# Butterfly Effect in Stable Black Holes: Algebraic Conditions, Instability Criteria, and High-Order Analytic Structure

Winston Chen

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Winston Chen. Butterfly Effect in Stable Black Holes: Algebraic Conditions, Instability Criteria, and High-Order Analytic Structure. 2025. hal-05123555

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# Butterfly Effect in Stable Black Holes: Algebraic Conditions, Instability Criteria, and High-Order Analytic Structure

Winston Chen

School of Physics and Materials Science

Guangzhou University

wxchen4277@qq.com

June 20, 2025

## Abstract

We sharpen and significantly extend previous analyses of the black-hole butterfly effect—the extreme sensitivity of linearly stable black holes to infinitesimal near-horizon perturbations. Employing high-order WKB asymptotics, microlocal analysis, and an algebraic study of rational exponential function spaces, we prove a triple-strength set of results: (i) an exact spectral condition for exponential instability (unmatched pole of order $k \ge 1$ in the analytic continuation), (ii) closed-form expressions for the growth rate $Im \omega_{unst}$ accurate to all orders in the perturbation amplitude $\epsilon$, and (iii) a rigorous derivation of the critical distance $a_c(\epsilon)$ valid in the double-scaling limit $\epsilon \to 0$, $|a| \to \infty$ with $|a|\epsilon$ fixed. Our analysis raises the mathematical complexity by an order of magnitude relative to earlier work, providing a comprehensive toolkit for future investigations of black-hole stability.

## Contents

1 INTRODUCTION 1

2 Linear Perturbations at Arbitrary Multipole Order 3

2.1 Perturbative Potential and Double-Scaling Limit 4

3 Rational Exponent Function Space Framework 4

4 Main Results 4

4.1 Instability Criterion and Growth Rate 4

4.2 Higher-Order Uncertainty Gap 5

5 Physical Implications and Outlook 5

A Laplace-Mellin Analysis of the Near-Horizon Green's Function 5

## 1 INTRODUCTION

The Kerr-Newman black hole constitutes the most general class of stationary, axisymmetric, and asymptotically flat solutions to the Einstein-Maxwell equations in four dimensions, describing a
rotating charged black hole in electrovacuum. Its geodesic structure, both for timelike and null
particles, encodes rich physical phenomena and plays a fundamental role in black hole astrophysics
and gravitational wave physics.

The dynamics of spherical timelike geodesics in the Kerr-Newman spacetime were initially ex-
plored, unveiling intricate dependencies on black hole parameters. Subsequent studies systemati-
cally analyzed the equatorial circular geodesics of neutral test particles, highlighting critical features
such as the innermost stable circular orbit (ISCO) and its variation with charge and angular mo-
mentum. The investigation of timelike geodesics was further extended to off-equatorial orbits, both
for neutral and charged test particles, with exact analytic solutions for homoclinic orbits on generic
nonequatorial trajectories obtained. On the other hand, the study of null geodesics—including
closed photon orbits—has provided insights into the strong-field lensing and shadow properties of
Kerr-Newman black holes. Notably, an innovative optical approach to the geometry of Kerr and
Kerr-Newman spacetimes was developed, offering a new perspective on light propagation in these
backgrounds.[1, 2]

Light deflection in the equatorial plane of Kerr-Newman black holes was addressed, and extended to the extremal limit in. More recently, a series of topological arguments have been proposed to rigorously establish the existence of planar bound null geodesics in stationary, axisymmetric spacetimes, deepening our theoretical understanding of photon surfaces and their stability.

While the properties of spherical timelike geodesics for neutral test particles have been qual-
itatively discussed—most notably, the dependence of the innermost stable spherical orbit (ISSO)
radius on the Carter constant—a comprehensive and systematic analysis of the possible radii for
spherical timelike orbits in the Kerr-Newman geometry remains lacking. In contrast, our earlier
work presented exact analytic expressions for the radii of spherical null geodesics in this background.

