Cheers Nabeshin,
I had started this post before you posted your response, so I thought I might as well finish it and you or others might still find something useful in it.
Nabeshin said:
..., ultimately I am interested in this equation in a curved space time.
Consider a stationary observer located on a tower a distance r from the centre of a non rotating gravitational body. If he were to measure the linear velocity of a passing satellite with a circular orbit of radius r, then yes, he could calculate the angular velocity ω of the satellite using the Newtonian equation ω = v/r. An observer at infinity would measure the linear velocity to be:
v \sqrt{1-\frac{r_s}{r}} =\ r\omega \sqrt{1-\frac{r_s}{r}}
and the angular velocity as :
\frac{v}{r} \sqrt{1-\frac{r_s}{r}} =\ \omega\sqrt{1-\frac{r_s}{r}}
An observer on board the satellite (considering himself to be stationary), would measure the instantaneous velocity of the observer on the tower to be v (or rω) as he passed by. When he multiplies the velocity by the period it takes the tower to complete one full rotation, he works out the circumference of the path taken by the tower observer to be:
2\pi r \sqrt{1-v^2/c^2}
so effectively he measures the circumference to be length contracted and the geometry does not appear Euclidean to the observer on board the satellite. Note that both the observer on the tower and the observer at infinity measure the circumference to be simply 2pi*r.
The Newtonian formula for the velocity of an body of mass m orbiting a gravitational body of mass M is:
v = \sqrt{\frac{GM^2}{(M-m)r}
which for very small m compared to M approximates to:
v = \sqrt{\frac{GM}{r}}
This is the linear orbital velocity as measured by an observer at infinity. (It is slightly odd that the Newtonian equation for angular velocity is equivalent to the local angular velocity in the Schwarzschild metric while the Newtonian equation for orbital velocity is equivalent to the measurement made by the observer at infinity in the same metric.)
Relative to the speed of light the last equation can be restated as:
\frac{v}{c} = \sqrt{\frac{GM}{rc^2}} = \sqrt{\frac{r_s}{2r}}
The relativistic linear orbital velocity v_r/c as measured by a stationary observer at radius r is time dilated so:
\frac{v_r}{c} \sqrt{1-\frac{r_s}{r}} = \sqrt{\frac{r_s}{2r}}
For an orbiting photon, v_r/c =1 so the orbital radius r_p of a photon can be determined by:
\sqrt{1-\frac{r_s}{r_p}} = \sqrt{\frac{r_s}{2r_p}}
1-\frac{r_s}{r_p} = \frac{r_s}{2r_p}
r_p = \frac{3r_s}{2}
