OK, section 2 talks about two rulers with clocks mounted along them moving past each other at constant velocity. This is exactly the model that Einstein used to define the physical meaning of a given observer's coordinate system, except that he also defined a synchronization scheme involving light-signals for the clocks on each ruler. You may want to look at my thread An illustration of relativity with rulers and clocks where I showed exactly what relativity predicts about what different clocks on each ruler would read at the time they meet, complete with diagrams.Perspicacious said:JesseM,
It's all answered in section 2 (The Definition of Time), which is less than a page and a half in length. I don't believe that the graph and Greek symbols can be easily copied and pasted here.
But what I don't see in section 2 is a clear definition of how an observer sitting on one ruler would define the "rate" that a clock on another ruler is ticking. In terms of the diagram in section 2, suppose that when clock A which sits on the 0 meter mark of the top ruler [tex]\Gamma '[/tex] is next to clock B on the 0 meter mark of the bottom ruler [tex]\Gamma[/tex], both clocks read "0 seconds"; then when clock A is next to clock C on the 3 meter mark of ruler [tex]\Gamma[/tex], clock A reads "1 second" while clock C reads "2 seconds". Does this mean that from the perspective of ruler [tex]\Gamma[/tex], clock A is ticking at half the rate of its own clocks, and that it's moving at 3/2 marks per second? If not, how do you define the rate of ticking and speed of clock A from the perspective of ruler [tex]\Gamma[/tex], in terms of measurements made by that ruler/clock system?