Dear Frank,
I just finished reading your essay (for the first time, at least!) Here are a few questions and thoughts:
1. Just to be sure, on page 1, by "length," do you always mean "spacetime interval with respect to the Minkowski metric?"
2. Similarly, when you write,"all world lines of this set lead from event A to event B and (secondly) the number of time units determined by counting time units along one random of these world lines is different from the number of the time units determined by counting time units along any other of these world lines," do you mean counting proper time units along each world line (I think this is what you mean), or time units in some arbitrary fixed frame?
3. You write, "the ratio of the length of each randomly chosen pair of world lines of this set is equal to the ratio of the numbers of time units counted along these very world lines." Right! The proper time along such a world line is equal to minus the spacetime interval.
4. When you say, "we must be allowed to assume that time units counted along these different world lines are actually all of equal length," the immediate question is "in what frame of reference?" If you started with two identical clocks y and z ticking at the same rate in a particular frame X, then boosted each into different frames Y and Z, the rate of y measured in Y would be the same as the rate of z measured in Z, but the rates of y and z in a third frame, for instance, X, would be different.
5. Right... the "spacelike components" of light paths with common initial and terminal events, measured in any chosen frame, will be equal.
6. Equation 2 worries me, essentially because of the "twin paradox." The "spacelike components" of the light paths are equal as measured in any chosen frame, but not necessary as measured along the worldlines. For instance, suppose A and B are pure timelike-separated in a given frame (e.g. A is my desk at 5 PM and B is my desk at 8 PM). The light path for a light clock that remains in this frame will be 3c long measured in this frame, but the light path for a light clock that travels outward near the speed of light and then returns will be shorter as measured on the "travelling" worldline. In the ratios L_i/n_i and L_j/n_j, it seems that L_i and L_j have to be measured in a single frame, but n_i and n_j are measured along worldlines. I don't know if I'm making any sense here...
Let me think about it a bit more... I am trying to finish reading all the essays people have asked to read before tomorrow, and I am experiencing a bit of information overload. Also, SR is supposed to become "easy" if you think about it long enough, but this never seems to happen, at least not with me! Anyway, let me know what you think! Take care,
Ben Dribus