Self-force effects on the marginally bound zoom-whirl orbit in Schwarzschild spacetime

Abstract

For a Schwarzchild black hole of mass $M$, we consider a test particle falling from rest at infinity and becoming trapped, at late time, on the unstable circular orbit of radius $r=4GM/c^2$. When the particle is endowed with a small mass, $μ\ll M$, it experiences an effective gravitational self-force, whose conservative piece shifts the critical value of the angular momentum and the frequency of the asymptotic circular orbit away from their geodesic values. By directly integrating the self-force along the orbit (ignoring radiative dissipation), we numerically calculate these shifts to $O(μ/M)$. Our numerical values are found to be in agreement with estimates first made within the Effective One Body formalism, and with predictions of the first law of black-hole-binary mechanics (as applied to the asymptotic circular orbit). Our calculation is based on a time-domain integration of the Lorenz-gauge perturbation equations, and it is a first such calculation for an unbound orbit. We tackle several technical difficulties specific to unbound orbits, illustrating how these may be handled in more general cases of unbound motion. Our method paves the way to calculations of the self-force along hyperbolic-type scattering orbits. Such orbits can probe the two-body potential down to the "light ring", and could thus supply strong-field calibration data for eccentricity-dependent terms in the Effective One Body model of merging binaries.

Figures (6)

• Figure 1: The Zero-(binding-)energy zoom-whirl geodesic orbit (ZEZO), depicted here in the orbital plane. The inset shows the radial effective potential $V(r;4\mu M)$ [cf. Eq. (\ref{['radEq']})], with the dashed line representing the radial range of the ZEZO orbit. The orbit asymptotes to the innermost bound circular orbit (IBCO) at $r=4M$, corresponding to the maximum of the effective potential.
• Figure 2: Treatment of junk radiation, illustrated here for the mode $(i,\ell,m)=(1,2,2)$ of the perturbation (other modes exhibit a similar behavior). In the oZEZO case (green) we release the particle at $r_{\rm init}=(4+10^{-11})M$, letting junk radiation dissipate away while the particle is still in a tight circular whirl around the black hole; we then discard the $r_{\rm p}< (4+10^{-4})M$ portion of the data, which is dominated by junk. In the iZEZO case (red), the particle is released from $r=133M$, giving usable junk-free data for $r\lesssim 90M$. The thick horizontal dashed line marks the value of the perturbation mode on a strictly circular geodesic at $r=4M$ (the IBCO); reassuringly, the perturbations along both iZEZO and oZEZO asymptotically approach this value, as desired.
• Figure 3: Raw numerical data for the monopole ($\ell=0$) mode along the iZEZO orbit, as it settles into a circular whirl (at around $t\sim 920M$). The upper and lower panels show our numerical variables $\bar{h}^{(i)00}$ (refer to the first paragraph of Sec. \ref{['review']}] and their first time derivatives on the particle, respectively. During the whirl, we expect the metric perturbation to assume a constant value on the orbit (in any reasonable gauge); we see instead the characteristic behaviour of a linear-in-time gauge mode, evidently present in the data.
• Figure 4: Numerical filtering of the monopole solution along the iZEZO. The plot shows our numerical metric variables, evaluated on the orbit, after filtering out the linear-in-t gauge mode present in the raw data. Horizontal dashed lines mark the (constant) values of the metric variables on the IBCO. Reassuringly, these values are approached as the orbit settles into a circular whirl at the IBCO. (Note the variable $h^{(2)}$ is zero for the IBCO; the residual value of the numerical $h^{(2)}$ solution can serve as an error estimate.)
• Figure 5: Numerical filtering of the even-parity dipole mode for the iZEZO. The plot shows the dipole field along the orbit, after subtraction of a suitable gauge mode with a generator of the form \ref{['Dlin']}. Horizontal dashed lines mark the (constant) absolute values of the metric functions along the IBCO. Reassuringly, our filtered dipole solution approaches these values as the iZEZO settles into a circular whirl at the IBCO.