Transient Resonances in the Inspirals of Point Particles into Black Holes

We show that transient resonances occur in the two-body problem in general relativity for spinning black holes in close proximity to one another when one black hole is much more massive than the other. These 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. At such points, the adiabatic approximation to the orbital evolution breaks down, and there is a brief but order unity correction to the inspiral rate. The resonances cause a perturbation to orbital phase of order a few tens of cycles for mass ratios $\sim {10}^{-6}$, make orbits more sensitive to changes in initial data (though not quite chaotic), and are genuine nonperturbative effects that are not seen at any order in a standard post-Newtonian expansion. Our results apply to an important potential source of gravitational waves, the gradual inspiral of white dwarfs, neutron stars, or black holes into much more massive black holes. Resonances’ effects will increase the computational challenge of accurately modeling these sources.

Figure 1

[Top] The adiabatic inspiral computed from our approximate post-Newtonian self-force, for a mass ratio $\epsilon =\mu /M=3\times {10}^{-6}$, with black hole spin parameter $a=0.95$, with initial conditions semilatus rectum $p=9.0M$, eccentricity $e=0.7$, and orbital inclination ${\theta }_{\mathrm{inc}}=1.20$. The bottom curve is $e$, the middle curve is ${\theta }_{\mathrm{inc}}$, and the top curve is ratio of frequencies ${\omega }_{\theta }/{\omega }_r$, shown as functions of $p$. [Middle] The fluctuating, dissipative part of the first-order self-force causes a strong resonance when ${\omega }_{\theta }/{\omega }_r=3/2$ at $p=8.495$. Shown are the corrections to the energy $E$, angular momentum $L_z$ and Carter constant $Q$, as functions of $p$, scaled to their values at resonance and divided by the square root $\sqrt{\mu /M}$ of the mass ratio. The sudden jumps at the resonance are apparent, with the largest occurring for the Carter constant. [Bottom] The lower curve is the correction to the number of cycles $\varphi /(2\pi )$ of azimuthal phase of the inspiral caused by the fluctuating, dissipative part of the first order self-force. The sharp downward kick due to the resonance at $p=8.495$ can be clearly seen. The resonant corrections to the number of cycles of $r$ and $\theta$ motion are similar. These phase shifts scale as $\sqrt{M/\mu }$. The upper curve is the postadiabatic phase correction due to the conservative piece of the first order self-force, which is considerably smaller and is independent of the mass ratio.