Self-force with ($3+1$) codes: A primer for numerical relativists

Prescriptions for numerical self-force calculations have traditionally been designed for frequency-domain or ($1+1$) time-domain codes which employ a mode decomposition to facilitate in carrying out a delicate regularization scheme. This has prevented self-force analyses from benefiting from the powerful suite of tools developed and used by numerical relativists for simulations of the evolution of comparable-mass black hole binaries. In this work, we revisit a previously-introduced ($3+1$) method for self-force calculations and demonstrate its viability by applying it to the test case of a scalar charge moving in a circular orbit around a Schwarzschild black hole. Two ($3+1$) codes originally developed for numerical relativity applications were independently employed, and in each we were able to compute the two independent components of the self-force and the energy flux correctly to within $<1\%$. We also demonstrate consistency between the $t$ component of the self-force and the scalar energy flux. Our results constitute the first successful calculation of a self-force in a ($3+1$) framework, and thus open opportunities for the numerical relativity community in self-force analyses and the perturbative modeling of extreme-mass-ratio inspirals.

Figure 1

Equatorial profile of the new effective source, $S_{\mathrm{eff}}$, at $\tilde{t}=0$. The axes are defined simply by $x=r\sin \theta \cos \varphi$ and $y=r\sin \theta \sin \varphi$, where $r$, $\theta$, $\varphi$ are just the Schwarzschild coordinates (or the Kerr-Schild coordinates of Sec. 5a). The charge in this plot is located at $X=10M$ and $Y=0$, where the $C^0$ behavior of the source is not apparent on this scale. Note that much of the structure induced by the new window function is between the charge and the event horizon.

Figure 2

$S_{\mathrm{eff}}$ zoomed in at the location of the charge.

Figure 3

The broken line shows $F_t$ for an outer boundary located at $R_{\mathrm{out}}=210M$. We can see reflections arriving at the location of the particle at around $390M$. If the outer boundary is moved out to $R_{\mathrm{out}}^{\prime }=400M$ (solid line), no such effects can be observed during an evolution time of $600M$.

Figure 4

Differences in $\psi$ at the particle location for runs with different angular resolutions given by $N_{\theta }$. All three cases are for $N_r=29$ and $N_{\varphi }=3N_{\theta }/4$. If we scale the difference between medium and high resolutions by 3.21, the two curves coincide. This corresponds to convergence of an order between 4 and 5.

Figure 5

$F_t$ computed at different angular resolutions of the multiblock code. The resolutions are described by the number of angular grid points per patch, so that $40\times 40$ corresponds to 40 grid points in both $\theta$ and $\varphi$ directions within each patch. The noise in this plot is due to the $C^0$ nature of the effective source. The frequency of this noise corresponds to the particle travel time from one grid point to the next, and the low-frequency modulation corresponds to the particle travel time between patch boundaries.

Figure 6

Comparing $F_t$ results from the multiblock and SGRID codes. The multiblock result was achieved with $60\times 60$ angular resolution and $\Delta r=M/10$ radial resolution, as in Fig. 5. For the SGRID result shown here, the number of collocation points in the angular directions were $N_{\theta }=64$ and $N_{\varphi }=48$. In the $r$ direction, $N_r^{\mathrm{in}}=53$, for the three inner shells. The two outer shells were always set to use $N_r^{\mathrm{out}}=161$ collocation points.

Figure 7

Comparing $F_r$ results from the multiblock and SGRID codes.

Figure 8

$\dot{E}$ computed at $40\times 40$ angular resolution of the multiblock code. The energy flux includes the contribution through the event horizon and an outer boundary defined by $R$. Shown here are results from two outer extraction radii, $R=50M$ and $R=300M$. Energy fluxes are expressed as $F_t$, according to Eq. (22). Also plotted is the local calculation of $F_t$ at the same resolution.

Figure 9

$\dot{E}$ with the SGRID code from the same run described in Fig. 6. This made use of an outer extraction radius of $R=150M$. Also plotted is the result of the local $F_t$ calculation.

Figure 10

Dependence of $\dot{E}$ on various extraction radii. Farther extraction radii are observed to yield better results.

Figure 11

$\dot{E}$ computed with the multiblock code and extrapolated to infinite extraction radius.

Figure 12

Scaled errors for second order convergence in $\rho$ at $t=3$ with a $C^0$ source.

Figure 13

Scaled errors for fourth order convergence in $\rho$ at $t=3$ with a $C^{\infty }$ source.