Michael Lee's Mathematics and Physics
Derivation of the Lorentz Factor, Time Dilation and Length Contraction
I added a "romantic" feel to the work below by placing a male observer in inertial frame S and a female observer in frame S'. This will help with using our language to describe things.
Inertial frames S (his) and S' (hers) Her frame S' is moving to the right with uniform velocity v whereas his frame S is stationary. Each observer has his or her own accurate and reliable clock and metre sticks (rods).
Shortly after synchronizing the clocks at the origin, an event happens at point A (e.g. a supernova).
From the point of view of frame S (his frame) the event took place according to time t and at a distance x using his clock and his metre sticks.
From the point of view of frame S' (her frame) the event took place according to time t' and at a distance x' using her clock and her metre sticks.
According to Galileo, both his and her clocks ( t and t' respectively ) run at the same speed, but Einstein showed that's not the case. So let,
Method II Here is an overview; it's as if he was suspended in a cherry picker above the tracks and she is aboard the train moving to the right with uniform velocity v. Both he and she are equipped with their own "light clocks." Think of a "light clock" as a laser that shoots out a pulse of light and then hits a mirror and bounces back to a receiver and it records the time. Much like the one Galileo designed when trying to measure the speed of light.
The Lorentz Factor. The way I remember the behaviour of this function is as v increases towards the speed of light, the value of gamma increases as well. That is, velocity is directly proportional to gamma.