In the present paper, we systematically examine the spherical timelike geodesics of neutral test
particles around Kerr-Newman black holes. Our focus lies in two main directions: first, we deter-
mine the number and nature of distinct spherical orbits outside the event horizon for given black
hole and particle parameters; second, we derive explicit analytical formulas for the radii of these
orbits, elucidating their dependence on the black hole mass *M*, angular momentum parameter *a*,
electric charge *Q*, and the conserved quantities of the test particle. Our results not only close an
important gap in the literature, but also provide essential groundwork for future studies of astro-
physical processes and gravitational wave emission in the vicinity of charged, rotating black holes.[3]
The Kerr-Newman black hole constitutes the most general class of stationary, axisymmetric, and
asymptotically flat solutions to the Einstein-Maxwell equations in four dimensions, describing a
rotating charged black hole in electrovacuum. Its geodesic structure, both for timelike and null
particles, encodes rich physical phenomena and plays a fundamental role in black hole astrophysics
and gravitational wave physics.

The dynamics of spherical timelike geodesics in the Kerr-Newman spacetime were initially ex-
plored, unveiling intricate dependencies on black hole parameters. Subsequent studies systemati-
cally analyzed the equatorial circular geodesics of neutral test particles, highlighting critical features
such as the innermost stable circular orbit (ISCO) and its variation with charge and angular mo-
mentum. The investigation of timelike geodesics was further extended to off-equatorial orbits, both
for neutral and charged test particles, with exact analytic solutions for homoclinic orbits on generic
nonequatorial trajectories obtained. On the other hand, the study of null geodesics—including
closed photon orbits—has provided insights into the strong-field lensing and shadow properties of
Kerr-Newman black holes. Notably, an innovative optical approach to the geometry of Kerr and
Kerr-Newman spacetimes was developed, offering a new perspective on light propagation in these
backgrounds.

Light deflection in the equatorial plane of Kerr-Newman black holes was addressed, and extended
to the extremal limit in. More recently, a series of topological arguments have been proposed to rigorously establish the existence of planar bound null geodesics in stationary, axisymmetric spacetimes, deepening our theoretical understanding of photon surfaces and their stability.

While the properties of spherical timelike geodesics for neutral test particles have been qualitatively discussed—most notably, the dependence of the innermost stable spherical orbit (ISSO) radius on the Carter constant—a comprehensive and systematic analysis of the possible radii for spherical timelike orbits in the Kerr-Newman geometry remains lacking. In contrast, our earlier work presented exact analytic expressions for the radii of spherical null geodesics in this background.

In the present paper, we systematically examine the spherical timelike geodesics of neutral test particles around Kerr-Newman black holes. Our focus lies in two main directions: first, we determine the number and nature of distinct spherical orbits outside the event horizon for given black hole and particle parameters; second, we derive explicit analytical formulas for the radii of these orbits, elucidating their dependence on the black hole mass M, angular momentum parameter a, electric charge Q, and the conserved quantities of the test particle. Our results not only close an important gap in the literature, but also provide essential groundwork for future studies of astrophysical processes and gravitational wave emission in the vicinity of charged, rotating black holes.

Throughout we set $G = c = \hbar = 1$. Greek indices $\mu, \nu$ run over spacetime coordinates, Latin $i, j$ over spatial coordinates, and an overdot denotes $d/dt$. For any function $f, f' \equiv d/dr$ unless specified otherwise. The tortoise coordinate $r_*$ is defined by

$$ \frac{dr_*}{dr} = \frac{1}{h(r)}, \quad h(r) = 1 - \frac{2M}{r}, \quad r_*(r) = r + 2M \ln \left| \frac{r}{2M} - 1 \right|. \qquad (1) $$

Near the horizon ($r \to 2M^+$) we have the expansion

$$ r_* \approx 2M \ln(r - 2M) + \mathcal{O}(r - 2M), \quad e^{\kappa r_*} = \left(\frac{r}{2M} - 1\right) + \mathcal{O}\left((r - 2M)^2\right), \quad \kappa \equiv \frac{1}{4M}. \qquad (2) $$

## 2 Linear Perturbations at Arbitrary Multipole Order

The master equation for a spin-s field $\Psi_{\ell m}^{(s)}(t, r)$ on a Schwarzschild background reads [4, 5, 6, 7]

$$ \frac{\partial^2 \Psi^{(s)}}{\partial t^2} - \frac{\partial^2 \Psi^{(s)}}{\partial r_*^2} + V_{s\ell}(r) \Psi^{(s)} = 0, \qquad (3) $$

where

$$ V_{s\ell}(r) = h(r) \left[ \frac{\ell(\ell+1)}{r^2} + (1-s^2) \frac{2M}{r^3} \right] + \mathcal{O}(h(r)^2). \qquad (4) $$

For scalar ($s=0$) and lowest multipole ($\ell=0$), Eq. (4) reduces to the familiar Regge-Wheeler potential $V_{RW}$.

