Dear Ted,

Before anything, I agree that the mentioned space-like geosdesic
should be perpendicular to the world-line of the observer, as I
corrected in a followup to my original post.

Your answer was illustrative. I learned that all that matters is what
we really measure; it's then our job to find what quantity in the
space-time we have actually measured. 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?