The silence of binary Kerr

TL;DR

This work examines whether gravitational 2→2 scattering can generate entanglement in spinning-particle systems, focusing on the Eikonal regime and incorporating Hilbert-space matching via Thomas–Wigner rotations. By computing the leading spin-dependent amplitude and its Fourier transform to impact parameter space, the authors define the relative entanglement entropy $\Delta S$ and study its dependence on Wilson coefficients that encode spin multipoles. They find a striking result: minimal coupling, corresponding to Kerr black holes with all $C_{a,n}=C_{b,n}=1$, yields near-zero entanglement across spins 1–3, while deviations from unity increase $\Delta S$, with the $C_{2}$ channel driving much of the effect. These findings suggest a deep link between the classical Kerr solution and quantum entanglement suppression in gravitational scattering, and motivate extensions to NLO effects and to other BH-like configurations such as Kerr–Newman or fuzzballs.

Abstract

A non-trivial $\mathcal{S}$-matrix generally implies a production of entanglement: starting with an incoming pure state the scattering generally returns an outgoing state with non-vanishing entanglement entropy. It is then interesting to ask if there exists a non-trivial $\mathcal{S}$-matrix that generates no entanglement. In this letter, we argue that the answer is the scattering of classical black holes. We study the spin-entanglement in the scattering of arbitrary spinning particles. Augmented with Thomas-Wigner rotation factors, we derive the entanglement entropy from the gravitational induced $2\rightarrow 2$ amplitude. In the Eikonal limit, we find that the relative entanglement entropy, defined here as the \textit{difference} between the entanglement entropy of the \textit{in} and \textit{out}-states, is nearly zero for minimal coupling irrespective of the \textit{in}-state, and increases significantly for any non-vanishing spin multipole moments. This suggests that minimal couplings of spinning particles, whose classical limit corresponds to Kerr black hole, has the unique feature of generating near zero entanglement.

Figures (7)

• Figure 1: We consider the $2\rightarrow 2$ scattering of two spinning objects exchanging gravitons. (I) Process in the leading order of the Newton constant $G$. (II) Eikonal approximation, which re-sums the ladder diagrams.
• Figure 2: (I) Relative entanglement entropy $\Delta S$ and (II) the entanglement power $\mathcal{E}_a$ for massive spin-1 particles. The initial state is set to $\lvert{\rm in}\rangle=\lvert\mathbin\uparrow\space\uparrow\rangle$ and the kinematic parameters are given by $|\vec{p}_a| = |\vec{p}_b| = |\vec{p}|$, $m_a = m_b = m$, $\vec{b} = (b, 0,0)$, $Gm^2 = 10^{-4}$, $|\vec{p}|b = 1000$, $|\vec{p}|/m = 100$. The minimum, represented by the black point, corresponds to the Wilson coefficient value $(C_{a,2}, C_{b,2})=(1,1)$, $\Delta S \approx 1.54\times10^{-9}$ and $\mathcal{E}_a \approx 1.10\times10^{-10}$.
• Figure 3: Relative entanglement entropy for massive spin-2 particles. The initial state is set to $\lvert{\rm in}\rangle=\lvert\mathbin\uparrow\space\uparrow\rangle$ and the kinematic parameters are given by $|\vec{p}_a| = |\vec{p}_b| = |\vec{p}|$, $m_a = m_b = m$, $\vec{b} = (b, 0,0)$, $Gm^2 = 10^{-4}$, $|\vec{p}|b = 1000$, $|\vec{p}|/m = 100$. The planes corresponds respectively to $(C_{a,2},C_{b,2})$, $(C_{a,3},C_{b,3})$ and $(C_{a,4},C_{b,4})$, while all others Wilson coefficients are set to one. In any of the cases, the minimum, represented by the black point, corresponds to the Wilson coefficients set to one and $\Delta S \approx 5.84\times10^{-9}$.
• Figure 4: Relative entanglement entropy for massive spin-3 particles. The initial state is set to $\lvert{\rm in}\rangle=\lvert\mathbin\uparrow\space\uparrow\rangle$ and the kinematic parameters are give by $|\vec{p}_a| = |\vec{p}_b| = |\vec{p}|$, $m_a = m_b = m$, $\vec{b} = (b, 0,0)$, $Gm^2 = 10^{-4}$, $|\vec{p}|b = 1000$, $|\vec{p}|/m = 100$. The Wilson coefficients $(C_{a, i \neq 2},C_{b, j \neq 2})$ are set to one. The minimum, represented by the black point, is at $\Delta S \approx 1.26\times10^{-8}$ and corresponds to the Wilson coefficient value $(C_{a,2},C_{b,2})=(1,1)$.
• Figure 5: Relative entanglement entropy for massive spin-3 particles. The initial state is set to $\lvert{\rm in}\rangle=\lvert\mathbin\uparrow\space\uparrow\rangle$ and the kinematic parameters are given by $|\vec{p}_a| = |\vec{p}_b| = |\vec{p}|$, $m_a = m_b = m$, $\vec{b} = (b, 0,0)$, $Gm^2 = 10^{-4}$, $|\vec{p}|b = 1000$, $|\vec{p}|/m = 100$, $C_{a,2}=C_{b,2}=C_2$, $C_{a,3}=C_{b,3}=C_3$. All others Wilson coefficients are set to one. The minimum is at $\Delta S \approx 1.26\times10^{-8}$ and corresponds to the Wilson coefficient value $(C_{2},C_{3})=(1,1)$ .