Self-force via $m$-mode regularization and $2\mathbf{+}1\mathbf{D}$ evolution. II. Scalar-field implementation on Kerr spacetime

This is the second in a series of papers aimed at developing a practical time-domain method for self-force calculations in Kerr spacetime. The key elements of the method are (i) removal of a singular part of the perturbation field with a suitable analytic “puncture” based on the Detweiler-Whiting decomposition, (ii) decomposition of the perturbation equations in azimuthal ($m$-)modes, taking advantage of the axial symmetry of the Kerr background, (iii) numerical evolution of the individual $m$-modes in $2+1$ dimensions with a finite-difference scheme, and (iv) reconstruction of the physical self-force from the mode sum. Here we report an implementation of the method to compute the scalar-field self-force along circular equatorial geodesic orbits around a Kerr black hole. This constitutes a first time-domain computation of the self-force in Kerr geometry. Our time-domain code reproduces the results of a recent frequency-domain calculation by Warburton and Barack, but has the added advantage of being readily adaptable to include the backreaction from the self-force in a self-consistent manner. In a forthcoming paper—the third in the series—we apply our method to the gravitational self-force (in the Lorenz gauge).

Figure 1

Grid for finite-difference method. The left plot shows the $2+1\mathrm{D}$ grid with linear spacings ($△t$, $△r_{*}$, $△\theta$). The evolution proceeds from initial data on the surface $t=0$ (shaded). The right plot illustrates a grid cell for the method-of-lines finite-difference method (with the $\theta$ direction suppressed for clarity). To obtain the value at point ($j$, $n+1$), we first use the values at the 5 grid points $(j-2,n),\dots ,(j+2,n)$ to obtain estimates on an intermediate time slice (unfilled circles); cf. Eq. (58). The domain of dependence of the grid point at ($j$, $n+1$) is shown with (densely) dotted lines.

Figure 2

Field modes ($m=0$, 1, 2, 5) on a constant time slice (at $t=t_{\max }$) for circular orbits at $r_0=10M$, for a range of Kerr parameters ($a/M=-0.9$, $-0.5$, 0, 0.5, 0.9). The left plots show field modes at fixed $\theta =\pi /2$ and the right plots show field modes at fixed $r=r_0$. Inside the worldtube we show both the residual field ${\widehat{\Psi }}_{\mathcal{R}}^m$ (forming the “trough”) and the full retarded field ${\widehat{\Psi }}^m$, which is divergent on the worldline. We note that the rotation rate $a/M$ has only a subtle effect on the field profile.

Figure 3

Typical time series data for the modes of the field ${\widehat{\Psi }}_{\mathcal{R}}$ (left column), the radial SF $F_r$ (center column) and the angular SF $F_{\varphi }$ (right column) for a range of $m$ modes (top to bottom: $m=0$, 2, 4, 6). These data are taken from simulations with radius $r_0=10M$ and Kerr parameter $a=0.5M$, with numerical parameters $△t=△r_{*}=M/32$, $△\theta =\pi /192$ and ${\Gamma }_{r*}=2.5M$, ${\Gamma }_{\theta }=\pi /4$. Note that $F_{\varphi }^{m=0}$ is identically zero.

Figure 4

Convergence of the FD scheme and Richardson extrapolation. The left plot shows the $m=2$ mode of the residual field on the worldline, ${\widehat{\Psi }}_{\mathcal{R}}^m$, for $a=0$ and a circular orbit at $r_0=6M$, as a function of time. The lines show the results from simulations with various grid resolutions, $x=1/16$, $1/20$, $1/24$, $1/28$ and $1/32$, where $x\equiv △t/M=△r_{*}/M=6△\theta /\pi$. The right plot shows the values at $t_{\max }=300M$ as a function of $x$. The red circles on the upper line show the values taken from the simulations in the left plot. The blue squares on the lower line show the values from the $uv\theta$ method (which only works for $a=0$). The two data sets have been fitted with cubic polynomial models, ${\widehat{\Psi }}_{\mathcal{R}}^m=c_0+c_2{x}^2+c_3{x}^3$ (red and blue lines). Reassuringly, the values $c_0$ extrapolated from the two data sets (which correspond to the zero grid spacing limit) are found to be in very good agreement.

Figure 5

Convergence of modal contributions as a function of $m$, for a fourth-order puncture implementation ($r_0=10M$, $t_{\max }=300M$). The upper left plot shows (minus) the $m$-modes of the angular (dissipative) component $F_{\varphi }$, on a semilog scale. We observe exponential convergence with $m$. The upper right plot shows the $m$-modes of the radial (conservative) component $F_r$, on a log-log scale. The dotted reference line is $\propto m^{-4}$. The lower plot shows the same data but rescaled by the expected power law for a fourth-order puncture, i.e., $m^4{F}_r^m$. We show a range of values of the Kerr parameter, $a/M=-0.9$, $-0.5$, 0, 0.5, 0.9 and we observe that rotation does not affect the convergence rate.