For an observer on a circular orbit around a Schwarzschild black hole, we can first define his distance

to the black hole scaled by the Schwarzschild radius

. The second parameter

defines his velocity

with respect to a static observer at the current position of the moving observer scaled by the velocity of light

. Then the orbit is parametrized by the proper time

of the moving observer. For one full orbit, the moving observer needs

, with

. After this time, the local reference frame of the observer has undergone a rotation of

, which follows from the Fermi–Walker transport. The corresponding precession frequency reads

. If the local reference frame does not rotate, the velocity

has to be

; however, this is only valid for

.

[1] T. Müller and S. Boblest, "Visualizing Circular Motion around a Schwarzschild Black Hole," forthcoming publication in the

*American Journal of Physics*.

[2] T. Müller and F. Grave, "Motion4D—A Library for Lightrays and Timelike Worldlines in the Theory of Relativity,"

*Computer Physics Communications*,

**180**(11), 2009 pp. 2355–2360.

[3] D. Bini, et. al., "Physical Frames along Circular Orbits in Stationary Axisymmetric Spacetimes",

*General Relativity and Gravitation*,

**40**(5), 2008 pp. 985–1012.