Spin-geodesic deviations in the Kerr spacetime

The dynamics of extended spinning bodies in the Kerr spacetime is investigated in the pole-dipole particle approximation and under the assumption that the spin-curvature force only slightly deviates the particle from a geodesic path. The spin parameter is thus assumed to be very small and the back reaction on the spacetime geometry neglected. This approach naturally leads to solve the Mathisson-Papapetrou-Dixon equations linearized in the spin variables as well as in the deviation vector, with the same initial conditions as for geodesic motion. General deviations from generic geodesic motion are studied, generalizing previous results limited to the very special case of an equatorial circular geodesic as the reference path.

Figure 1

Reference path: equatorial circular geodesic. Panels (a) and (b) show the behavior of the perturbed radius and polar angle (red curves) as functions of the proper time, respectively (unperturbed quantities in dashes). Panel (c) shows the behavior of the frame components of the deviation vector $Y$ as functions of the proper time ($Y^{\widehat{1}}$ red, $Y^{\widehat{2}}$ black, $Y^{\widehat{3}}$ blue). Panel (d) shows the behavior of the magnitude of the displacement vector $\xi$ as a function of the proper time. The associated reference geodesic is an equatorial circular orbit at $r_0=8M$, so that $E\approx 0.94$, $L\approx 3.32$ and $K\approx 8.1$, the black hole rotation parameter being set to $a/M=0.5$. The spin parameter [only necessary for Figs. (a) and (b)] has been chosen as $\sigma /M=0.05$ and the spin orientation has the general orientation $N^{\widehat{1}}=0.5=N^{\widehat{3}}$, $N^{\widehat{2}}\approx -0.71$. Note that the selected value for $\sigma$, even if representing a reference value, is not so far from actual astrophysical ones (see discussion at the end of Sec. 2). The choice of initial conditions is the following: $t_{(g)}(0)=0$, $r_{(g)}(0)=8M$, ${\theta }_{(g)}(0)=\pi /2$, ${\varphi }_{(g)}(0)=0$, ${\xi }^{\widehat{a}}(0)=0=d{\xi }^{\widehat{a}}/d{\tau }_{(g)}(0)$, $\psi (0)=0$.

Figure 2

Reference path: equatorial noncircular geodesic. Panel (a) shows the projection of the perturbed trajectory on the $X\mathrm{\text{−}}Y$ plane (red curve), where $X=r\cos \varphi$ and $Y=r\sin \varphi$ are Cartesian-like coordinates (unperturbed orbit in dashes). Panel (b) shows the behavior of the perturbed polar angle as a function of the proper time (unperturbed orbit in dashes). Panel (c) shows the behavior of the frame components of the deviation vector $Y$ as functions of the proper time ($Y^{\widehat{1}}$ red, $Y^{\widehat{2}}$ black, $Y^{\widehat{3}}$ blue). Panel (d) shows the behavior of the magnitude of the displacement vector $\xi$ as a function of the proper time. The parameters of the reference equatorial geodesic are chosen so that $E=0.95$, $L=3.1$, $K\approx 6.9$ and ${ϵ}_r=-1$. The values of black hole rotation parameter, spin orientation and spin parameter as well as initial conditions are the same as in Fig. 1. The geodesic path crosses the horizon (black disk at $r_{+}\approx 1.867M$) at ${\tau }_{(g)}\approx 39.7$, whereas the path of the spinning particle becomes nearly circular at $r\approx 3.71M$. Correspondingly the deviation components take the values $Y^{\widehat{1}}\approx -7.56$, $Y^{\widehat{2}}\approx -0.67$ and $Y^{\widehat{3}}\approx -45.9$.

Figure 3

Reference path: equatorial noncircular geodesic. Panel (a) shows the projection of the perturbed trajectory on the $X\mathrm{\text{−}}Y$ plane (red solid curve; unperturbed orbit in dashes). Panel (b) shows the behavior of the perturbed polar angle as a function of the proper time (unperturbed quantity in dashes). Panel (c) shows the behavior of the frame components of the deviation vector $Y$ as functions of the proper time ($Y^{\widehat{1}}$ red, $Y^{\widehat{2}}$ black, $Y^{\widehat{3}}$ blue). Panel (d) shows the behavior of the magnitude of the displacement vector $\xi$ as a function of the proper time. The parameters of the reference equatorial geodesic are chosen so that $E=0.95$, $L=5$, $K\approx 12.4$ and ${ϵ}_r=-1$. The values of black hole rotation parameter, spin orientation and spin parameter as well as initial conditions are the same as in Fig. 1.

Figure 4

Reference path: equatorial noncircular geodesic. Panels (a) shows the projection of the perturbed trajectory on the $X\mathrm{\text{−}}Y$ plane (red solid curve; unperturbed orbit in dashes). Panels (b) shows the behavior of the perturbed polar angle as a function of the proper time (unperturbed quantity in dashes). Panel (c) shows the behavior of the frame components of the deviation vector $Y$ as functions of the proper time ($Y^{\widehat{1}}$ red, $Y^{\widehat{2}}$ black, $Y^{\widehat{3}}$ blue). Panel (d) shows the behavior of the magnitude of the displacement vector $\xi$ as a function of the proper time. The parameters of the reference equatorial geodesic are chosen so that $E=0.95$, $L=3.3$, $K\approx 8$ and ${ϵ}_r=-1$. The values of black hole rotation parameter, spin orientation and spin parameter as well as initial conditions are the same as in Fig. 1.

Figure 5

Reference path: nonequatorial circular geodesic. Panels (a) and (b) show the projections of the perturbed trajectory on the $X\mathrm{\text{−}}Y$ plane and $X-Z$ plane (red solid curves; unperturbed orbits in dashes), respectively, where $X=r\sin \theta \cos \varphi$, $Y=r\sin \theta \sin \varphi$ and $Z=r\cos \theta$ are Cartesian-like coordinates. Panel (c) shows the behavior of the frame components of the deviation vector $Y$ as functions of the proper time ($Y^{\widehat{1}}$ red, $Y^{\widehat{2}}$ black, $Y^{\widehat{3}}$ blue). Panel (d) shows the behavior of the magnitude of the displacement vector $\xi$ as a function of the proper time. The background spin parameter has been set to $a/M=0.5$. The associated reference geodesic is a spherical geodesic at $r_0=8M$ with ${ϵ}_{\theta }=-1$ and $K=5$, so that $E\approx 0.94$ and $L\approx 5.56$. The values of black hole rotation parameter, spin orientation and spin parameter as well as initial conditions are the same as in Fig. 1.

Figure 6

Reference path: nonequatorial noncircular geodesic. Panels (a) and (b) show the projections of the perturbed trajectory on the $X\mathrm{\text{−}}Y$ plane and $X\mathrm{\text{−}}Z$ plane (red solid curves; unperturbed orbits in dashes), respectively. Panel (c) shows the behavior of the frame components of the deviation vector $Y$ as functions of the proper time ($Y^{\widehat{1}}$ red, $Y^{\widehat{2}}$ black, $Y^{\widehat{3}}$ blue). Panel (d) shows the behavior of the magnitude of the displacement vector $\xi$ as a function of the proper time. The parameters of the reference geodesic are chosen so that $E=0.9$, $L=3$, $K=10.5$, ${ϵ}_r=-1$ and ${ϵ}_{\theta }=1$. The values of black hole rotation parameter, spin orientation and spin parameter as well as initial conditions are the same as in Fig. 1.