Gyroscope precession along bound equatorial plane orbits around a Kerr black hole

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 the 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.

Figure 1

The behavior of the dimensionless orbital frequencies $M{\mathrm{\Omega }}_r$ and $M{\mathrm{\Omega }}_{\varphi }$ versus each other (left panel) and their ratio versus $M{\mathrm{\Omega }}_{\varphi }$ (right panel) is shown for $a=0.5M$ and selected values of the eccentricity $e$. The plots are made parametrically in $u_p$ ranging from $u_p=0$ (at the left end points corresponding to radial infinity where both frequencies approach each other and zero) to the eccentricity-dependent values of $u$ corresponding to the marginally stable orbits for corotating geodesics. Only the circular orbit case ($e=0$) reaches the horizontal axis.

Figure 2

The behavior of the average dimensionless precession frequency $⟨M{\mathrm{\Omega }}_{\mathrm{prec}}⟩$ versus the dimensionless azimuthal orbital frequency $M{\mathrm{\Omega }}_{\varphi }$ is shown for $a=0.5M$ and selected values of the eccentricity $e$ for corotating orbits, as in Fig. 1.