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

We present a numerical code for calculating the self-force on a scalar charge moving in a bound (eccentric) geodesic in the equatorial plane of a Kerr black hole. We work in the frequency domain and make use of the method of extended homogeneous solutions [Phys. Rev. D 78, 084021 (2008)], in conjunction with mode-sum regularization. Our work is part of a program to develop a computational architecture for fast and efficient self-force calculations, alternative to time-domain methods. We find that our frequency-domain method outperforms existing time-domain schemes for small eccentricities, and, remarkably, remains competitive up to eccentricities as high as $\sim 0.7$. As an application of our code, we (i) compute the conservative scalar-field self-force correction to the innermost stable circular equatorial orbit, as a function of the Kerr spin parameter; and (ii) calculate the variation in the rest mass of the scalar particle along the orbit, caused by the component of the self-force tangent to the four-velocity.

Figure 1

Behavior of the $l$-mode contributions to the mode sum for $F_t$ (left panel) and $F_r$ (right panel), shown separately for the conservative and dissipative pieces, as well as for their sum. Here we have set $(a,p,e)=(0.5M,10M,0.2)$ and the SSF is calculated at $\chi =\pi /2$. The straight solid line is a reference line $\propto l^{-2}$. From standard mode-sum theory we expect the $l$-mode contributions to the full SSF and to the conservative component of the SSF to fall off as $l^{-2}$ for large $l$. The curved solid line is an exponential reference line. We expect the $l$-mode contributions to the dissipative component of the SSF to decay exponentially with $l$. Numerical round-off error dominates the behavior of the dissipative modes for $l\gtrsim 15$.

Figure 2

Same as in Fig. 1, this time showing the $l$-mode contributions to the SSF at $\chi =0.010472$ ($=2\pi /300$), i.e., very close to periastron at $r=r_{\min }$. Near the orbital turning points the contribution to the full $F_t$ (triangles, left panel) transitions from an exponential decay to an $\propto l^{-2}$ fall off. This transition (in this example around $l\approx 8$) makes extrapolating the large-$l$ tail of the full SSF difficult in this portion of the orbit. This problem is completely avoided if we separate the SSF into conservative and dissipative pieces, calculate the large-$l$ tail contribution to the conservative piece, and then add the two pieces together to recover the full SSF. Similar behavior is observed for the $F_{\phi }$ component (not shown). No transition is observed for $F_r$ (right panel) as in this case the conservative piece dominates the SSF even near the orbital turning points.

Figure 3

Sample SSF results. The top left panel shows orbits with $(p,e)=(10M,0.5)$ in the equatorial plane, for three different black-hole spins: $a=0$ (dotted, blue curve), $a=0.9M$ (solid, red curve), and $a=-0.5M$ (dashed, green curved). For each orbit we show one complete revolution, from one periastron to the next, with markers indicating the points taken for the sample data in Table . The other three panels show (reading clockwise) the $F_r$, $F_{\phi }$, and $F_t$ components of the SSF, for the three orbits shown in the top left panel. $\chi =0$ corresponds to a periastron passage.

Figure 4

Shown in the left panel is a “zoom-whirl”-type orbit with parameters $(a,p,e)=(0.2M,6.15M,0.4)$. The right panel shows the corresponding components of the SSF along the orbit. $\chi =0$ is a periastron of the orbit. Markers indicate the location of the data points shown in Table .

Figure 5

Computational cost. We show the total time required to calculate up to (a fiducial) $l_{\max }=15$, leading to fractional accuracies of order $\sim {10}^{-4}$ in the SSF. In the Schwarzschild case ($a=0$) we can compute the SSF for eccentricities up to $e=0.7$ in around 12 hours. For $a=0.9M$ the calculation requires more time, primarily because the coupling between the spheroidal and spherical-harmonic modes necessitates calculation of higher spheroidal-harmonic modes, which is computationally expensive. At low eccentricities, our frequency-domain algorithm is vastly faster than any existing time-domain method.

