Measuring Coordinates in Strong Gravity: Schwarzschild Metric

[victorvmotti]

We know that Schwarzschild metric describes an asymptotically flat spacetime. This means that far away from the event horizon we can safely interpret the $r$ coordinate as distance from the center.

But when close enough to the event horizon the curvature becomes significant and our common sense of $r$ breaks.

The question is that what is understood as measurement of the coordinates near very strong gravity?

 

[Dale]

Coordinates are defined, not measured. You can of course define them such that they have some defined relationship to measurements. In that case you perform the measurements, apply the defined relationship and obtain the coordinate.

 

[victorvmotti]

Is it reasonable to define and "measure" the $r$ coordinate like this:

$r$ is in the Schwarzschild metric the integral of the proper length along the curve $$dt = dr = d\theta = 0$$ for $\phi$ from $0$ to $2\pi$ divided by $2\pi$?

Is this definition and measurement valid inside the event horizon as well?

 

[Dale]

Yes. Although usually it is stated in terms of measuring the area, but with spherical symmetry what you wrote is equivalent