On Fridays, I'm working through Einstein's groundbreaking 1905 essay where he proposed his special theory of relativity.
I. Kinematical Part
The essay is divided into two parts, the "kinematical" part is the part that has to do with bodies in motion. The second part is the "electrodynamical" part, which has to do with charges in motion.
1. Definition of Simultaneity
Einstein's purpose in this section is to come up with a definition for simultaneous events in a "stationary" system. [To call it a "stationary" system is a bit of a misnomer, since movement is really relative, as Galileo indicated three hundred years earlier. Any "system" moving at a constant velocity might be considered stationary in terms of all the elements in that system. Such a system constitutes an "inertial frame of reference."]
If something is at rest relative to a stationary system, you can identify its location easily using x, y, z coordinates that you locate from some point of origin in that system. It gets a little more complicated if you want to describe the motion of some material object in that system, its change in location in relation to time. Then you will have to be clear about what you mean by "time."
If we say the velocity of an object means that it is at one point at time A and another point at time B, we are assuming, in effect, that if there were clocks at both points A and B, they would be moving forward in sync with each other. [After all, what is time, really, at this point, other than to say that 7 o'clock means that the small hand on my watch is pointing toward the 7?]
How can we make sure the clocks are in sync? Einstein defines synchronous and simultaneous in the following way, namely, that the amount of time it takes for light to get from stationary point A to stationary point B is the same as the amount of time it takes for light to get back from B to A. Since the speed of light is constant in any medium, the time should be the same given that the distance is fixed in this system.
[Since Einstein is defining simultaneity here, rather than demonstrating it, I still find this move on his part a little puzzling. Is he saying that these two clocks are synchronized if the hands on clock A read twice the amount of time when the light reflects back as the hands on clock B read when the light arrived there from clock A? That would satisfy his definition.]
Einstein further mentions two truths that follow from this assumed definition. 1) If the clock at B synchronizes with the clock at A, then the clock at A will synchronize with the clock at B. 2) If there is some other clock C that synchronizes with A, then it will also synchronize with B.
Finally, we can express the speed of light as the distance traveled per time. The distance from A to B and back is 2AB. The time passed is the time it took from when the the light began at point A (ta) till its return back to point A (t'a). Thus c (the speed of light) = 2AB/(t'a - ta).