Session L11: Binary Inspiral and Gravitational Wave Kicks

Modeling gravitational recoil from precessing highly-spinning unequal-mass black-hole binaries

We measure the gravitational recoil for unequal-mass-black- hole-binary mergers, with the larger BH having spin $a/m^H=0.8$, and the smaller BH non-spinning. We choose our configurations such that, initially, the spins lie on the orbital plane. The spin and orbital plane precess significantly, and we find that the out-of plane recoil (i.e. the recoil perpendicular to the orbital plane around merger) varies as $\eta^2/(1+q)$, in agreement with our previous prediction, based on the post-Newtonian scaling.   

Post-Newtonian Approximation in a Maxwell-Like Form for Use in Interpreting Binary-Black-Hole Simulations

Recent numerical-relativity simulations of binary-black-hole mergers have revealed large gravitational recoils. These results motivate us to explore the distribution and flow of linear momentum inside compact binaries. A powerful tool in our explorations is a formulation of the first post-Newtonian approximation to general relativity in a ``gravitoelectromagnetic'' Maxwell-like form that facilitates physical intuition. Relying heavily on work of Damour, Soffel and Xu, we have fleshed out this formulation, including all nonlinearities. We focus especially on density and flux of gravitational momentum, as expressed in terms of the Landau-Lifshitz pseudotensor, which we bring into forms that are almost identical to those for the electromagnetic field.   

Momentum Flow in Inspiraling Black-Hole Binaries

Numerical-relativity simulations of binary black-hole mergers in the extreme-kick configuration have revealed bobbing of the orbital plane during inspiral, and ``superkicks'' after merger. As part of our project to explore momentum flow in such binaries, we explain the bobbing as due to an exchange of linear momentum between the holes and the surrounding, near-field curved spacetime. Using the Landau-Lifshitz pseudotensor in Maxwell-like form, we demonstrate that, when the holes are moving synchronously upward due to frame dragging and spin-curvature-coupling, the nearby curved spacetime contains an equal and opposite downward momentum, and conversely. Although our formalism is gauge-dependent, it is powerful for developing physical intuition about the nonlinear dynamics of curved spacetime.   

Momentum flow in numerical simulations of binary black hole mergers

Most research on extracting science from binary-black-hole simulations has adopted a ``scattering matrix'' perspective: given the binary's initial parameters, what are the final hole's parameters and the emitted gravitational waveform? In contrast, we are using binary-black-hole simulations to explore the nonlinear dynamics of curved spacetime. We use the Landau-Lifshitz pseudotensor to describe the density and flux of a binary's linear momentum. Focusing on the head-on plunge, merger, and ringdown of a binary black hole with antiparallel spins, we explore numerically the momentum flow between the holes and the surrounding spacetime. To investigate the gauge dependence of our results, we compare simulations in several different gauges, and we also compare our simulations with the Maxwell-like post-Newtonian approximation.   

Recoil velocities from black hole mergers using perturbation theory

Comparable mass black hole binaries radiate gravitational energy as they spiral into each other and merge. An integration of the momentum carried away by gravitational waves from asymmetric binaries results in a non-zero recoil velocity of the merged object. We have recently developed a numerical toolkit to study gravitational radiation from the viewpoint of black hole perturbation theory, where the binary consists of a massive central black hole (mass=M1) and a much smaller companion (mass=M2). The central engine of our numerical toolkit is a finite-difference based numerical algorithm to solve the inhomogeneous Teukolsky equation, which describes perturbations around black holes. Earlier calculations have shown that perturbation theory with M2/M1 = O(0.1) yields reliable estimates for the recoil velocity. Here, we use our numerical toolkit to improve earlier estimates for the recoil velocities from black hole mergers. Our numerical toolkit also allows us to compute fluxes of angular momentum during the merger. We compare these results with (3+1) dimensional numerical relativity. Such a comparison is both a calibration of the reliability of our code and a consistency check for numerical relativity.   

Progress in modeling of black hole binaries

Frontiers of numerical studies of black hole binaries include large mass ratios, analysis of gravitational waveform-dependence on spin and comparison with analytic models, qualitatively new effects of large precession, precise calculation of spin, general analytic fitting of radiative recoil, and modeling of accreting matter. Results of simulations pertinent to these areas are presented, as well as some discussion of relevant numerical methods.   

Transient resonances in the inspirals

We show that the two body problem in general relativity in the highly relativistic regime has a qualitatively new feature: the occurence of transient resonances. The resonances occur when the ratio of polar and radial orbital frequencies, which is slowly evolving under the influence of gravitational radiation reaction, passes through a low order rational number. The resonances make the orbit more sensitive to changes in the initial data (though not quite chaotic), and are genuine non-perturbative effects that are not seen at any order in the standard post-Newtonian expansion used for two body systems at large separation. Our results directly apply to an important potential source of gravitational waves, namely the gradual inspiral of compact objects into much more massive black holes. Exploiting observations of these gravitational waves to map the spacetime geometry of black holes is contingent upon accurate theoretical models (templates) of the binary dynamics. At present, only the leading order in the mass ratio gravitational waveforms can be computed. Corrections to the waveform's phase due to resonance effects scale as the square root of the inverse of the mass ratio and are characterized by sudden jumps in the time derivatives of the phase. We numerically estimate the net size of these corrections and find indications that the phase error is of order a few cycles for mass ratios $\sim 10^{- 3}$ but will be significant (of order tens of cycles) for mass ratios $\sim 10^{-6}$. Computations of these corrections will require the computation of pieces of the forcing terms in the equations of motion which are currently unknown.   

A self-force primer for numerical relativists

A small mass $\mu$ moving about a much more massive black hole $M$ travels along a world line that is most easily described as being a geodesic of the perturbed metric $g_{ab}+h_{ab}$, where $h_{ab} \sim O(\mu)$ is the metric perturbation suitably regularized at the location of the small mass. This motion is said to result from the gravitational self-force acting on $\mu$. A novel technique uses currently available methods of numerical relativity to calculate the regularized $h_{ab}$, the self-force acting back on $\mu$, and the effects of the self-force on the gravitational waves being emitted in the context of extreme mass ratio inspiral.