Einstein's Derivation of Lorentz transformations

Question

I have been reading the derivation of Lorentz transformations in the appendix of relativity:the special and general theory by Einstein and am stuck in the following spot:
He states
$$
x - c t = 0tag{1}
$$
implies
$$
x' - c t' = 0tag{2}
$$
and this implies
$$
x - c t = k (x' - c t')tag{3}
$$
where $k$ is a constant. I am not able to see where the third equation comes from. Will be grateful for any help.

Ali · Answer

Not sure what the context is but I will assume we are talking about travelling light beams.
Light travels at c in both reference frames.
Suppose both axes are lined up at the start.
A light beam travels at $c$ in the first frame $x=ct$
In the other frame, the light beam is also travelling at c. An observer in that frame measuring with their time is $x'=ct'$.
Now you have two equations
$$x-ct=0$$
$$x'-ct'=0$$
Both equations are zero but they don't necessarily have to be equal. The equations could differ by multiplication of a constant or a function. From this you can construct your third equation for arbitrary $k$.

Einstein's Derivation of Lorentz transformations

One Answer

Add your own answers!

Ask a Question