general relativity - Freedom in physically interpreting the Schwarzschild coordinates

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

1. Why is this physical interpretation justified? Can't I define $r= C/3 \pi$?


2. 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?

Answer

The freedom of physically interpreting the Schwarzschild coordinates gets significantly curtailed because of the symmetry assumptions involved in deriving the Schwarzschild metric. Recall we have some physical symmetry in the system:

1. 
   

   The system is stationary (does not change with time). The black hole just sits there, idly.

   
   


2. 
   

   The influence of the black hole is spherically symmetric. There is no preferred direction.

   
   

These two physical symmetries are implicit in the derivation of the Schwarzschild metric.

1. 
   
   

   The first physical symmetry above is implemented mathematically by not including any cross terms involving $dt~dx^i$. See Sean Carroll's book or any other reference. Hence, the variable $t$ has to be interpreted closely with time (or proper time). A careful interpretation of the variable $t$ (coordinate time) requires more work (see Exploring Black Holes by Taylor and Wheeler).

   
   


2. 
   

   The physical spherical symmetry in the bullet point (2) above is implemented by assuming $d\Omega^2 = d\theta^2 + \sin^2 \theta d\theta^2$ to be a part of the invariant interval $ds^2$. With this assumption, $\theta,~\phi$ are constrained to be interpreted as variables labeling directions in space. You can't interpret $\theta$ or $\phi$ as the radial distance from the black hole.

   
   

So, in summary, because of the physical symmetries of the system and the mathematical assumptions going into the derivation of the Schwarzschild metric, you no longer have total freedom to interpret the Schwarzschild coordinates.

With these interpretational constraints, the last variable $r$ is to be interpreted as a measure of the radial distance from the black hole. So, if you go around a black hole at the constant $r,~\theta,~t$ and measure the circumference $C= \int ds = r \int d\phi = 2 \pi$ . Hence, you can't set $C=3 \pi$.

Also see this related question