amounting to numbers of revolutions, which may also be expressed using radians, or degrees. Furthermore, I assumed that clocks C1 and C2 were synchronized to start with, and that they have very much the same kinds of mechanism, which caused their hands to move in a way that maps certain astronomical observations.

Now introduce a further term time to designate the number of revolutions the long hands of clocks C1 and C2 make in their journeys between positions P1 and P2 on the paths Q1 and Q2 which are very close to being the same path. Since for C1 this number is 1 and for C2 is 1.5, we can say that it takes more time for C2 to make the journey than it does for C1 to make the journey. This suggests that we can say that the long hand of C1 to have moved slower than the long hand of C2 during their journeys over the path from P1 to P2, where saying that one of the long hands moves slower than the other is another way of saying that it takes less time for C1 than for C2 to move from P1 to P2, i.e. the number of revolutions of C1ís long hand is smaller than the number of revolutions of C1ís long hand. A more colorful way of expressing this would be to say that C2ís time for traveling between P1 and P2 was dilated with respect to C1ís time, i.e. C2ís time for the journey was bigger than C1ís time for much the same journey.

I take it that the time dilation I have described here is not the time dilation of relativity theory. My time dilation occurs without regard to constant speeds along straight line paths, or considerations of inertia. The behavior of light has not been used in my derivation, although such behavior is fundamentally involved in the time dilation of relativity theory.

I note in passing that one might want to rename what I have been calling the long and short hands of the clocks I referred to in my derivation the minute hand and hour hand. However, I note also that I didnít do this before I introduced my definitions of minute and hour, since this might easily have led me to commit logical circularity, or at least to confusion for someone else reading this. My definitions of the long hand of one of my clocks would be that it makes 1 revolution and the short hand makes 1/12 of a revolution when certain astronomical movements between celestial objects are measured, using only measures of angles and units of distance.

This completes the example illustrating how the term time dilation can be introduced on the basis of a certain definition of time, using only angular measures and units of distance, although it also can be expressed using different names for these measures and units which are in use in connection with a different definition of time (or perhaps several definitions which are different from mine and from each other).

I have described here, by way of a generalizable example, a way of referring to rates of change of distance using perceptible standard motions of the hands of a common kind of clock. Such a clock may be said to measure distances traversed along arcs of a circle, as determined by angles between radii. On the other hand, astronomers still use a procedure of the kind presented here when they define right ascension, setting an hour to be 15˚ of arc along an imaginary circle (the hour circle) on an imaginary celestial sphere, one of whose great circles is a projection of the earthís equator (the celestial equator) and another of whose great circles, the ecliptic, traces out the apparent path of the sun as viewed from a position on earth). Geographers also do something like this when they use longitude and latitude on our earth, considered, say, as an imaginary sphere, and it has been known for some time that measures of some kind of time can be used to determine longitudes.

In fact, there are a multitude of different uses of the term time. For example:

(1) In 1687, Isaac Newton said about time:

Hitherto I have laid down the definitions of such words as are less known, and explained the sense in which I would have them to be understood in the following discourse. I do not define time, space, place and motion, as being well known to all. Only I must observe, that the vulgar

conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices, for the removing of which, it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common.

"I. Absolute, true, and mathematical time, of itself, and from its own nature flows equably without regard 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.

Newtonís grammar suggests that what he calls absolute time be regarded as a kind of non-material entity, known to everyone, which "flows equably". He doesnít talk about what this entity might be made of, nor where might be flowing in some steady way.

(2) In 1781, Immanuel Kant said about time:

Time is not a discursive, or what is called a general concept, but a pure form of sensible intuition. Different times are but parts of one and the same time; and the representation which can be given only through a single object is intuition.

(3) In 1876, James Clerk Maxwell said about time:

The idea of Time in its most primitive form is probably the recognition of an order of sequence in our states of consciousness. If my memory were perfect, I might be able to refer every event within my own experience to its proper place in a chronological series. But it would be difficult, if not impossible, for me to compare the interval between one pair of events and

that between another pair By our intercourse with other persons, and by our experience of natural processes which go on in a uniform or a rhythmical manner, we come to recognise the possibility of arranging a system of chronology in which all events whatever, whether relating to ourselves or to others, must find their places. . . . . .

Absolute, true, and mathematical Time is conceived by Newton as flowing at a constant rate, unaffected by the speed or slowness of the motions of material things. It is also called Duration. Relative, apparent, and common time is duration as estimated by the motion of bodies, as by days, months, and years. These measures of time may be regarded as provisional, for the progress of astronomy has taught us to measure the inequality in the lengths of days, months, and years, and thereby to reduce the apparent time to a more uniform scale, called Mean Solar Time.

Maxwellís "idea of Time" gives time somewhere to flow, namely in the consciousness and memories of a person, and alludes to how calendars can be constructed which allow events to be ordered. This evokes thoughts about how what may be called historical time, and about how it is that the terms past, present and future may function in connection with time, where the term time may be used in different ways by different people in talking about this.

(4) In 1916, Albert Einstein said about time:

We are thus led also to a definition of " time " in physics. For this purpose we suppose that clocks of identical construction are placed at the points A, B and C of the railway line (co-ordinate system) and that they are set in such a manner that the positions of their pointers are

simultaneously (in the above sense) the same. Under these conditions we understand by the

" time " of an event the reading (position of the hands) of that one of these clocks which is

in the immediate vicinity (in space) of the event. In this manner a time-value is associated

with every event which is essentially capable of observation.

There is a certain resemblance here between Einsteinís proposed use of the term time, and the use I have made here of the term. However, when Einstein used the term simultaneously in this passage he followed it with the parenthetical remark "in the above sense", which refers a reader back to a discussion by Einstein about how he will use the term simultaneous, a discussion not to be discussed here. Suffice it to say here that light plays a fundamental role in the way Einstein defines the term simultaneity.

(5) About 350 B.C., Aristotle said about time:

Not only do we measure the movement by the time, but also the time by the movement, because they define each other. The time marks the movement, since it is its number, and the movement the time. We describe the time as much or little, measuring it by the movement, just as we know the number by what is numbered, e.g. the number of the horses by one horse as the unit. For we know how many horses there are by the use of the number; and again by using the one horse as unit we know the number of the horses itself. So it is with the time and the movement; for we measure the movement by the time and vice versa. It is natural that this should happen; for the movement goes with the distance and the time with the movement, because they are quanta and continuous and divisible. The movement has these attributes because the distance is of this nature, and the time has them because of the movement. And we measure both the distance by the movement and the movement by the distance; for we say that the road is long, if the journey is long, and we say that the journey is long, if the road is long Ė the time [is long], too, if the movement [is long], and the movement [is long], if the time [is long].

I note that there are quite extensive resemblances between Aristotleís definition of time and the definition of time I made here. However, I will not discuss these resemblances now (or here?).