Lorentz Transformation

Prerequisite: physics, algebra 2

Einstein's book Relativity: The Special and the General Theory is not the usual kind of pie that I stomach, but hey, the cover states: "a clear explanation that anyone can understand". Anyone. Hmm.. It is not really anyone's fault, which Steven Pinker explains rather nicely.

Nonetheless, here is my shot at following the derivation of the Lorentz Transformation. I did manage to squeeze out the final result, although I may not have applied the correct maths or logic. If anyone can spot an error that would be great. Even better, tell me about it.

In the below diagram, the coordinate system k' is moving away from k. At the origin of each "point of view" is a photon. The k' photon is moving away from the k photon in the positive x direction at velocity v (which I did not label..).

First, we can agree that the distance a photon covers is (speed of light)*(time). This applies for both k and k'. The two are related by transformation constants λ and μ. In the former case, the photons are moving in the positive x direction; in the latter, negative x.

Add 3) and 4) to isolate x'; subtract to isolate ct'. Introduce constants a and b to unite λ and μ, and for convenience.

Pick the origin of k where x = 0 and substitute into 5) to get 6), recalling that velocity is displacement over time ( v = x/t ). For comparison between scale increments ∆x and ∆x', pick the points t = 0 and t' = 0 to substitute into 5). From k' point of view, ∆x = 1/a as seen in 7).

*The substitution of constant a is from rearrangement of 6).

And from k point of view, ∆x' = a(1-(v^2/c^2)) as seen in 7a). Although ∆x and ∆x' are not the same in different perspectives, they should. Setting the two equal yields 7b).

Substitute 6) and 7b) into 5) as needed, to get 8). Substitute 8) into (x'^2 -c^2t'^2) to confirm that it equals (x^2 - c^2t^2). Except, does (x'^2 -c^2t'^2) really equal (x^2 - c^2t^2)?

There is argument that (x-ct) and (x'-ct') are incorrectly squared, since the correct square roots are as shown below:

But maybe the squaring came beforehand, leading to that conclusion (sorry for the transgender-looking x):

So there you have it. Your final product is equation 8). If the teacher ever threw you the equation for Lorentz Transformation, now you know where it comes from. Kind of. Maybe it is true that "anyone" can understand such things, but it is just a matter of time. And choosing manageable perspectives. Got to keep working at it.