How was the Lorentz factor derived?

[Psyguy22]

So I know about the lorentz factor and how it describes time dialition, mass increasing etc.. but I was wondering how it was derived in the first place?

 

[Nugatory]

So I know about the lorentz factor and how it describes time dialition, mass increasing etc.. but I was wondering how it was derived in the first place?

You start with the assumption that the speed of light will be the same in two frames moving at a constant velocity relative to one another (this is a consequence of Einstein's two postulates).

Then if the coordinates in one frame are (x, t) and the coordinates of the other frame are (x', t'), the path of two flashes of light, one moving to the left and the other to the right will be:

x = ct (right-moving)
x = -ct (left-moving)

in the (x ,t) frame and

x' = ct' (right-moving)
x' = -ct' (left-moving)

in the (x', t') frame. We also know that (0,0) in the unprimed frame is (-vt',t') in the primed frame, (0,0) in the primed frame is (vt,t) in the unprimed frame, and that the ratio of the length of an object in the unprimed frame to its measured length in the primed frame must be equal to the ratio of the length of an object in the primed frame to its measured length in the unprimed frame.

From there, it's just algebra to find the relationship between x' and t' as functions of x and t, and vice versa. Time dilation, length contraction, and the Lorentz factor fall out of these relationships when you transform the times and places of two clock ticks at the same place and the two ends of a moving rod at the same time, from one frame to the other.

The complete derivation (and as I said, it is just algebra) is in Appendix A of Enstein's book "Relativity: The Special and General Theory" which is readily available online, for example at http://www.bartleby.com/173/

 

[bcrowell]

There are many ways of deriving it. Nugatory has given one. Here is another, completely different one: http://arxiv.org/abs/physics/0302045