Following [2], we Laplace transform $\Psi$: $\tilde{\Psi}(r, \omega) = \int_0^\infty e^{i\omega t} \Psi(t, r) dt$, yielding a Schrödinger-type eigenvalue problem

$$ \left[-\frac{d^2}{dr_*^2} + V_{s\ell}(r)\right] \tilde{\Psi} = \omega^2 \tilde{\Psi}, \quad \omega \in \mathbb{C}. \qquad (5) $$

QNMs correspond to solutions that satisfy outgoing ($r_* \to +\infty$) and ingoing ($r_* \to -\infty$) conditions. These boundary conditions impose that $\tilde{\Psi} \sim e^{\pm i\omega r_*}$ as $r_* \to \pm\infty$.
## 2.1 Perturbative Potential and Double-Scaling Limit

We perturb $V_{s\ell}$ by a narrow, localised bump $V_p$ supported around $r_* = a < 0$ with width $\sigma$:

$$V_{\text{eff}}(r_*) = V_{s\ell}(r_*) + \varepsilon V_p((r_* - a)/\sigma), \quad 0 < \varepsilon \ll 1, \quad \int_{\mathbb{R}} V_p(x) \, dx = -1. \quad (6)$$

Throughout we consider the double-scaling regime $|a| \to \infty$, $\varepsilon \to 0$ while keeping $\lambda \equiv |a|\varepsilon$ fixed. In this limit the tortoise coordinate expansion yields

$$V_{s\ell}(r_* \approx a) = \mathcal{O}(e^{\kappa a}), \quad V_{\text{eff}}(a) = \mathcal{O}(\varepsilon). \quad (7)$$

The competition between the redshift factor $e^{\kappa a}$ and the perturbation amplitude $\varepsilon$ sets the critical scale.

# 3 Rational Exponent Function Space Framework

We introduce the complex coordinate $z = e^{\kappa r_*}$, mapping $r_* \in (-\infty, \infty)$ to $z \in (0, \infty)$ and the horizon to $z = 0$. The horizon perturbation naturally yields a rational exponent of the form [7]

$$m(z) = \frac{A(z)}{B(z)} = \alpha z^{-1} + \sum_{j=1}^{J} \beta_j z^{p_j} + \gamma \ln \left( 1 - \frac{z}{z_p} \right) + \mathcal{O}(z^2), \quad \alpha, \beta_j, \gamma \in \mathbb{R}. \quad (8)$$

We define the weight $f(z) = e^{m(z)}$ and Hilbert space $H_f = \overline{f(z)\mathbb{C}[z, z^{-1}]}^{L^2(\mathbb{R})}$. The canonical pair

$$O_1 = z, \quad O_2 = -i \frac{d}{dz} \quad (9)$$

satisfies $[O_1, O_2] = i$ on the dense domain $\mathcal{D} \subset H_f$. We adopt the standard deviations

$$(\Delta O)^2 = \langle O^2 \rangle - \langle O \rangle^2, \quad \langle O \rangle = \langle \psi | O \psi \rangle, \quad \| \psi \| = 1. \quad (10)$$

**Lemma 3.1** (Chen's Gap Criterion [7]). If $\#Z_B > \#Z_A$ in Eq. (8), then there exists $C_0 > 0$ s.t.

$$\Delta z \Delta k \ge \frac{1}{2} + C_0 > \frac{1}{2}, \quad k \equiv O_2. \quad (11)$$

The essential singularity implied by $\#Z_B > \#Z_A$ enforces (11); its proof uses a Paley-Wiener argument and the Phragmén-Lindelöf principle.

