INTRODUCTION

First, I want to thank Mr. Don Hurd, who pointed out some errors in Part I of Appendix E. The errors resulted from a mislabeling of Figure E-1 that was overlooked in the editing of the original manuscript. But, after reviewing it thoroughly, and giving it some additional thought, I decided that instead of just correcting the errors, I would replace the derivation with one that I believe will be clearer and more effective. The following replaces a major portion of Appendix E, Part I starting with the last paragraph on Page 327, over to the end of Part I, on page 330.

REPLACEMENT

The speed of light is constant for all observers. This basic underlying principle of Einstein’s relativity has been demonstrated many times and many ways, starting with the famous Michelson-Morley experiment. The following experiment will enable us to see how the fact that light always travels at a constant speed, regardless of the motion of the observers, produces definite observable and measureable space and time distortions that are formalized in the Lorenz transformation equations.

This experiment is set up on a long, straight and level stretch of a high-speed train track. First, with the train standing absolutely still in a station that will be used as an observation point, observers on the train and in the station will calibrate and synchronize their measuring devices as follows: A light source on the train will send one photon at a time along a vertical measuring rod. Both the light source and the measuring rod are securely fastened to the rigid structure of the train. For the purposes of this experiment, the length of the measuring rod is defined as one unit of length, and the length of time it takes for a photon to traverse the length of the rod is defined as one unit of time. The light source will be designed to release photons at regular intervals. When one photon impacts a receptor at the end of the measuring rod, the next photon will be released. Another light source and measuring rod, identical to those on the train, will be aligned with them and fastened securely to the rigid structure of the station platform, so that observers in the station and on the train can see that they are synchronized, with the light sources releasing photons at exactly the same time.

Next, the train will retreat along the track to a starting point that will allow it to reach its maximum speed by the time it returns to the station, so that when it passes the observer in the station, it will be traveling at a constant high rate of speed. While the photon on the train is traversing the length of the rod, the train will have traveled a short distance along the track. An observer on the train will see the photon traveling vertically from point A to Point B (the length of the rod on the train) in one unit of time, while an observer on the station platform will see the photon traveling from point A to point C. Figure E-1 is a geometrical representation of these observations.

Based on our normal everyday experience, we expect the length of the rod on the station platform, and the length of time for a photon to traverse it, to coincide with the rod and photon on the train as the train passes through the station, just as they did when they were synchronized at the beginning of the experiment, with the train at rest in the station. This, however, we shall see, is not the case.

Let’s start by determining how the length of time elapsed for the observer on the station platform compares with the time elapsed for the observer on the train during the movement of the photon down the length of the rod from point A to point B. Referring to Figure E-1, we define the time elapsed for the observer in the station as ts, the time it takes for the train to move the distance from B to C. Since distance traveled equals velocity multiplied by time, the distance from B to C is v times ts, or vts. Similarly, the distance from A to C is cts, where c is the speed of light, and the distance from A to B is ctt, where tt is the time elapsed for the observer on the train. Since the train track is horizontal and the rod was fastened to the train vertically, the geometric figure ABC is a right triangle. In high school geometry we learned that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. In other words,

or

In order to compare the time elapsed for the observer on the train, tt, with the time elapsed for the observer in the station, ts, we can solve this equation algebraically for tt/ ts:

Transposing all ts terms to the right-hand side of the equation and dividing through by cts, we have

Cancelling like terms in the fractions on both sides gives us

Taking the square root of both sides and multiplying through by ts, we have:

This equation tells us that the time elapsed for the observer on the train, tt, as measured by the observer in the station, is smaller than ts, the time elapsed on the station platform, since v will always be smaller than c (trains travel only a fraction of the speed of light at the present time) and therefore, the value of the square root of one minus v2/c2 will always be less than one, and thus tt = ts times a factor that is less than one implies that, as a result of the constancy of the speed of light, tt is always smaller than ts when the train is moving.

Furthermore, since tt is smaller than ts, the length ctt is shorter than cts and the unit measuring rod on the train would appear shorter to the observer on the platform. Thus, a snapshot taken of the unit rod and a clock on the moving train, by an observer on the station platform, would show the rod on the train to be shorter than the rod on the platform by a factor of , and the clock on the train to be running slower than the clock on the platform.

These observations, while counter-intuitive, are undeniable results of constant light speed. We will use these facts in Part II of this appendix to derive the Lorentz transformation equations, and then we will apply them to a three-observer experiment.

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