Freedom in physically interpreting the Schwarzschild coordinates

Question

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 1

You can choose any coordinates you want to write down the metric, but some coordinates make more sense than others. In this case we choose $r = C/2\pi$ because as $M \to 0$, i.e. the flat space limit, the $r$ coordinate becomes the radial distance you'd measure using a tape measure.

Or to put this another way, consider measuring the radial distance between two spherical shells with the Schwarzschild $r$ coordinates $r_1$ and $r_2$. In the flat space limit and in the limit of infinite distance from the black hole the distance is simply $\Delta r = r_1 - r_2$.

So the Schwarzschild $r$ coordinate written in this way has a nice physical interpretation that makes it an obvious choice. Likewise the $t$ coordinate because in the flat space limit, or far from the black hole, it is just the time measured by the observer's clock.

But the Schwarzschild coordinates aren't always the best choice because they are singular at the event horizon. If we want to follow trajectories through the horizion we would probably choose the Kruskal-Szekeres coordinates. These make the maths easier, but the penalty we pay is that the KS coordinates don't correspond in any intuitive to any observable physical property.

So the point is that you choose coordinates that suit your purpose. This might mean choosing coordinates with an intuitive meaning, or it might mean choosing coordinates that are computationally convenient. It's up to you.

Answer 2

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

Answer 3

Relate coordinates to measurements in the local inertial frame of a given observer. For instance, measure tidal forces in 3 different spatial directions, compare a coordinate to rigid rulers held by an observer (note the 4-velocity is provided, as that's what I mean by observer at a given event), or their proper time. If your spacetime has nice properties such as being asymptotically flat or has symmetries such as not changing over time, you might be able to justify extending your interpretation of coordinates from beyond local measurements to that of a distant "observer at infinity" or the like.