Self-force via $m$-mode regularization and $2+1\mathrm{D}$ evolution: Foundations and a scalar-field implementation on Schwarzschild spacetime

To model the radiative evolution of extreme mass-ratio binary inspirals (a key target of the LISA mission), the community needs efficient methods for computation of the gravitational self-force (SF) on the Kerr spacetime. Here we further develop a practical “$m$-mode regularization” scheme for SF calculations, and give the details of a first implementation. The key steps in the method are (i) removal of a singular part of the perturbation field with a suitable “puncture” to leave a sufficiently regular residual within a finite worldtube surrounding the particle’s worldline, (ii) decomposition in azimuthal ($m$) modes, (iii) numerical evolution of the $m$ modes in $2+1\mathrm{D}$ with a finite-difference scheme, and (iv) reconstruction of the SF from the mode sum. The method relies on a judicious choice of puncture, based on the Detweiler-Whiting decomposition. We give a working definition for the “order” of the puncture, and show how it determines the convergence rate of the $m$-mode sum. The dissipative piece of the SF displays an exponentially convergent mode sum, while the $m$-mode sum for the conservative piece converges with a power law. In the latter case, the individual modal contributions fall off at large $m$ as $m^{-n}$ for even $n$ and as $m^{-n+1}$ for odd $n$, where $n$ is the puncture order. We describe an $m$-mode implementation with a 4th-order puncture to compute the scalar-field SF along circular geodesics on Schwarzschild. In a forthcoming companion paper we extend the calculation to the Kerr spacetime.

Figure 1

Illustration of a typical effective source $S_{\mathrm{eff}}^{[4]}$ derived via Eq. (43) from the 4th-order puncture field ${\Phi }_{\mathcal{P}}^{[4]}$ [given in Eq. (62)]. The plots show $M^3{q}^{-1}{S}_{\mathrm{eff}}^{[4]}$, for a particle on a circular orbit at $r_0=7M$ on Schwarzschild spacetime, as a function of the coordinate differences $\delta r$, $\delta \theta$, $\delta \phi$, with the worldline being at $\delta r=\delta \theta =\delta \phi =0$. The source is shown on the spatial slices (clockwise from top left) $\delta \phi =0$, $\delta \theta =0$ and $\delta r=0$, with $t=\mathrm{const}$. As expected, the 4th-order source is continuous but not differentiable at the worldline.

Figure 2

Visualization of the worldtube in $2+1\mathrm{D}$ domain for a particle on a circular orbit. Here the $t$ axis runs vertically, and the worldtube $\mathcal{T}$ is shown as a boxed domain of fixed width $\{{\Gamma }_r,{\Gamma }_{\theta }\}$, centered on the worldline $\gamma$ at fixed $r=r_0$, $\theta =\pi /2$. Inside the tube $\mathcal{T}$ we evolve ${\Phi }_{\mathcal{R}}^m$, outside the tube we evolve ${\Phi }_{\mathrm{ret}}^m$, and across the tube boundary $\partial \mathcal{T}$ we convert between the two using ${\Phi }_{\mathrm{ret}}^m={\Phi }_{\mathcal{R}}^m+{\Phi }_{\mathcal{P}}^m$.

Figure 3

$2+1\mathrm{D}$ finite-difference scheme. The left plot shows the grid in $u$, $v$ and $\theta$. The right plot shows a single “cell” for the finite-difference method. These plots are reproduced from [59], Figs. 1, 2.

Figure 4

Field modes on constant time slices (at $t=t_{\max }/2$) for a circular orbit at $r_0=7M$ ($r_{*0}\approx 8.8326M$) with the 2nd-order puncture scheme. The left plots show field modes at fixed $\theta =\pi /2$ and the right plots show field modes at fixed $r=r_0$, for a range of modes $m=0$, 1, 2, and 5. Inside the worldtube (visible as the central “trough”), the dashed (red) line shows the full retarded field ${\tilde{\Psi }}_{\mathrm{ret}}^m$ and the solid (blue) line shows the residual field, ${\tilde{\Psi }}_{\mathcal{R}}^m$. The numerical parameters are $\{h=M/8,\Delta =\pi /40,{\Gamma }_{\theta }=\pi /2,{\Gamma }_{r_{*}}=5M\}$.

Figure 5

Modes of the residual field, ${\tilde{\Psi }}_{\mathcal{R}}^m$, as a function of time, evaluated along the worldline $r_{*}=r_{*0}$, $\theta =\pi /2$ [slice (iii) in text]. The initial burst of “junk radiation” (due to the imperfect initial condition) radiates away, and the field approaches a steady state in the vicinity of the worldline.

