Time-domain metric reconstruction for self-force applications

We present a new method for calculation of the gravitational self-force (GSF) in Kerr geometry, based on a time-domain reconstruction of the metric perturbation from curvature scalars. In this approach, the GSF is computed directly from a certain scalarlike self-potential that satisfies the time-domain Teukolsky equation on the Kerr background. The approach is computationally much cheaper than existing time-domain methods, which rely on a direct integration of the linearized Einstein’s equations and are impaired by mode instabilities. At the same time, it retains the utility and flexibility of a time-domain treatment, allowing calculations for any type of orbit (including highly eccentric or unbound ones) and the possibility of self-consistently evolving the orbit under the effect of the GSF. Here we formulate our method, and present a first numerical application, for circular geodesic orbits in Schwarzschild geometry. We discuss further applications.

Figure 1

Relaxation of the numerical solution at late time. Shown here are numerical results for ${\varphi }_{20}^{\pm }(t,r_0)$ (divided by $\mu /M^2$), i.e. the $(\ell ,m)=(2,0)$ mode of the IRG Hertz potential along the particle’s orbit (where it is discontinuous), as a function of time. The orbit is a circular geodesic with $r_0=7M$. The early part of the solution is dominated by nonphysical junk radiation. The solution relaxes at late time to a stationary value, shown to be in agreement with that of the analytical solution (85), indicated as a dashed line. The inset shows, for ${\varphi }_{20}^{-}$, how the agreement with the analytical solution improves with increasing numerical resolution: shown, on a semilogarithmic scale, is the magnitude of relative difference between the numerical data and the analytic value for each of $h=\{\frac{1}{2},\frac{1}{4},\frac{1}{8}\}M$, along with (in the dashed line) a Richardson extrapolation to $h\to 0$, which assumes quadratic convergence in $h$. As a rough error bar on the extrapolated value one may take the magnitude of its difference with the $h=M/8$ result, which in relative terms is $\sim 3\times {10}^{-5}$. We see that the extrapolated value agrees with the analytical result to within that error bar.

Figure 2

The axisymmetric piece of the Hertz potential on the particle, for different $\ell$ values. We show here $|{\varphi }_{\ell 0}^{-}(t,r_0)|$ (divided by $\mu /M^2$) for $\ell =3$, 4, 5, 10 and $r_0=7M$. The corresponding known analytic values are shown in dashed lines, for comparison. Note how modes of higher $\ell$ relax faster, as expected.

Figure 3

Same as in Fig. 2, this time showing the behavior of $|{\varphi }_{\ell 0}^{\pm }|$ as a function of $r$ at some fixed (late) time $t=t_0$. We show individual numerical data points, with the analytic solutions (dashed lines) displayed for comparison. The fields ${\varphi }_{\ell 0}^{-}$ and ${\varphi }_{\ell 0}^{+}$ do not agree on the particle [recall Eq. (87)], which is clearly manifest in the $\ell =3$ case; higher-$\ell$ modes have smaller jumps, which are harder to resolve by eye in this figure.

Figure 4

Results for $(\ell ,m)=(2,1)$. This figure is organized the same as Fig. 1, but for convenience we show here the quantities $|{\varphi }_{21}^{\pm }(t,r_0){{}_{-2}Y}_{21}(\pi /2,{\phi }_{\mathrm{p}}(t))|$ (divided by $\mu /M^2$), which are $t$ independent in the physical solution. Again, $r_0=7M$. Each one-sided numerical solution settles down to a constant value, which appears to be in agreement with solutions obtained using the frequency-domain approach of Ref. [47] (dashed line). The inset explores the agreement for ${\varphi }_{21}^{-}$ in more detail: it shows the relative difference between our numerical results and the frequency-domain ones for three choices of step size $h$, with a Richardson extrapolation to $h\to 0$ (dashed line). As a rough error bar on the extrapolated value we take the magnitude of its difference with the $h=M/8$ result, which in relative terms is $\sim 3\times {10}^{-5}$. We see that the extrapolated value agrees with the frequency-domain result to within that error bar (the numerical error of the frequency-domain value is known to be much smaller).

Figure 5

A $t=\mathrm{const}$ slice of the same solution shown in Fig. 4. We show the real part of the IRG field ${\varphi }_{21}$ as a function of $r_{*}$ on some (late) time slice. The noisy features at both extremes are remnants of initial junk radiation, to be discarded.

Figure 6

The $(\ell ,m)=(2,0)$ mode of the time-domain reconstructed IRG metric perturbation $h_{\alpha \beta }^{\mathrm{rec}\pm }$, in the vicinity of the orbit (a circular geodesic at $r_0=7M$). We show, for illustration, only four of the components (with $h_{tt}$ and $h_{rr}$ divided by $\mu /M$ and $h_{\theta \theta }$ and $h_{\phi \phi }$ divided by $\mu M$), on a $t=\mathrm{const}$ slice. Data points correspond to numerical data, and the solid lines come from applying the reconstruction formula (22) to the analytical solution (85). This reconstructed perturbation can be used directly as input for a self-force calculation, using the mode-sum procedure reviewed in Sec. 2c.

Figure 7

Our $1+1\mathrm{D}$ double-null uniform grid in $(u,v)$ coordinates. The numerical integration starts with characteristic initial conditions on $v=v_0$ and $u=u_0$ and proceeds along successive lines of constant $v$. The particle’s worldline is represented by the vertical dashed line at $v-v_0=u-u_0$, and grid points are labeled in accordance with their position with respect to it. In the text we describe the finite-difference formula applied for each type of grid point.