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

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 nonrotating Schwarzschild black hole case are also discussed.

Figure 1

The relative observer boost plane of the two 4-velocities $u$ and $U$ within the tangent space showing the relative velocity and rapidity hyperbolic decomposition of the projections parallel and perpendicular to $u$. The relative speed is $||\nu (U,u)||=\tanh \alpha$, illustrated here with value 0.5, while the gamma factor is $\gamma (U,u)=\cosh \alpha$.

Figure 2

The relative observer plane, the relative velocities and the associated relative observer maps between the local rest space directions in that plane, for a vector $X\in LR{S}_U$.

Figure 3

The geodesic orbit [here $x=(r/M)\cos \varphi$ and $y=(r/M)\sin \varphi$ are Cartesian-like coordinates] is shown for $L>0$, $\widehat{a}=0.5$, $e=2$ [so that ${\chi }_{(\max )}=2\pi /3$] and $u_p=0.05$ (left panel) and $u_p=0.1$ (right panel). In the former case we have $\widehat{x}\approx 4.7410$, $L/M\approx 5.2831$, $E\approx 1.0841$; in the latter case instead $\widehat{x}\approx 4.0384$, $L/M\approx 4.6398$, $E\approx 1.2028$. The solid circles correspond to the outer horizon.

Figure 4

Left panel: The quantity $\varphi (\chi )-\chi$ is plotted in degrees as a function of $\chi$ for $e=2$ and $u_p=1/15$ and the selected values $\widehat{a}=[0,\pm 0.1,\pm 0.5,\pm 1]$. Right panel: The quantity $\varphi ({\chi }_{(\max )})-{\chi }_{(\max )}$ is plotted as a function of $\widehat{a}$ for $e=2$ and $u_p=1/15$.

Figure 5

The precession frequency ${\mathrm{\Omega }}_{(\mathrm{prec})}$ of a test gyroscope in Kerr spacetime is shown as a function of $\chi$ for $\widehat{a}=[0,\pm 0.5,\pm 1]$ and the same orbital parameters as in Fig. 3 (i.e., $e=2$ and $u_p=0.1$, implying that ${\chi }_{\max }=\arccos (-1/e)=2\pi /3=120°$). The closeup plot (b) here shows the negative values of ${\mathrm{\Omega }}_{(\mathrm{prec})}$, which vary on a much smaller scale compared to the positive values. For instance, for $\widehat{a}=0.5$ corresponding to the orbit of Fig. 3 the precession frequency vanishes at ${\chi }_0\approx \pm 1.63\approx \pm 93.45°$, i.e., at the radius $r_0\approx 11.37M$ compared to the periastron $r_{\mathrm{per}}\approx 3.33M$.

Figure 6

The precession frequency ${\mathrm{\Omega }}_{(\mathrm{prec})}$ of a test gyroscope in Kerr spacetime as a function of $\tau$ for the same parameter choice as in Fig. 5, but with $\widehat{a}=0.5$.

Figure 7

A plot of ${\mathrm{\Omega }}_{(\mathrm{prec})}$ versus ${\mathrm{\Omega }}_{(\mathrm{orb})}=d\varphi /d\tau$ for $u_p=1/15$ and $e=2$ for selected values of $\widehat{a}$. Positive values of ${\mathrm{\Omega }}_{(\mathrm{prec})}$ correspond to a forward precession (in the same sense as the orbital direction), and negative values to a backward precession. Note that the origin corresponds to $u=0$ ($r\to \infty$) where both frequencies go to 0.

Figure 8

The total precession angle $\mathrm{\Delta }{\varphi }_{(\mathrm{gyro})}$, the deflection angle $\mathrm{\Delta }{\varphi }_{(\mathrm{orb})}$ and their difference $\delta$ (in degrees) for orbits with $\dot{\varphi }>0$ are shown for $e=2$ and $u_p=1/15$ as functions of $\widehat{a}$. Dash-dotted horizontal lines are equally spaced by 180 degrees.