We know that Schwarzchild 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?

  • $\begingroup$ Your question seems to be essentially 'explain great chunks of the differential geometry which underlies GR to me': that's too broad. $\endgroup$ – tfb Sep 25 '16 at 12:02
  • $\begingroup$ @tfb I am more or less familiar with GR math. Any hint by you will be helpful. I am here interested in the measurements as different from just theory. $\endgroup$ – user56963 Sep 25 '16 at 12:09

The Schwarzschild $r$ coordinate at a point is defined as the circumference of the circle passing through that point and centred on the mass divided by $2\pi$. This definition applies at all distances even inside the event horizon.

To see this take the Schwarzschild metric:

$$ ds^2 = -\left(1-\frac{r_s}{r}\right)dt^2 + \frac{dr^2}{1-\frac{r_s}{r}}+r^2d\theta^2 + r^2\sin^2\theta d\phi^2 $$

and integrate the proper length along the circle I've just described. Along this line $dt = dr = d\theta = 0$ and $\theta=\pi/2$, so the metric reduces to:

$$ ds^2 = r^2 d\phi^2 $$

And therefore the length of the circle is:

$$ s = \int_0^{2\pi} rd\phi = 2\pi r $$

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  • $\begingroup$ You must have meant $\theta=\pi/2$ $\endgroup$ – user56963 Sep 25 '16 at 12:40

You seem to have some serious conceptual misunderstandings about the subject. Co-ordinates are not "measured" they are defined on a space time manifold. Proper time on the other hand can be measured. So please think about the concepts correctly. I will recommend reading Einsteins works on the subject.

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  • $\begingroup$ The answer has been given above. Coordinates are functions to the real line defined for each point in the manifold. So the can both be defined and related to measurement. $\endgroup$ – user56963 Sep 25 '16 at 15:32
  • $\begingroup$ @VictorVMotti I don't understand what you are saying, what real line are you talking about. Co-ordinates describe events on a manifold. Formally c0-ordinates are maps from a local patch on the manifold to R^N, provided some patching conditions are satisfied. What John Rennie said is correct. Co-ordinates are defined on a manifold, events can be described by co-ordinates. Language is very important. So please be clear about what you are measuring. $\endgroup$ – Prathyush Sep 25 '16 at 15:49
  • $\begingroup$ A single coordinate like $r$ is defined as a map or function from the point on the 4 dimensional manifold to the real line. My question is about the physical aspects of the underlying geometry. $\endgroup$ – user56963 Sep 25 '16 at 16:15
  • $\begingroup$ @VictorVMotti That is correct a single co-ordinate is a map from the manifold to R. Now that co-ordinates are defined the word measurement cannot apply to them. What is meaningful to talk about is the co-ordinate values for an event for instance where and when did the light bulb glow. So long as the event occurs outside the horizon. It can been seen by an outside observer given a sensitive apparatus and sufficient time and its co-ordinates can be found. Then it is a matter of instrumentation. But definition of the co-ordinate system apples the entire manifold including inside the black hole. $\endgroup$ – Prathyush Sep 25 '16 at 16:30
  • $\begingroup$ Not of course the singularity inside the black hole which is not in the manifold. $\endgroup$ – user56963 Sep 25 '16 at 16:34

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