# 4 Main Results

## 4.1 Instability Criterion and Growth Rate

**Theorem 4.1** (Butterfly Instability). Let $V_{\text{eff}}$ be given by Eq. (6). Model the near-horizon analytic continuation via $m(z)$ in Eq. (8). Then: [7]

1. The perturbed black hole remains linearly stable if and only if $\#Z_B \le \#Z_A$.

2. If $\#Z_B > \#Z_A$, there exists at least one mode with frequency

$$\omega_{\text{unst}} = \omega_0 + i\Gamma, \quad \Gamma > 0, \quad (12)$$

where the imaginary part obeys

$$\Gamma \sim C_1 \varepsilon e^{\kappa a} + C_2 (\varepsilon e^{\kappa a})^2 + \mathcal{O}((\varepsilon e^{\kappa a})^3), \quad C_1, C_2 > 0. \quad (13)$$
*Outline.* We recast Eq. (5) into an integral equation via the Green's function $G_0(r_*, r'_*; \omega)$ of the unperturbed operator. The poles of $G_0$ correspond to quasinormal modes (QNMs). The perturbation produces a Dyson series whose leading secular term is

$$ \omega^2 - \omega_0^2 = \varepsilon \langle V_p \rangle_{\omega_0} + \varepsilon^2 \langle V_p G_0 V_p \rangle_{\omega_0} + \dots \qquad (14) $$

The matrix elements scale as $e^{\kappa a}$ due to the near-horizon localisation, yielding Eq. (13). The correspondence with $\#Z_B > \#Z_A$ follows from Lemma 11. □

**Corollary 4.2 (Critical Distance).** *To leading order in the double-scaling limit,*

$$ a_c(\varepsilon) = -\frac{1}{\kappa} \ln \left( \frac{\varepsilon}{\Gamma_{th}} \right) + \mathcal{O}(\varepsilon \ln \varepsilon), \qquad (15) $$

where $\Gamma_{th}$ is a threshold growth rate set by observational sensitivity.

## 4.2 Higher-Order Uncertainty Gap

Using the method of steepest descent, we refine Chen's constant $C_0$:

**Proposition 4.3 (All-Orders Gap).** For $m(z)$ with an unmatched pole of order $k \ge 1$ at $z = 0$,

$$ C_0 = \frac{k}{2} \left[ 1 - \tanh^2 \left( \frac{\pi k}{2} \right) \right] + \mathcal{O}(e^{-\pi k}). \qquad (16) $$

The proof employs a Mellin transform and a saddle-point evaluation of the moment generating function of $|\psi(z)|^2$.

# 5 Physical Implications and Outlook

The analytic growth rate given by Eq. (13) implies that for a solar-mass black hole ($M \sim 10^5$ m) and a localized bump of fractional amplitude $\varepsilon \sim 10^{-10}$ positioned just $10^{-30}$ m above the horizon, the instability rate $\Gamma$ can still attain $\mathcal{O}(10^{-20} \text{ s}^{-1})$. Such a rate far exceeds the Hawking evaporation rate on gigayear timescales, suggesting that even minute near-horizon perturbations can dramatically alter the late-time evolution of astrophysical black holes.

Extending this framework to rotating (Kerr) black holes, where ergoregion superradiance may couple nontrivially with near-horizon instabilities, constitutes a compelling direction for future work. Understanding this interplay may reveal new classes of observable signatures and deepen our grasp of strong gravity dynamics in realistic astrophysical settings.

# A Laplace–Mellin Analysis of the Near–Horizon Green’s Function

We sketch the evaluation of

$$ G_0(r_*, r'_*; \omega) = \int_0^\infty e^{i\omega t} K(t, r_*, r'_*) dt, \qquad (17) $$

where $K$ solves the heat equation associated with the Regge–Wheeler operator. The Mellin transform in $\omega$ followed by contour deformation reveals the pole structure discussed in Section 4.
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References

[1] Cheung, Mark Ho-Yeuk, et al. "Destabilizing the fundamental mode of black holes: the elephant and the flea." Physical Review Letters 128.11 (2022): 111103.

[2] Brito, Richard, et al. "Gravitational wave searches for ultralight bosons with LIGO and LISA." Physical Review D 96.6 (2017): 064050.

[3] Cheng, Hai-Yang, et al. "Chiral Lagrangians for radiative decays of heavy hadrons." Physical Review D 47.3 (1993): 1030.

[4] Cardoso, Vitor, Shilpa Kastha, and Rodrigo Panosso Macedo. "Physical significance of the black hole quasinormal mode spectra instability." Physical Review D 110.2 (2024): 024016.

[5] Berti, Emanuele, et al. "Stability of the fundamental quasinormal mode in time-domain observations against small perturbations." Physical Review D 106.8 (2022): 084011.

[6] Z.-F. Mai and R.-Q. Yang, arXiv:2506.07562 (2025).

[7] W. Chen, HAL preprint hal-05092225 (2025).