Figure 6

Calculation of $F_{r\mathrm{is}}^1$, $F_{t\mathrm{is}}^1$, and $F_{\phi \mathrm{is}}^1$ by extrapolation along three paths in the $e\mathrm{\text{−}}p$ plane for the case of a Schwarzschild black hole ($a=0$). The sampling paths in the $e\mathrm{\text{−}}p$ plane are shown in the bottom right panel. The other three panels show numerical data points for ${\widehat{F}}_r^1$, ${\widehat{F}}_{\phi }^1$, and ${\widehat{F}}_t^1$ as extracted from the conservative piece of the SSF using Eqs. (106, 107). Solid curves are cubic interpolations of the numerical data points, and the extrapolated values at $p=6$ (which, in theory, should not depend on the choice of curve) represent our numerical predictions for $F_{r\mathrm{is}}^1$, $F_{t\mathrm{is}}^1$, and $F_{\phi \mathrm{is}}^1$. The small variance in these extrapolated values serves as a rough measure of error. The thick dot on the vertical axis marks the values found by Diaz-Rivera et al. [19]. For $F_{r\mathrm{is}}^1$ and $F_{\phi }^{1\mathrm{is}}$ we find a close agreement with their results, but for $F_t^{1\mathrm{is}}$ there is a discrepancy at a level ($\sim 2\%$) which we cannot explain. (The code used by Diaz-Rivera et al. cannot be retrieved to allow a careful examination of this discrepancy [36]; we are, however, quite confident in our results given the good agreement between the values extrapolated from the different curves.)

Figure 7

Same as in Fig. 6 but for $a=0.9M$. We use the paths shown on the graphs to extract the values of $F_{r\mathrm{is}}^1$, $F_{t\mathrm{is}}^1$, and $F_{\phi \mathrm{is}}^1$ via extrapolation to the ISCEO, whose location at $r_{\mathrm{is}}=2.32088M$ is marked by the vertical line. (The results for $F_{\phi \mathrm{is}}^1$, for brevity not shown here, are qualitatively similar to those of $F_{t\mathrm{is}}^1$.) Sample numerical values for $F_{r\mathrm{is}}^1$, $F_{t\mathrm{is}}^1$, and $F_{\phi \mathrm{is}}^1$, and the resulting ISCEO shifts for different spins $a$, can be found in Table .

Figure 8

The conservative SSF effect upon the ISCEO location and frequency. We plot here the numerical results shown in Table  as a function of the Kerr spin parameter $a$ (curves are cubic interpolations). Vertical error bars indicate the estimated numerical error. Notice in the graph that two separate scales are being used for $\Delta r_{\mathrm{is}}$ (left-hand scale) and $\Delta {\Omega }_{\phi \mathrm{is}}/{\Omega }_{\phi \mathrm{is}}$ (right-hand scale). The radial shift is monotonically increasing with $a$ and changes sign around $a=0.8M$. The frequency shift similarly changes its sign (and turns negative) at large spin values. Note that although the change of sign in the radial and frequency shifts occur near the same spin value ($a=\sim 0.8M$) the error bars on our results exclude the possibility of the sign change occurring at the same spin value. The Schwarzschild ISCO shift results of Diaz-Rivera et al. [19] are marked (green, thick dots) for comparison. The small discrepancy is discussed briefly in the caption of Fig. 6.

Figure 9

Rest-mass change due to the SSF for a scalar charge in an eccentric equatorial orbit about a Kerr black hole. (Left panel) The variation in the rest mass, for orbits about a Schwarzschild black hole, as a function of $\chi$, is strongly dependent on the orbit’s eccentricity. For $e$ and $p$ far away from the separatrix, the mass initially increases and then returns to its original value at apastron before further decreasing and then once again returning to the original value at periastron. For a zoom-whirl-type orbit, the mass is observed to decrease slightly initially. (Right panel) Results for Kerr. The change in mass is weakly dependent on the black-hole spin.