Geodesic Precession on a Timelike Circular Orbit around a Schwarzschild Black Hole

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 .

Detailed discussions of the circular motion around a Schwarzschild black hole can be found in the following references.

References

[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.