Antikick Relation in High-Energy Head-On Collisions of Spinning Black Holes

Abstract

The collision of black holes at relativistic speeds probes gravity in its most extreme dynamical regime. While the maximum gravitational recoil from \emph{grazing} high-energy collisions ($\approx 28562$~km/s, i.e., $\sim 0.1c$) and the maximum radiated energy $E_{\mathrm{rad}}$ and remnant spin $\hat{I}{\pm }_f^{\max }$ from such encounters ($E_{\mathrm{rad}}/M_{\mathrm{ADM}}\approx 32\%$ where $M_{\mathrm{ADM}}$ is the ADM mass, and $\hat{I}{\pm }_f^{\max }\approx 0.987$) have been established previously~\cite{Healy:2022jbh,Healy:2024lhl}, here we focus on the \emph{head-on} high-energy collision of equal-mass spinning black holes and on the detailed structure of the resulting recoil. Performing a sequence of full numerical simulations for spin magnitudes $s=0.5,0.65$, and $0.8$ over a range of initial momenta ${\hat{I}}^{3}v$, we characterize the peak recoil $V_p$, the final recoil $V_f$, and the antikick $\hat{I}”V\equiv V_f-V_p$, and we provide phenomenological fits of their dependence on ${\hat{I}}^{3}v$ and $s$. We complement these results with a zero-frequency-limit (ZFL) analysis of the radiated energy and momentum, a quasinormal-mode model of the antikick, and a superposed boosted double-Kerr close-limit estimate. We find that in the relativistic regime (${\hat{I}}^{3}v>1$) the peak and final recoil are directly proportional, $V_p\approx 7.4V_f$ (equivalently $\hat{I}”V\approx -6.4V_f$), largely independent of both the initial momentum and the spin magnitude, pointing to a common post-merger relaxation. While the ZFL predicts a leading linear-in-spin dependence, the close-limit analysis predicts a leading $s^3$ dependence of the recoil amplitude; with the three spin magnitudes studied here the empirical exponent is $s^{1.27\pm 0.08}$, motivating an even higher energy collision spin sequence study.

9 pages, 5 figures, 3 tables

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