Derive the Lorentz Transformation

The fastest way is to generalize Galilean relativity like this:

X = A(x - vt)
T = B(t - Cvx)

where the constants A, B, and C are to be determined. The reverse transformation is:

x = (X/A + vT/B) / (1 - Cv^2)
t = (T/B + CvX/A) / (1 - Cv^2)

These equations have to work if x is swapped with X and t is swapped with T so:

A = B = 1 / sqrt(1-Cv^2)

The constant C puts an upper limit on the velocity v so it should be expressed in terms of a speed limit:

C = 1 / c^2

The value of ‘c’ is pretty much arbitrary because it’s really just a unit conversion factor. Any non-zero real number will do the trick. You could set it equal to one with no loss of generality. You just have to measure space and time in the same units.

The last step, which was Einstein’s deep insight, is to notice that the same is true of Maxwell’s equations. i.e. The phase speed of an electromagnetic wave in the absence of any external fields is also unity if you measure space and time in the same units so the (vacuum) speed of light is a practical way of doing that.

Exercise for the reader: calculate X/T when x=ct.

Note also that setting C=0 (i.e. c=infinity) recovers Galilean relativity, but that breaks Maxwell’s equations. And setting C < 0 (i.e. c=imaginary) breaks causality.


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