Hybrid method for understanding black-hole mergers: Head-on case

Black-hole-binary coalescence is often divided into three stages: inspiral, merger, and ringdown. The post-Newtonian (PN) approximation treats the inspiral phase, black-hole perturbation (BHP) theory describes the ringdown, and the nonlinear dynamics of space-time characterize the merger. In this paper, we introduce a hybrid method that incorporates elements of PN and BHP theories, and we apply it to the head-on collision of black holes with transverse, antiparallel spins. We compare our approximation technique with a full numerical-relativity simulation, and we find good agreement between the gravitational waveforms and the radiated energy and momentum. Our results suggest that PN and BHP theories may suffice to explain the main features of outgoing gravitational radiation for head-on mergers. This would further imply that linear perturbations to exact black-hole solutions can capture the nonlinear aspects of head-on binary-black-hole mergers accessible to observers far from the collision.

Figure 1

This figure depicts a space-time diagram of a black-hole collision, modeled after Price’s description of stellar collapse. We choose the trajectory of the two holes as a way to separate the space-time into an interior and an exterior region. The exterior region is a perturbed, black-hole space-time, whereas the interior is that of a post-Newtonian (PN) black-hole-binary system [shaded in yellow (light gray)]. The red (dark gray) region of the diagram shows the place at which the effective potential of the black-hole is significantly greater than zero. This formalism allows us to divide the waveform into three sections: inspiral (or infall), which extends from the beginning of the binary’s evolution until when the $l=2$ effective potential of the exterior BHP space-time starts to be exposed; merger, which extends from the end of inspiral to when the majority of the exterior potential is revealed; and ringdown, which represents the remainder. We overlay the even-parity, ($l=2$, $m=0$) mode of the waveform.

Figure 2

This figure shows, at a given moment in time, the Schwarzschild and PN radial coordinates, the binary separation, and the position of the shell where we match the two theories.

Figure 3

This figure shows the trajectory of the boundary shell as the solid blue (black) curve labeled by $r_s(t)$. The other two curves show the early- and late-time behavior of the shell. The red (gray) dashed curve labeled by $r_{s,(N)}(t)$ shows the trajectory of a shell that follows the Newtonian equations of motion for a plunge from rest. The green (light gray) dashed and dotted curve [denoted by $e^{-t/(2M)}$] shows the exponential convergence to the horizon at the rate expected in a Schwarzschild space-time. The upper inset shows how the shell agrees with a Newtonian plunge from rest at early times, and the lower inset shows how the shell converges exponentially to the horizon at the expected rate.

Figure 4

The blue (dark gray) dashed curve is ${\Psi }_{\mathrm{even}}^{2,2}$ from our hybrid method, whereas the black solid curve is the same quantity in full numerical relativity. The red (light gray) dashed vertical line corresponds to the retarded time at which the shell in the hybrid method reaches the light ring of the final black hole, $r=3M$. We shift the numerical-relativity waveform so that the peaks of the numerical and hybrid ${\Psi }_{\mathrm{even}}^{2,2}$ waveforms align.

Figure 5

The blue (dark gray) dashed curve is ${\Psi }_{\mathrm{odd}}^{(2,1)}$ from our hybrid method, whereas the black solid curve is the same quantity in full numerical relativity. The red (light gray) dashed vertical line corresponds to the retarded time at which the shell in the hybrid method reaches the light ring of the final black hole, $r=3M$. We shift the numerical-relativity waveform so that the peaks of the numerical and hybrid ${\Psi }_{\mathrm{even}}^{2,2}$ waveforms align.

Figure 6

The blue (dark gray) dashed curve is the velocity of the final black hole as a function of time (inferred from the gravitational radiation) from our hybrid method, using only the $l=2$ modes of the wave. The black solid curve is the equivalent quantity in numerical relativity, computed from the full Weyl scalar, ${\psi }_4$. The red (light gray) dashed vertical line corresponds to the retarded time at which the shell in the hybrid method reaches the light ring of the final black hole, $r=3M$. As before, the numerical-relativity velocity is shifted so that the peaks of the numerical and hybrid ${\Psi }_{\mathrm{even}}^{2,2}$ waveforms align.