Hello Ted:

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.


I don't think this is a great question because it depends only on the
coordinate system chosen. One could just as well use isotropic
coordinates. It appears that theorists use the Schwarzschild
coordinates because they simplify calculations a bit. Experimentalists
use Taylor series expansions of the metric in isotropic coordinates.

Sorry to go off on a tangent here, but I was just looking at the
isotropic form, eq. 31.22 in MTW:

ds^2 = -((1 - M/2R)/(1 + M/2R))^2 dt^2

+ (1 + M/4R)^4[dr^2 + dr^2(dtheta^2 + sin^2 theta dphi^2)

If R = M/2, then the coefficient for dt^2 goes to zero, but the radial
term does not become underfined as is the case for this metric in the
Schwarzschild metric. What is the usual take on this?


doug
quaternions.com