In article ,
Mahdiarnt wrote:

But what still bothers me, is
the implementation of the notion of distance in the results of general
relativity. For example, in the Schwarzschild solution, when we derive
the equations of geodesics, we arrive at an equation like that of the
Newtonian theory with a small correction. But the similarity to the
Newtonian case is achieved only if we interpret the r coordinate as
distance from the "sun" to the "planet," and the phi coordinate as the
angle. To me these are mere coordinates. Could you tell me the result
of what measurements are r and phi?

That's a very good question.

If we use Schwarzschild coordinates to describe the spacetime around a
black hole, then the coordinate r of a given point is not (in any
meaningful sense) the distance from the black hole to the point. It
is (1/2pi) times the circumference of a circle centered on the black
hole and passing though that point. And the coordinate phi is just
(1/r) times the arc length along such a circle.

In fact, I think that Taylor and Wheeler, in their introductory
book "Exploring Black Holes" go to the trouble of referring to r
as the "reduced circumference" coordinate rather than the
radial coordinate, in order to emphasize this distinction.

For both r and phi, the measurements are to be taken along curves at a
fixed time. This is a meaningful thing to say because the
Schwarzschild geometry is static. (That is, there's a preferred time
coordinate such that nothing about the metric changes in those
coordinates.)

That's the operational, in-principle-measurable description of
what the Schwarzschild coordinates r,phi mean.

Now, if we're using Schwarzschild geometry to describe spacetime
outside of the Sun rather than outside of a black hole (which is
perfectly correct, in the excellent approximation that the geometry
is static), then it does make sense to talk about the distance from
the center of the Sun to a given point. After all, because
we're assuming things are static, there's a preferred notion
of things being "at the same time," so there's a unique
spacelike geodesic that goes from the given point to the center
of the Sun at constant time. The length of that geodesic will not
be exactly the same as the Schwarzschild r coordinate of the point.
For a system like the Sun, for which the curvature is always pretty
weak (the Sun's quite far from being a black hole), the difference
would be tiny. For something like a neutron star, though, it
could be significant.

-Ted

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