Black Hole Mergers and Unstable Circular Orbits

TL;DR

This work shows that equal-mass, eccentric, non-spinning binary black holes exhibit a threshold of immediate merger, where near-threshold evolutions undergo a prolonged whirl before deciding to merge or separate, with the whirl duration scaling as $e^{n} \propto |p-p^{*}|^{-\gamma}$ and $\gamma \approx 0.35$. To interpret this, the authors construct a geodesic analogue in Kerr spacetime, demonstrating that equatorial geodesics near threshold also approach unstable circular orbits and share a consistent scaling relation $\gamma = \omega_0/(2\pi\lambda)$, where $\lambda$ is the instability exponent. The close quantitative agreement between full numerical relativity results and the geodesic predictions supports a deep link between near-threshold binary dynamics and geodesic motion, enabling rough estimates of energy emission and cross sections in ultra-relativistic black-hole scattering. The findings have potential implications for high-energy collisions and extra-dimensional scenarios at the LHC, suggesting that substantial gravitational radiation could occur near threshold and that geodesic methods may help bound production cross sections and energy losses. Overall, the paper advances our understanding of strong-field gravity in mergers and proposes a useful bridge to test-particle dynamics in Kerr backgrounds for high-energy gravitational interactions.

Abstract

We describe recent numerical simulations of the merger of a class of equal mass, non-spinning, eccentric binary black hole systems in general relativity. We show that with appropriate fine-tuning of the initial conditions to a region of parameter space we denote the threshold of immediate merger, the binary enters a phase of close interaction in a near-circular orbit, stays there for an amount of time proportional to logarithmic distance from the threshold in parameter space, then either separates or merges to form a single Kerr black hole. To gain a better understanding of this phenomena we study an analogous problem in the evolution of equatorial geodesics about a central Kerr black hole. A similar threshold of capture exists for appropriate classes of initial conditions, and tuning to threshold the geodesics approach one of the unstable circular geodesics of the Kerr spacetime. Remarkably, with a natural mapping of the parameters of the geodesic to that of the equal mass system, the scaling exponent describing the whirl phase of each system turns out to be quite similar. Armed with this lone piece of evidence that an approximate correspondence might exist between near-threshold evolution of geodesics and generic binary mergers, we illustrate how this information can be used to estimate the cross section and energy emitted in the ultra relativistic black hole scattering problem. This could eventually be of use in providing estimates for the related problem of parton collisions at the Large Hadron Collider in extra dimension scenarios where black holes are produced.

Figures (12)

• Figure 1: The number of orbits $n$ versus logarithmic distance of the initial boost parameter $k$ from the immediate merger threshold ${k^*}$, for evolutions that did result in a merger. Results from the three resolutions are plotted, and a least-squares fit to each set of data assuming the relation (\ref{['n_gamma']}).
• Figure 2: Data as in Fig. \ref{['fig_gamma_merge']}, though here from evolutions that did not merge during the time of the simulation (i.e. $k>{k^*}$).
• Figure 3: Plots of the orbital motion from the two higher resolution simulations ($h/2$) tuned closest to threshold (only the coordinate motion of a single black hole---initially at positive $x$---is shown for clarity). The dashed curve is the case resulting in a merger, and the curve ends once a common apparent horizon is first detected, while for the solid curve the black holes separate again and here the curve ends when the simulation was stopped.
• Figure 4: The total energy radiated in gravitational waves plotted as a function of logarithmic distance from the immediate merger threshold. We have overlayed the data from both super and sub critical cases, though for clarity have only added the linear regression line for the cases that merged.
• Figure 5: The gravitational waves emitted during a merger event. Here we show the real part of the dominant spin weight -2, $\ell=2$, $m=2$ spherical harmonic component of $\Psi_4$. The solid curve is the merger case tuned closest to threshold (from the higher resolution simulations). The dotted curve was computed by taking the coordinate motion of the AH's from this simulation (see Fig.\ref{['orbit_example']}), assuming they represent the locations of point particles of mass $m/2$, and plugging the result into the quadrupole formula; this waveform ends once a common horizon is formed. The dashed curve shows the same information as the dotted curve, but here the data is from the non-merger case tuned closest to threshold. Note that the two curves from the quadrupole formula were shifted in time to account for the propagation time to the sphere at $r=50m$ where $\Psi_4$ was measured.