For the Schwarzschild metric
$$c^2 d \tau^2 = (1-\frac{r_s}{r}) c^2 dt^2 - (1-\frac{r_s}{r})^{-1} dr^2 - r^2(d\theta^2 + \sin^2{\theta}~ d\phi^2)$$
most references (Landau-Lifshitz and Schutz) say that the physical meaning of the coordinate $r$ is $C/2 \pi$ where $C$ is the circumference around the central massive object. My questions are
- Why is this physical interpretation justified? Can't I define $r= C/3 \pi$?
- Are we free to adopt any physical interpretation (operational definition) of the coordinates (not just $r$) in general relativity as we please? I say so because the physical interpretation of $r$ changes for the Kerr metric. So who has the authority to decide the physical interpretation of coordinates?