Why can coefficient $a$ between spacetime intervals depend on absolute relative velocity between the systems?

Question

I read Landau & Lifshitz' Classical Theory of Fields book (page 14-15) (see pic below) and I was confused when I saw in proof that coefficient $a$ between spacetime intervals $(ds)^2$ and $(ds')^2$ can only depend on the absolute relative velocity between the systems.

I.e. $$(ds)^2=a(ds')^2.\tag{p.5}$$

I proved that coefficient $a$ should be. But I don't understand what should it depend on and why?

And why can coefficient $a$ depend on absolute relative velocity between the systems? i.e. $a=a(V)$?

Answer 1

The authors are trying to argue that the factor $a$ is identically 1. They start with the assumption that it could in principle depend on the location in spacetime, or on the velocities of the two frames that are measuring the interval. But they then continue to argue that:

• It can't actually depend on the location in spacetime, since we are assuming that spacetime is homogeneous.
• It can't depend on the direction of the relative velocities of the frames, since we are assuming that spacetime is isotropic.

The only thing left that it could depend on is the magnitude of the relative velocity between the frames.

The authors then consider three reference frames $K$, $K_1$, and $K_2$ moving non-colinearly. From the relations that must hold between the $a$ factors in these transforamtions, they conclude that the factor $a$ must actually be 1, i.e., $ds^2 = {ds'}^2$.