Gravitational self-force on eccentric equatorial orbits around a Kerr black hole

This paper presents the first calculation of the gravitational self-force on a small compact object on an eccentric equatorial orbit around a Kerr black hole to first order in the mass ratio. That is the pointwise correction to the object’s equations of motion (both conservative and dissipative) due to its own gravitational field, which is treated as a linear perturbation to the background Kerr spacetime generated by the much larger spinning black hole. The calculation builds on recent advances on constructing the local metric and self-force from solutions of the Teukolsky equation, which led to the calculation of the Detweiler-Barack-Sago redshift invariant on eccentric equatorial orbits around a Kerr black hole in a previous paper. After deriving the necessary expression to obtain the self-force from the Weyl scalar ${\psi }_4$, we perform several consistency checks of the method and numerical implementation, including a check of the balance law relating the orbital average of the self-force to the average flux of energy and angular momentum out of the system. Particular attention is paid to the pointwise convergence properties of the sum over frequency modes in our method, identifying a systematic inherent loss of precision that any frequency domain calculation of the self-force on eccentric orbits must overcome.

Figure 1

Illustration of the convergence of the frequency modes for large radial mode number $|n|$. The plotted lines correspond to the sup-norms of $F_{lm\omega }^t$ obtained for an orbit with $(a,p,e)=(0.9,5.5,0.3)$ and normalized by the sup-norm of the complete time domain mode $F_{lm}^t$. Solid lines represent values obtained from the outside field, whereas dashed lines represent inside values. The different colors indicate various combinations of $l$ and $m$. The diagonal gridlines indicate a reference decay of $e^{|n|}$.

Figure 2

Illustration of the loss of precession in construction the $l$ modes from the frequency domain modes as a function of $l$ for an orbit with parameters $(a,p,e)=(0.9,5.5,0.3)$. The loss of precision is measured by taking the maximal value of the sup-norms of all the frequency domain modes $F_{lm\omega }^t$ contributing to a certain $l$ mode $F_l^t$ normalized by the sup-norm of the complete $l$ mode. The diagonal gridlines show a reference growth proportional to $(\frac{x_{\max }}{x_{\min }}{)}^l$ with $x=(r-r_{+})$.

Figure 3

The anatomy of a typical homogeneous solution to the $s=+2$ Teukolsky equation. The particular solutions plotted are for $a=0.9$, $l=20$, $m=1$, and $\omega =0.5$ (as solid lines) and $\omega =0$ (as dashed lines). Qualitatively, other solutions look the same. At large radii, the $\omega \ne 0$ modes scale as $z^{-5}$ for the outside modes and ${\zeta }^{-1}$ for the inside mode. In contrast, the static modes scale as ${\zeta }^{-l-3}$ and ${\zeta }^{l-2}$ respectively. Below a radius $\zeta \sim 12$ the oscillating modes become similar to the static modes initially showing the same ${\zeta }^{-l-3}$ and ${\zeta }^{l-2}$ behavior. Near to the horizon, all modes scale as ${\zeta }^{-2}$. (For the sake of this figure all modes have been normalized to numerically agree in this limit.)

Figure 4

An illustration of the precision loss that occurs when summing over all frequency modes as a function of the orbital phase $q_r$. Shown are both the maximal contributions from the frequency modes $F_{20m\omega }^{r,\pm }$ to the $l$ mode $F_{20}^{r,\pm }$, and the final $l$ mode (both the one side values and the two-sided average). The “$+$” modes lose most precision at periastron and the “$-$” modes lose most precision at apastron. The values are obtained for our standard reference orbit with parameters $(a,p,e)=(0.9,5.5,0.3)$.

Figure 5

The $l$ modes of the self-force on an orbit with parameters $(a,p,e)=(0.9,5.5,0.3)$ (before regularization) at a generic point $q_r=\pi /2$ along the orbit. The inside and outside values follow the expected $\mathcal{O}(l)$ growth (indicated by the diagonal gridlines), while the two-sided averages asymptote to a constant value.

Figure 6

The same $l$ modes as in Fig. 5 after subtracting the regularization parameters. All components follow the expected $\mathcal{O}(l^{-2})$ behavior.

Figure 7

Convergence of the (regularized) $l$ modes at apapsis ($q_r=0$). Despite the $t$ and $\varphi$ components of the self-force being regular at this point, the $l$ mode decay only as $\mathcal{O}(l^{-2})$ due to the nonsmoothness of the extension.

Figure 8

Similar, to Fig. 7 but now evaluated at periapsis ($q_r=\pi$). Again we see that the $t$ and $\varphi$ components show convergence no faster than $\mathcal{O}(l^{-2})$ due to the nonsmoothness of the extension.

Figure 9

Self-force loops for an orbit with parameters $(a,p,e)=(0,7,0.25)$ calculated in two different gauges using the radiation gauge method of the paper and the Lorenz gauge code of [35]. We find partial overlap in the loops for $F^t$ as is necessitated by the balance law. However, the $F^r$ loops for different gauges are completely disjoint, stressing the gauge dependence of the self-force.

Figure 10

“Self-force loops” for a variety of orbits. On each plot the horizontal axis shows the radial position $r$ of the particle, and the vertical axis shows a component of the self-force, rescaled by an appropriate power of $r$ ($r^3$ for $F^t$ and $F^r$, and $r^5$ for $F^{\varphi }$). Each column shows one component of the self-force ($F^t$ on the left $F^r$ in the middle, and $F^{\varphi }$ on the right), while each row shows orbits with a fixed spin $a$ and semilatus rectum $p$. Complete data for these plots is available as supplementary data [101].