Last revised on Tuesday, 02-Feb-2010 07:06:55 EST
©1998--2016 Creative Consulting for Research and Education
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.
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