D3 Derivation of the Lorentz Transformations from Basic Phenomena
Consider again two coordinate systems, as described at the beginning of D1. An event E takes place for red at time t' at the point (x', y', z'). We now know that it takes place for black at the point (x, y, z) with y = y' and z = z'. Still black's values for t and x are to be determined for this event.
For black the ‘master clock’ of red at location B (like each clock of red) runs too slowly, thus
t = t(A) = t’(B)/√ or t • √ = t’(B)
A clock at location x' of red, in addition, shows for black a desynchronization to the clock at B of
∆t’ = -x’ • v/c2
Thus the red clock at B already shows t'(B) = t'(x') - ∆t' = t' + x' • v/c2, when E takes place. Thus we obtain the corresponding clock state for A from the expression
t(A) • √ = t’(B) = t’ + x’ • v/c2
For t = t (A) itself we obtain t = (t' + x' • v/c2) / √
It was precisely this expression for t that we derived in D2!
We still must clarify, at which x-coordinate the event E takes place for black. Red thinks that the distance d' from A to the location x' of the x'-coordinate of E has the following value:
d’ = x’(E) - x’(A) = x’ + v • t’(B) = x’ + v • t’(x’) = x’ + v • t’
Here we have used the fact that for red the clocks at B and at x' are synchronized. Thus for red the event E has the distance d' from A along the x-axis, as well as along the x'-axis. For black all measurements of red in the x-direction are Lorentz contracted. The distance of the event from A must therefore be for black d = d'/√, and we are finished:
x = d = d'/√ = (x' + v • t') / √
It was precisely this expression for x that we also derived in D2!
It is interesting to note that most high school text books about the STR do not derive these Lorentz transformations. Often they are assumed or simply stated and then time dilation and length contraction are derived from them. The reverse path from the basic phenomena to these more abstract transformations can be taken, only after desynchronization has been quantitatively dealt with.
The reason for introducing the Lorentz transformation is however the same with all authors: They provide a simple derivation of the correct formula for the addition of velocities. Here we have to deal in principle with three inertial frames: B moves with v relative to A, and C moves with w' relative to B. What speed does C then have for A? The angle φ between the time axes can be the same, e.g. 60º, for both A and B and for B and C. If one combines rather naively the two corresponding Epstein diagrams into one with 3 time axes, then the angle between the time axes of A and C is already 120º! Even I, as Epstein evangelist, was not spared until recently the derivation of the formula for the addition of velocities in the STR via the Lorentz transformations.
Alfred Hepp made me aware of the fact that Epstein shows in the second edition of  in appendix A, how one can construct with compass and straight-edge the correct tilting angle for the speed of w from those for v and w'. Thanks to the sketch in that section (which also first requires understanding!) we could finally find a quite simple proof for the addition formula (red box on the following page), which is based only on Epstein diagrams, and with which one can completely avoid Lorentz transformations. I present this proof in the appendix K7.