special relativity
The equation explained
In wordsHalf the sum of the time the ray leaves the origin of the moving system and the time it arrives back there equals the time it is reflected at the far point, all three read on clocks at rest in the moving system.OverviewThe clocks of the moving system are set by the rule of § 1: a ray sent out and reflected back must take the same time each way, by those clocks. This line is that rule, written for a ray sent along the axis of the motion.Full explanationThe three times are readings of clocks that rest in and travel with it. The ray leaves 's origin at , is reflected at by a point that rests in a fixed distance along the axis of the motion, and returns to the origin at . The rule of § 1 says the outward and the return leg take equal times, so lies halfway between and . Nothing here is yet a statement about the stationary system: it is a condition the function must satisfy. What turns it into an equation for is the next display, which writes each of the three times as at a place and a stationary time, and those stationary times are fixed by the velocity of light in the stationary system.Every stepThe rule of § 1: out and back take the same time, judged by the clocks of the system the ray travels in, here .Add to both sides.Halve both sides, which is the line as the paper prints it.Historian's marginThe equality is a stipulation, not a finding: § 1 defines what synchronous means by this very rule, after saying that the time a ray takes each way is not something the paper knows in advance (p. 894). The same definition was applied to the stationary system in § 1 and is applied here, unchanged, to the moving one, which is the step the principle of relativity licenses.Also