Gravitational self-force on generic bound geodesics in Kerr spacetime

In this work we present the first calculation of the gravitational self-force on generic bound geodesics in Kerr spacetime to first order in the mass ratio. That is, the local correction to equations of motion for a compact object orbiting a larger rotating black hole due to its own impact on the gravitational field. This includes both dissipative and conservative effects. Our method builds on and extends earlier methods for calculating the gravitational self-force on equatorial orbits. In particular we reconstruct the local metric perturbation in the outgoing radiation gauge from the Weyl scalar ${\psi }_4$, which in turn is obtained by solving the Teukolsky equation using semianalytical frequency domain methods. The gravitational self-force is subsequently obtained using (spherical) $l$-mode regularization. We test our implementation by comparing the large $l$-behavior against the analytically known regularization parameters. In addition we validate our results by comparing the long-term average changes to the energy, angular momentum, and Carter constant to changes to these constants of motion inferred from the gravitational wave flux to infinity and down the horizon.

Figure 1

The $l$-modes of the various components of the GSF on a geodesic with $(a,p,e,z_{\max })=(0.9,10,0.1,0.1)$, shown at the point along the orbit identified by $(q_r,q_z)=(\pi /3,\pi /6)$. This point is indicative of the generic behavior (certain special points such as periapsis and apapsis will show better convergence behavior). The gray lines are reference lines of $L=l+1/2$. As expected the $\pm$ parts of the $t$ and $r$ components diverge with $L$. The parameters $A_z$ and $A_{\varphi }$ vanish [56, 69]. Consequently, we see $\pm$ parts of the $r$ and $\varphi$ $l$-modes converge to a constant, just like all the two-side average parts.

Figure 2

The same $l$-modes as in Fig. 1 after subtracting the Lorenz gauge regularization parameters. At large $l$, all components of the GSF conform with $L^{-2}$ behavior indicated by the gray reference lines. This is a stringent check on the validity of our method and numerical implementation.

Figure 3

Time series data of the GSF on an orbit with $(a,p,e,z_{\max })=(0.9,10,0.1,0.5)$. Because of the biperiodic nature of the GSF none of the modulation patterns ever repeat.

Figure 4

GSF as a function on the torus for an orbit with $(a,p,e,z_{\max })=(0.9,10,0.1,0.1)$. The horizontal axis displays changing $q_r$, while the vertical axis displays $q_z$.

Figure 5

GSF as a function on the torus for an orbit with $(a,p,e,z_{\max })=(0.9,10,0.1,0.3)$. The horizontal axis displays changing $q_r$, while the vertical axis displays $q_z$.

Figure 6

GSF as a function on the torus for an orbit with $(a,p,e,z_{\max })=(0.9,10,0.1,0.5)$. The horizontal axis displays changing $q_r$, while the vertical axis displays $q_z$.

Figure 7

GSF as a function on the torus for an orbit with $(a,p,e,z_{\max })=(0.9,10,0.1,0.7)$. The horizontal axis displays changing $q_r$, while the vertical axis displays $q_z$.

Figure 8

GSF as a function on the torus for an orbit with $(a,p,e,z_{\max })=(0.9,10,0.1,0.9)$. The horizontal axis displays changing $q_r$, while the vertical axis displays $q_z$.

Figure 9

Fourier spectrum of the GSF of an orbit with $(a,p,e,z_{\max })=(0.9,10,0.1,0.1)$. The horizontal axis varies the radial mode number $n$, while the vertical axis displays the polar mode number $k$. Modes with amplitudes below the error threshold are shown as white.

Figure 10

Fourier spectrum of the GSF of an orbit with $(a,p,e,z_{\max })=(0.9,10,0.1,0.3)$. The horizontal axis varies the radial mode number $n$, while the vertical axis displays the polar mode number $k$. Modes with amplitudes below the error threshold are shown as white.

Figure 11

Fourier spectrum of the GSF of an orbit with $(a,p,e,z_{\max })=(0.9,10,0.1,0.5)$. The horizontal axis varies the radial mode number $n$, while the vertical axis displays the polar mode number $k$. Modes with amplitudes below the error threshold are shown as white.

Figure 12

Fourier spectrum of the GSF of an orbit with $(a,p,e,z_{\max })=(0.9,10,0.1,0.7)$. The horizontal axis varies the radial mode number $n$, while the vertical axis displays the polar mode number $k$. Modes with amplitudes below the error threshold are shown as white.

Figure 13

Fourier spectrum of the GSF of an orbit with $(a,p,e,z_{\max })=(0.9,10,0.1,0.9)$. The horizontal axis varies the radial mode number $n$, while the vertical axis displays the polar mode number $k$. Modes with amplitudes below the error threshold are shown as white.