Figure 6

Sample numerical results for the 4th-order puncture scheme at $r_0=7M$. Here we show ${\tilde{\Psi }}_{\mathrm{ret}}^m$ (outside the worldtube) and ${\tilde{\Psi }}_{\mathcal{R}}^m$ (inside the worldtube) as a function of $r_{*}$ and $\theta$ at late time $t=200M$ for $m=0$, 1, 5 and 10. The worldtube is visible as a thin rectangle of fixed width ${\Gamma }_{\theta }$, ${\Gamma }_{r*}$ around the particle position at $\theta =\pi /2$, $r_{*}=r_{*}(r_0)\approx 8.83M$.

Figure 7

Test of convergence. The left plot shows a mode of the residual field on the worldline, ${\tilde{\Psi }}_{\mathcal{R}}^m$, as a function of time, for resolutions $n_{\mathrm{res}}=16$, 24, 32, 48, 56, and $64$ [here $h=M/n_{\mathrm{res}}$, $\Delta =\pi /(10n_{\mathrm{res}})$, $m=2$, $r_0=6M$, with a 4th-order puncture]. The right plot shows the field extracted at late times ($t=300M$) as a function of grid resolution. The line shows the best fit extrapolation model, $(-1.07487+1.08598h^2-1.0056h^3)\times {10}^{-2}$. See also Fig. 8.

Figure 8

Finite worldtube size effect and extrapolation to zero grid spacing. The plots show typical ($m=2$, $r_0=6M$, 4th-order puncture) modal contributions to the residual field (left) and radial SF (right) as a function of grid resolution ($h=M/n_{\mathrm{res}}$, ${\alpha }_{\mathrm{res}}=10$), for worldtubes of three different widths, i.e., (i) narrow: ${\Gamma }_{r*}=1.25M$, ${\Gamma }_{\theta }=\pi /8$, (ii) medium: ${\Gamma }_{r*}=2.5M$, ${\Gamma }_{\theta }=\pi /4$, and (iii) wide: ${\Gamma }_{r*}=5M$, ${\Gamma }_{\theta }=\pi /2$. Numerical results from runs at six resolutions are shown ($h=M/n_{\mathrm{res}}$ where $n_{\mathrm{res}}=24$, 32, 48, 56, 64 and ${\alpha }_{\mathrm{res}}=10$) as data points. The best fits to the model $X_0+A{h}^2+B{h}^3$ are shown as lines. The extrapolated values for the field (radial SF) vary only within $\sim 0.0004\%$ ($\sim 0.0005\%$), which is considerably less than the relative error of $\sim 0.024\%$ (0.036%) obtained by comparing the highest resolution result with the extrapolated value.

Figure 9

Relaxation towards equilibrium of the $m=0$ modes for a range of orbital radii ($r_0=6M$, $10M$ and $20M$). The left plots show the $m=0$ mode of the field, ${\tilde{\Psi }}_{\mathcal{R}}^{m=0}$, evaluated on the worldline, and the right plots show the same mode of the radial SF, $F_r^{m=0}$, as a function of time $t$, for a low-resolution long time run ($n_{\mathrm{res}}=16$, ${\alpha }_{\mathrm{res}}=10$, $t_{\max }=2000M$). In the appropriate late-time regime, the data are well fitted by simple power-law relaxation models (see Fig. 10).

Figure 10

Power-law relaxation of the $m=0$ mode. The left-hand plot shows the local power-law index $\eta$ of the relaxation of the $m=0$ mode of the field, determined from $\eta (t)=-t{\ddot{\Psi }}_{\mathcal{R}}^{m=0}/{\dot{\Psi }}_{\mathcal{R}}^{m=0}-1$ (where overdot denotes differentiation with respect to $t$, and derivatives are evaluated numerically). The right-hand plot shows $\eta (t)$ for the radial SF mode $F_r^{m=0}$. In the case of the field (left), the index tends towards $\eta =2$, as expected for a noncompact $l=0$ perturbation. In the case of the radial SF (right), the index tends towards $\eta =3$, although for large $r_0$ it takes a long time to reach this asymptotic regime. The higher index in the right plot (i.e., $\eta =3$) is due to the fact that the slowest-decaying part of the monopole is spatially constant [74].

Figure 11

Three-stage multigrid refinement method. A “crude” run (1) provides initial data for an “intermediate” run (2) which in turn provides initial data for a “fine” run (3). Here we show a single slice of the grid at $\theta =\mathrm{const}$.

