precessing GR conic sections
in the Kerr equatorial plane
geodesic motion problem

bob jantzen, Villanova University

Elliptical orbits of planets in Newtonian gravitational theory precess due to the other planets etc:
https://en.wikipedia.org/wiki/Apsidal_precession
In general relativity, these orbits precess due to the theory itself. This Maple exercise generalizes the above Newtonian animation to the GR case. The precession of the orbit of Mercury due to GR is famous as one of the 3 crucial tests which convinced people that GR was the correct theory.

In Newtonian theory the spin of a gyroscope following an elliptical orbit has its direction remain fixed with respect to the distant sky. In GR the spin instead precesses along the orbit. The component perpendicular to the plane of the orbit remains fixed but the component in the plane of the orbit precesses, and hence can be represented by a pair of orthogonal axes in the plane which rotate with the spin precession angular velocity.

Theory developed in these two articles:

143. Gyroscope precession along bound equatorial plane orbits around a Kerr black hole
D. Bini, A. Geralico, and R.T. Jantzen
Physical Review D 94, 064066 (2016); http://arxiv.org/abs/1607.08427, 18 pages (2016) [pdf].
The precession of a test gyroscope along stable bound equatorial plane orbits around a Kerr black hole is analyzed and the precession angular velocity of the gyro's parallel transported spin vector and the increment in precession angle after one orbital period is evaluated. The parallel transported Marck frame which enters this discussion is shown to have an elegant geometrical explanation in terms of the electric and magnetic parts of the Killing-Yano 2-form and a Wigner rotation effect.
144. Gyroscope precession along unbound equatorial plane orbits around a Kerr black hole
D. Bini, A. Geralico, and R.T. Jantzen
submitted to Physical Review D 94, 124002 (2016); https://arxiv.org/abs/1610.06513, 16 pages (2016) [pdf].
The precession of a test gyroscope along unbound equatorial plane geodesic orbits around a Kerr black hole is analyzed with respect to a static reference frame whose axes point towards the ``fixed stars." The accumulated precession angle after a complete scattering process is evaluated and compared with the corresponding change in the orbital angle. Limiting results for the non-rotating Schwarzschild black hole case are also discussed.

Equatorial plane corotating elliptical orbits in a a/m = 0.5 Kerr black hole spacetime, event horizon in gray:

precessing elliptical GR Kerrr orbit elliptical precessing GR Kerr orbit with spin precession
GRorbit (red, blue dot) with precessing ellipse (green),
6 radial oscillations, 1 full spin revolution
one radial oscillation with (boosted) precessing spin axes,
1/6 spin revolution

Shown are the spin vector axes boosted from their local rest space (in relative motion with respect to the stationary gravitational field)  to the static frame field whose directions are locked to those of the "distant stars", removing the effect of stellar aberration in the comparison of the local gyro spin directions with those "distant stars". The principal spin precession effect is a forward or prograde rotation of the spin anchored axes in the same sense as the rotational direction of the orbit around the black hole (like the precession of the orbit itself), with a much smaller increase or decrease of the angular velocity due to the rotation of the hole itself depending on whether the orbit corotates or counterrotates with respect to the spin of the hole.

The motion in time in these videos are parametrized by the precession angle, not by the proper time along the orbit, which requires another layer of Maple code to transform the independent variable using a numerical inversion. Still to do.

hyperbola precessing GR Kerr orbit

 

Here is a corotating precessing hyperbolic orbit in the same Kerr spacetime. Spinning version still to do.

Note also that any central force can have its orbits described by precessing conic sections, but for GR it is particularly useful since in normal circumstances the precession is small and slightly corrects our Newtonian picture, while in strong fields we see how the idea bridges the gap to the fast precessing case.

Maple worksheets: