Self-force on a scalar charge in Kerr spacetime: Circular equatorial orbits

We present a calculation of the scalar-field self-force (SSF) acting on a scalar-charge particle in a strong-field orbit around a Kerr black hole. Our calculation specializes to circular and equatorial geodesic orbits. The analysis is an implementation of the standard mode-sum regularization scheme: We first calculate the multipole modes of the scalar-field perturbation using numerical integration in the frequency domain, and then apply a certain regularization procedure to each of the modes. The dissipative piece of the SSF is found to be consistent with the flux of energy and angular-momentum carried by the scalar waves through the event horizon and out to infinity. The conservative (radial) component of the SSF is calculated here for the first time. When the motion is retrograde this component is found to be repulsive (outward pointing, as in the Schwarzschild case) for any spin parameter $a$ and (Boyer-Lindquist) orbital radius $r_0$. However, for prograde orbits we find that the radial SSF becomes attractive (inward pointing) for $r_0>r_c(a)$, where $r_c$ is a critical $a$-dependent radius at which the radial SSF vanishes. The dominant conservative effect of the SSF in Schwarzschild spacetime is known to be of third post-Newtonian (3PN) order (with a logarithmic running). Our numerical results suggest that the leading-order PN correction due to the black hole’s spin arises from spin-orbit coupling at 3PN order, which dominates the overall SSF effect at large $r_0$. In PN language, the change of sign of the radial SSF is attributed to an interplay between the spin-orbit term ($\propto -a{r}_0^{-4.5}$) and the Schwarzschild term ($\propto r_0^{-5}\log r_0$).

Figure 1

Coupling of spheroidal and spherical modes, illustrated here for $a=0.9M$ and $r_0=4M$. Shown are the contributions from a given $\widehat{l}$ mode to $b_{lm}^{\widehat{l}}$ (left panel) and $F_{r+}^{(\mathrm{full})l}$ (right panel), for various spherical-harmonic $l$ modes. (Note that $b_{lm}^{\widehat{l}}=0$ identically for odd values of $l-\widehat{l}$.) The width of the $l$ distribution depends mainly on the magnitude of the spheroidicity parameter, $|{\sigma }^2|=a^2{\omega }^2=a^2{m}^2{\Omega }_{\varphi }^2$; the two cases shown, $(\widehat{l},m)=(44,34)$ and $(\widehat{l},m)=(44,10)$, have spheroidicities ${\sigma }^2=-11.821$ and $-1.022$, respectively. The point of this illustration is to note that, in practice, one only needs to calculate a handful more spheroidal $\widehat{l}$ modes than the desired maximum spherical $l$ mode, especially for smaller $a$ and/or larger $r_0$, where the coupling is weaker than demonstrated above.

Figure 2

Left panel: The regularized modes $F_r^{l(\mathrm{reg})}$ as a function of $l$ for $r_0=5M$ and $a=0.5M$. The solid reference line is $\propto 1/l^2$. The regularized modes demonstrate an asymptotic $\propto 1/l^2$ behavior at large $l$, as expected from theory (note the log-log scale). Right panel: The regularized modes $F_t^{l(\mathrm{reg})}$ as a function of $l$ for $r_0=5M$ and $a=0.8M$. The solid reference line is exponentially decreasing with $l$. The regularized modes of the $t$ component show a clear exponential decay at large $l$, as expected from theory (note the semilog scale). Similar behavior is observed for other values of $r_0$ and $a$.

Figure 3

The horizon flux of scalar-field energy, ${\dot{E}}_{-}$, as a percentage of the total flux for different orbital radii $r_0$ and spin parameters $a$. The curves are interpolations based on the numerical data points shown. Superradiance behavior (${\dot{E}}_{-}<0$) is manifest whenever the horizon’s angular velocity ${\Omega }_{+}$ is greater than that of the particle.

Figure 4

Time component of the SSF: comparison with Gal’tsov’s slow-motion formula. Plotted is the relative difference between our full SSF $F_t$ and Gal’tsov’s weak-field/slow-motion analytic approximation (58) as a function of orbital radius $r_0$. Solid lines are interpolations of the data points shown. We show results for $a=\pm 0.998M$; similar agreement between $F_t$ and $F_t^{{\mathrm{Gal}}^{\prime }\mathrm{tsov}}$ at large $r_0$ is manifest for other values of $a$, too.

Figure 5

The radial component of the SSF, multiplied by $r_0^5$ for convenience, across the $a$, $r_0$ parameter space. Contour lines are lines of fixed $r_0^5{F}_r$, with labels giving the value of $(M/q{)}^2(r_0/M{)}^5{F}_r$. The near-vertical thick line indicates the location of the ISCO, while the near-horizontal thick line marks the curve $r_0=r_c(a)$ along which the radial SSF vanishes. The two lines intersect at $a\simeq 0.461M$; for $a\gtrsim 0.461M$ all stable circular geodesics experience an attractive radial SSF.

Figure 6

Left panel: Radial component of the SSF as a function of $a$ for various fixed values of the orbital radius $r_0$. Right panel: Radial component of the SSF as a function of $r_0$ for various fixed values of the spin parameter $a$. In both panels dots represent numerical data points, and solid lines are interpolations.

Figure 7

Comparison of numerical data for $F_r$ (dots) with the PN fit model (61) (solid lines). For prograde orbits with $a\lesssim 0.461M$ the radial SSF changes sign at $r_0=r_c(a)$; cf. Figs. 5, 6.