On the Concepts of the Special Theory of Relativity

v 1.0 (5 Oct 2002)

© 2002 Krishna Myneni


The special theory of relativity, as with all theories of fundamental importance in science, deals with a problem of measurement. Central to this theory is the rethinking of our assumptions about time, and how a measurement of time may be made. Einstein's original paper on this theory, On the electrodynamics of moving bodies, published in 1905, begins with a plainly worded, but very concise, description of the problem in comparing time measurements made at different locations in space. He reveals the logical inconsistency in the attempt to establish by measurement the synchronization of two clocks at distant points in space. His argument, based on certain imaginary situations, or what have come to be called thought experiments, is rather brief and yet is central to a basic understanding of the more well-known aspects of special relativity, such as the discrepancies in times and lengths for two observers who are moving relative to each other. In this paper, which is a work in progress and hence has an associated version number, I will introduce and extend these thought experiments with the goal of aiding the reader in understanding the problem at the base of special relativity, the measurement of time. Hopefully, this paper can serve as a basis on which to begin the reader's introduction to the special theory of relativity.

Clock Synchronization

Initially we start with one observer, myself, standing at some location, call it point A, with two identical clocks. I make sure the two clocks are set to the same time, and after observing both of them run for a number of ticks, I am satisfied that the two clocks are synchronized --- both clocks are reading the same time after a large number of clock ticks have occured on the first clock.

Next, I give the second clock to a second observer, my friend, and ask him to go to a distant location, point B. We can both communicate with each other using cell phones. Now I ask the simple-minded question: How can I tell if the clock at point B has the exact same reading as at clock A? Before you, the reader, try to answer this question, you might object, "you have already established that the two clocks were synchronized when they were together at point A; why would you assume that the two clocks are not synchronized now that they are at separate locations?" I reply that just because I knew that the two clocks were once synchronized, I could not assume that they remained synchronized after one has moved away to a different location. This may seem to be a rather picky objection and a waste of time on my part, but you decide to humor me. My friend and I discuss this problem and decide to set up a way to transmit the clock reading at point B to my location at point A. The clock at B will send me its time reading by rapidly switching on and off a laser that is aimed at my location. The on and off switching of the laser will be done in such a way that clock B's current time will be encoded in the sequence of pulses. I have a receiving instrument at point A which can sense the sequence of laser pulses and decode the time value to instantly give me the reading of the clock. Now I can test whether or not clock B is reading the same as the clock at A.

I begin the test and start writing down in my notebook a table with two columns. In the first column I record the clock B time which I receive from the laser and which my instrument decodes. At the same time, in the second column I write down the current clock A time. Since this is an imaginary situation, we allow my being able to write down both numbers, the received clock B time and the current clock A time, without any pause whatsoever. After recording a dozen or so pairs of numbers, I compare the two columns and notice that the received time from clock B always lags the recorded clock A time by a small amount.

My friend from point B and I get together at a restaurant to discuss the results of the experiments. I am puzzled by the discrepancy between the two recorded columns of time, but my friend quickly points out that the sequence of laser pulses, or pulse train, takes some time to go from point B to point A. The discrepancy, he suggests, is simply the propagation time of light in free space. "We have measured the propagation time for a light signal to go from point B to point A," declares my friend. But wait, I say, this would be true if we knew that your clock at point B was in fact synchronized with mine at point A. Establishing the synchronization of the two clocks was the purpose of the experiment in the first place. If they were not synchronized, then the lag which we noted in our measurements of clock A and clock B would not truly represent the propagation time for light to go from B to A. My friend considers this for some time, as he finishes his apple pie, and says, "What we need is another method of measuring the propagation time of a light signal to go from point B to point A! Then, we can correct your received clock B readings by adding to them the true propagation time. Thus we can determine whether or not our two clocks are synchronized." Great! I say. Let's set up a measurement for the propagation time for the laser signal. We agree to do this the next day.

The next morning we meet to work out the details of the experiment. We agree to do the following. My friend will once again be located at point B, with his clock. He will modify his clock and laser transmitter to send only a single short pulse of light, periodically. Whenever a pulse is transmitted, he will write down in his notebook the current clock B reading. I will be located at point A, as before, and will modify my laser receiver to alert me upon the arrival of a single laser pulse. At that instant I will write down in my notebook the current clock A reading. After repeating this some fixed number of times, we will get together and compare my pulse arrival times, determined using clock A, with his pulse departure times, recorded using clock B. Suddenly, as we think through this experiment, we both realize, just as you the reader probably has recognized, we have a problem once again. In order to measure the propagation time for the laser to go from point B to point A with the experiment we have devised, we once again must be sure that the two clocks are synchronized with each other. We seem to be caught in a vicious circle. In order to establish that clock A is synchronous with clock B, we must know the propagation time for the light signal to go from B to A. But in order to measure the propagation time for the light signal, we must ensure synchronization of the clocks at A and B.

Now we must look into the matter more carefully. Does a method exist to measure the propagation time of light from point B to point A, without the use of a second clock at point B? We realize that we must remove the second clock from the picture, so as to avoid the question of synchronization. My friend devises the following experiment: The laser pulse starts from point A and goes to point B, at which is placed a mirror. The light is reflected back to point A. Now, since I'm sitting at point A with a clock, I can record both times with a single clock: the time at which the light pulse was sent and the time at which it returned to A. Then, I can take the time difference between my two readings and divide by two, since the light propagated twice the distance from point B to point A. It would seem that we can therefore independently measure the propagation time for a laser signal to go from point B to point A, without worrying about synchronization of a clock at B. With this information, we can revisit our original problem of determining whether or not a clock at B is synchronized with a clock at A, since we now know the correction for the time reading being sent on a laser signal from point B.

Let's pause for a moment and see how it is that we managed to escape the vicious circle of determining both a propagation time for light to go from point B to point A, and whether or not the clock at point A is synchronized with the clock at point B. Did we make any assumptions? In fact we did make two assumptions in our measurement of the propagation time for a light signal. Remember that we divided by two the total time it takes for light to from point A to point B and back from point B to point A, in order to arrive at the one-way propagation time from point B to point A. By doing so, we have implicitly assumed that light propagates in the same elapsed time in going one way, point A to point B, as it does in the other way, from point B to point A. This assumption is not able to be tested --- again we are led to the vicious circle of first having to show synchronization of the two clocks at A and B! The second assumption, which is independently testable, is that light takes no time to reverse its propagation direction at B, when reflecting from the mirror. This assumption is testable by allowing my friend to be located at point B, observing his clock while watching the beam reflect. It turns out that he cannot see any significant time for the light to reverse direction so it appears we don't have to worry about the second assumption.

It would seem that our attempt to check whether the clock at point A is synchronized with the clock at point B can only be successfull if we make an untestable assumption about the propagation of light signals between point A and point B:

A light signal propagates from point A to point B in the same amount of time as it propagates from point B to point A.

An untestable assumption really amounts to a definition of the subject of the assumption. In this case, the subject is time, and the above statement amounts to a definition of time. According to the statement above, time itself is defined by the propagation of light between two points. The reader should consider these latter statements carefully and convince him/herself that only by asserting this definition of time can we arrive at a way to determine, free of logical inconsistencies, whether or not two clocks, located at distinct points in space, are synchronized.

Comments regarding this paper may be emailed to krishna.myneni@ccreweb.org