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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?
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  • $\begingroup$ You find the actual circumference by integrating $ds$ along a circle, so you don't have any freedom. $\endgroup$
    – Javier
    Commented Nov 18, 2017 at 15:34

2 Answers 2

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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.

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  • $\begingroup$ In this case we choose $r = C/2\pi$ because as $M \to \infty$, i.e. the flat space limit, the $r$ coordinate becomes the radial distance you'd measure using a tape measure. Do you mean $r\to\infty$ or $M\to0$? $\endgroup$
    – user4552
    Commented Apr 18, 2018 at 15:23
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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.

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