Figure 12

Sample results comparing multigrid and unigrid evolutions of the $m=0$ mode. The left-hand plots show the evolution of the field mode ${\tilde{\Psi }}_{\mathcal{R}}^{m=0}$ [upper] and radial SF mode $F_r^{m=0}$ [lower] at $r_0=6M$. The right-hand plots show similar data at $r_0=30M$. The dashed lines show unigrid evolutions at three different resolutions, $n_{\mathrm{res}}=4$, 8 and 16. The solid lines show the results of the multigrid scheme. Refinements in resolution occur at $t=500M$ ($n_{\mathrm{res}}=4$ to $n_{\mathrm{res}}=8$) and $t=750M$ ($n_{\mathrm{res}}=8$ to $n_{\mathrm{res}}=16$). After a period of transition dominated by junk radiation, we find that the “refined” data asymptotes to the unigrid data. Note that the multigrid evolution is approximately 9 times faster than the unigrid evolution at equivalent resolution.

Figure 13

Modes of the residual field, ${\tilde{\Psi }}_{\mathcal{R}}^m$, at $r_0=7M$, for puncture orders $n=2$, 3 and 4. The upper plot shows that the small-$m$ modes may take either sign. The lower plot shows power-law falloff ${\tilde{\Psi }}_{\mathcal{R}}^m\sim \mathcal{O}(m^{-\zeta })$ at large $m$, with exponent $\zeta =2$ for the 2nd-order puncture, and $\zeta =4$ for the 3rd- and 4th-order punctures. The dotted lines are reference lines $\propto m^{-2}$ and $\propto m^{-4}$.

Figure 14

Modes of the radial (conservative) SF component, $F_r^m$, at $r_0=7M$, for puncture orders $n=2$, 3 and 4. The upper plot shows that small-$m$ modes may take either sign. The lower plot shows power-law falloff $F_r^m\sim \mathcal{O}(m^{-\zeta })$ at large $m$, with exponent $\zeta =2$ for the 2nd and 3rd-order punctures, and $\zeta =4$ for the 4th-order puncture. The dotted lines are reference lines $\propto m^{-2}$ and $\propto m^{-4}$.

Figure 15

Modes of the angular (dissipative) SF component, $F_{\phi }^m$, at $r_0=7M$ for puncture orders $n=2$, 3 and 4. The plot shows the modal contributions $-F_{\phi }^m$ on a semilog scale. It illustrates that (i) the modes $F_{\phi }^m$ do not depend on the order of the puncture to within numerical error (note that the points in the plot are superimposed upon one another), and (ii) the modal contributions diminish exponentially fast with $m$.

Figure 16

Modal contributions to the radial component $F_r^{\mathrm{self}}$ for a range of orbital radii, with a 4th-order puncture. Plotted here on a log-log scale is the absolute magnitude of $F_r^m$ as a function of $m$. The modal contributions diminish in magnitude as $r_0$ increases (see also Fig. 18), and it appears that all modes are scaled by approximately the same factor. For large $m$, the amplitude of the modal contributions falls off as $\sim m^{-4}$. The asymptotic regime begins around $m\sim 10$ for all radii.

Figure 17

Modal contributions to the angular component $F_{\phi }^{\mathrm{self}}$ for a range of orbital radii. Plotted here on a semilog scale is $-F_{\phi }^m$ (as all modes are negative) as a function of $m$. The modal contributions $F_{\phi }^m$ fall off exponentially with $m$ and the decay rate increases with $r_0$.

Figure 18

Comparing the $m=0$ mode ${\tilde{\Phi }}_{\mathcal{R}}^{m=0}$ (left plot) and $F_r^{m=0}$ (right plot) with the totals ${\Phi }_R$ and $F_r^{\mathrm{self}}$, across a range of orbital radii $r_0$. The dashed (red) line shows the magnitude of the monopole component for our 2nd-order puncture, and the dotted (blue) line shows the same for our 4th-order puncture. The solid (black) line shows the magnitude of the total radiative field (left) and radial SF (right), which scale as $r_0^{-3}$ and $r_0^{-5}$ (respectively) in the limit $r_0\to \infty$. The ratios ${\tilde{\Phi }}_{\mathcal{R}}^{m=0}/{\Phi }_R$ and $F_r^{m=0}/F_r^{\mathrm{self}}$ increase as $r_0$ increases, and hence the accuracy of the mode sum degrades at large $r_0$ due to “mode cancellation error” (see text for discussion).