Time-domain metric reconstruction for hyperbolic scattering

Self-force methods can be applied in calculations of the scatter angle in two-body hyperbolic encounters, working order by order in the mass ratio (assumed small) but with no recourse to a weak-field approximation. This, in turn, can inform ongoing efforts to construct an accurate model of the general-relativistic binary dynamics via an effective-one-body description and other semianalytical approaches. Existing self-force methods are to a large extent specialized to bound, inspiral orbits. Here, we develop a technique for (numerical) self-force calculations that can efficiently tackle scatter orbits. The method is based on a time-domain reconstruction of the metric perturbation from a scalarlike Hertz potential that satisfies the Teukolsky equation, an idea pursued so far only for bound orbits. The crucial ingredient in this formulation is certain jump conditions that (each multipole mode of) the Hertz potential must satisfy along the orbit, in a $1+1$-dimensional multipole reduction of the problem. We obtain a closed-form expression for these jumps, for an arbitrary geodesic orbit in Schwarzschild spacetime, and present a full numerical implementation for a scatter orbit. In this paper, we focus on method development and go only as far as calculating the Hertz potential; a calculation of the self-force and its physical effects on the scatter orbit will be the subject of forthcoming work.

Figure 1

Sketch of the $1+1\mathrm{D}$ characteristic grid used in our numerical evolution of the no-string Hertz potentials ${\varphi }_{\pm }^{\ell m}(t,r)$ outside a Schwarzschild black hole. The grid lines are uniformly spaced in Eddington-Finlkelstein coordinates $u$, $v$. Initial conditions are set on the rays $u=u_0$ and $v=v_0$. The dashed line represents the particle’s worldline (or, equivalently, the $1+1\mathrm{D}$ reduction of the surface $\mathcal{S}$ interfacing between the vacuum regions ${\mathcal{S}}^{\gtrless }$) for a typical hyperbolic orbit. The evolution proceeds along successive $u=\text{const}$ rays, with appropriate jump conditions imposed across $\mathcal{S}$.

Figure 2

Results from evolution of the $(\ell ,m)=(2,0)$ mode of the vacuum $1+1\mathrm{D}$ BPT equation with $s=-2$. The evolution is seeded with a narrow Gaussian near $(t,r_{*})\sim (0,9M)$. We show, on a log-log scale, the field amplitude $|{\varphi }_{20}^{-}|$ sampled along slices of constant $r_{*}=10M$ (top), $u=500M$ (lower left) and $v=500M$ (lower right). The dashed lines ($\text{const}\times t^4$, $\text{const}\times v^4$ and $\text{const}\times u^4$, respectively) are shown for reference.

Figure 3

Results for the evolution of the $(\ell ,m)=(2,0)$ mode of the vacuum $1+1\mathrm{D}$ BPT equation with $s=+2$. Other details are as in Fig. 2, except here the scale is semilogarithmic. The dashed lines are for reference.

Figure 4

The modulus of the jump $[X_{\ell m}]$ in the no-string field $X_{\ell m}$ along the geodesic scatter orbit depicted in Fig. 5, for the $(\ell ,m)=(2,0)$ and (2,2) modes. This is obtained by numerically integrating the first-order ODE (81) forward in time with the initial condition (87) at large negative $t$ and ${\phi }_{\mathrm{in}}=0$. As a check, the solution approaches the asymptotic value given in (87) with ${\phi }_{\mathrm{out}}\simeq 481°$ (obtained by integrating the geodesic equation). The inset plot demonstrates this for the real part of $[X_{22}]$, with dashed lines indicating the analytical asymptotic values. This jump function inputs into our $1+1\mathrm{D}$ characteristic evolution scheme, whose application is illustrated in Sec. 7.

Figure 5

The sample scatter geodesic orbit used for our numerical illustration, with parameters given in Eqs. (102) and (103). The orbit is plotted in the equatorial plane using Cartesian-like coordinates $(x,y)=(r\cos \phi ,r\sin \phi )$. The location of the innermost stable circular orbit (ISCO) is shown for reference. The deflection angle of this strong-field orbit is $\delta \phi \simeq 301°$.

Figure 6

The RW field $X_{\ell m}$ along the particle’s worldline for the orbit shown in Fig. 5 and a sample of $(\ell ,m)$ values. Here, we show $|X_{\ell m}^{>}(t,R(t))|$ as a function of time $t$ (lower scale) and orbital radius $R$ (upper scale). Curves are labeled with their $(\ell ,m)$ values, with $\ell =8$ data shown amplified by a factor $\times 600$. The periastron location at $t=0$ is indicated with a vertical line. The early part of the data is contaminated by initial junk radiation, and it is to be discarded.

Figure 7

Convergence test for the $(\ell ,m)=(2,2)$ numerical solution. The inset shows a detail from the $|X_{22}^{>}|$ (green) curve in Fig. 6, for a sequence of runs with decreasing grid spacing, $h=\{\frac{1}{8},\frac{1}{16},\frac{1}{32}\}M$. The main plot quantifies the convergence rate; it shows the ratio $\mathcal{R}≔|X_8^{>}-X_{16}^{>}|/|X_{16}^{>}-X_{32}^{>}|$ as a function of $t$ along the orbit, where a subscript 8 (for example) denotes a calculation with grid spacing $h=M/8$. A ratio of $\mathcal{R}=4$ is indicative of quadratic convergence.

Figure 8

Numerical results for $|X_{22}^{>}|$ on the particle’s worldline, as calculated with $R_{\mathrm{init}}=100M$ (blue) and with $R_{\mathrm{init}}=200M$ (orange). The comparison illustrates how, reassuringly, the clean portion of the data is insensitive to $R_{\mathrm{init}}$, up to a small error that dies off in time. The inset displays the relative difference between the two curves, showing a $t^{-4}$ falloff at late time, consistent with the theoretically predicted $t^{-\ell -2}$ decay rate for compact vacuum perturbation along a curve $r=R\propto t$ (see, for instance, Eq. (89) of Ref. [51]).

Figure 9

The modulus of the no-string IRG Hertz potential ${\varphi }_{22}^{>}$ along the particle’s worldline. The field falls off as $X\sim t^{-2}$ at large $R$. The inset shows the same data rescaled by a factor $(R/M{)}^2$. The field exhibits the lagging peak and postperiastron undulation features discussed in the text. (The multiplication by $R^2$ makes more distinct the undulation feature, only barely visible in the main plot.)

Figure 10

The modulus of the Hertz potential ${\varphi }_{22}^{>}$ as a function of $u$ at $v=\text{const}=499M$ (approximating ${\mathcal{I}}^{+}$). The vertical line represents $u$ at periastron. The inset shows the same data rescaled by a factor $(u/M{)}^2$ to again highlight the postperiastron undulation feature.

Figure 11

A particle cell is traversed by the particle’s worldline (dashed curve) in one of four possible ways: cases UU, VV, UV, and VU, illustrated here. A different variant of the FD formula applies in each case, as described in the text. The apex of the cell is the point $c$ at $(u,v)=(u_c,v_c)$, and the particle exits the cell at $(u_f,v_c)$ or $(u_c,v_f)$, depending on the case. The vacuum portions of the cell left and right of the worldline are $C^{<}$ and $C^{>}$, respectively.