Ergodic Hysteresis of the Kerr black hole spectrum

TL;DR

The paper reveals an infinite cascade of exceptional points (EPs) in the quasinormal-mode spectrum of massive scalar perturbations of Kerr black holes, enabling adiabatic mixing of overtones via geometric phases and leading to adiabatic ergodicity. It employs an isomonodromic framework, reducing radial/angular equations to the confluent Heun equation (CHE) and, at extremality, to the double-confluent Heun equation (DCHE), with the spectrum tied to Painlevé tau functions $\tau_V$ and $\tau_{III}$ through a Riemann-Hilbert map. An infinite sequence of EPs in the $(\ell,m)=(1,1)$ and $(2,2)$ sectors is identified, labeled by $(M\mu)^c_n$ and $(a/M)^c_n$, accumulating at $(a/M,M\mu)=(1,(M\mu)^c_\star)$ with $(M\mu)^c_\star\approx 0.16189$, each carrying a geometric phase that enables exchange between modes. The authors also propose an EP-counting method via closed parameter loops, implying potential EP-based control of mode distribution in black hole ringdowns and highlighting non-Hermitian physics in gravitational systems.

Abstract

We uncover a cascade of exceptional points (EPs) in the quasinormal mode spectrum of massive scalar perturbations of Kerr black holes, revealing an intricate non-Hermitian structure underlying their linear response. The cascade originates from a single damped mode that enters the extremal spectrum for sufficiently large field masses. We obtain evidence for an infinite sequence of EPs in the $(\ell,m)=(1,1)$ and $(2,2)$ sectors near the extremal limit, mediating the transition between damped and zero-damping modes. Each EP carries a geometric phase that enables adiabatic mode mixing across the entire overtone spectrum, a phenomenon we refer to as adiabatic ergodicity.

Figures (5)

• Figure 1: The behavior of the $(\ell, m) = (1,1)$ overtones, parametrized by $a/M$, near the associated EPs. Each dashed curve corresponds to a fixed scalar mass $M\mu \lesssim (M\mu)^c_n$, while each solid curve corresponds to $M\mu \gtrsim (M\mu)^c_n$. For $a/M < (a/M)^c_n$, the curves are indistinguishable. As the black hole spin increases (along the direction of the arrow), the EP is reached and the initially coincident curves branch out. Dashed curves are ZDMs, which approach $\mathrm{Im}\, (M\omega)=0$ as $a/M\rightarrow 1$. Solid curves are DMs, which spiral to a finite decay time in the extremal limit.
• Figure 2: The near-extremal behavior of the $n=3$ QNM across a range of $M\mu$ encompassing the $n=2$ and $n=3$ EPs (denoted by dots). The mode transitions from ZDM to DM at each EP. Within region (b), corresponding to the window $(M\mu)^c_3<(M\mu)<(M\mu)^c_2$, the $n=3$ mode exhibits DM behavior, whereas in regions (a) and (c) it behaves as a ZDM.
• Figure 3: The real (left) and imaginary (right) parts of the DM frequency as functions of $M\mu$ in the extremal regime $a/M=1$ are shown by the solid curve originating at the star. The star marks the frequency associated with the lowest scalar mass for which this DM exists, namely $M\mu = (M\mu)^c_{\star}\simeq 0.16189$. Red dots denote the DM frequencies at $a/M=1$ and $M\mu = (M\mu)^c_n$. The shaded band corresponds to the interval $(M\mu)^c_3 < M\mu < (M\mu)^c_2$, which partially overlaps with region (b) of Fig. \ref{['fig:surfacebranching']}. Vertical lines illustrate how the spacing between EPs progressively narrows as one approaches the critical point $(M\mu)^c_{\star}\simeq 0.16189$, signaling the termination of the EP cascade. For visual clarity, we display only the first nine red dots.
• Figure 4: The closed path ABCDEFA around the shaded region, beginning and ending at the origin, along which we adiabatically track the evolution of the Schwarzschild fundamental mode. The trajectories labeled (i), (ii), and (iii) illustrate different choices of $(M\mu)|_D=(M\mu)|_E$, each enclosing a different number of EPs within the shaded region.
• Figure 5: The real (circle dots) and imaginary (square dots) parts of the coefficient $A_n$ as a function of $n$, as given in Table \ref{['tab:AnBnl1m1']} -- see also Eqs. \ref{['eq:nearEPbehavior']} and \ref{['AnBn']}. The dashed and dotted lines correspond, respectively, to fits of $\mathrm{Re}(A_n)$ and $\mathrm{Im}(A_n)$ with respect to an harmonic sequence of the form $\alpha/(n + \beta)$.