Antikick Relation in High-Energy Head-On Collisions of Spinning Black Holes
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 ($\approx28\,562$~km/s, i.e., $\sim0.1c$) and the maximum radiated energy $E_{\rm rad}$ and remnant spin $\alpha_f^{\max}$ from such encounters ($E_{\rm rad}/M_{\rm ADM}\approx32\%$ where $M_{\rm ADM}$ is the ADM mass, and $\alpha_f^{\max}\approx0.987$) 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 $\gamma v$, we characterize the peak recoil $V_p$, the final recoil $V_f$, and the antikick $\Delta V\equiv V_f-V_p$, and we provide phenomenological fits of their dependence on $\gamma v$ 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 ($\gamma v>1$) the peak and final recoil are directly proportional, $V_p\approx7.4\,V_f$ (equivalently $\Delta V \approx-6.4\,V_f$), 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 $s^{1.27\pm0.08}$, motivating an even higher energy collision spin sequence study.