Chaos, Quasinormal Modes, and Pole Dynamics

Black-hole response and many-body chaos meet through analytic structure, but they are not the same observable. Quasinormal modes are poles of retarded correlators; hydrodynamic poles encode conservation laws; pole-skipping is a special nonuniqueness of a retarded Green function; shockwaves calculate horizon sensitivity; and out-of-time-order correlators diagnose operator growth. This chapter builds those links in that order and then asks how finite coupling, higher spin, and stringy Regge physics change them. Every claimed equality is conditional on channel, boundary conditions, contour, and approximation.

Helpful background. Gauge Theory and String-Theory Preparation supplies linear response and large-$N$ methods; Quantum Gravity supplies horizon scattering; Quantum Information supplies scrambling diagnostics; and Black-Hole Perturbation Theory supplies ingoing modes and quasinormal boundary conditions.

Three analytic continuations, three questions

With Fourier convention $e^{-i\omega t+i\mathbf k\cdot\mathbf x}$, an ingoing bulk solution has near-boundary coefficients $A(\omega,k)$ and $B(\omega,k)$. Schematically,

$$G_R(\omega,k)\propto\frac{B(\omega,k)}{A(\omega,k)},$$

so a quasinormal frequency satisfies $A=0$ with ingoing horizon and normalizable boundary conditions. Stable modes lie in the lower half-plane. Son and Starinets 2002, §§2–4, pp. 3–14 establishes this retarded prescription.

Hydrodynamic poles approach the origin with momentum because conservation laws create long-lived modes. Pole-skipping instead occurs at a special complex $(\omega,k)$ where a near-horizon constraint degenerates and both numerator and denominator can vanish. OTOCs live on a different time contour and, within an intermediate factorization window, may behave as

$$F(t,\mathbf x)=F_0-\varepsilon \exp\!\left[\lambda_L\left(t-\frac{|\mathbf x|}{v_B}\right)\right]+\cdots .$$

Identifying $\omega_*=i\lambda_L$ and $k_*=i\lambda_L/v_B$ requires the appropriate energy-density channel and a theory in which the near-horizon degeneracy is connected to the same chaotic mode. A generic QNM damping rate is not a Lyapunov exponent.

Read the chapter in this order

1. Quasinormal Modes as Retarded Poles derives the ingoing prescription and residue information.
2. Hydrodynamic Poles and Transport follows conserved densities into diffusion and sound and states the derivative-expansion domain.
3. Pole-Skipping analyzes the exceptional near-horizon point and the slope-dependent Green-function limit.
4. Shockwaves and the Butterfly Effect turns an early perturbation’s blueshift into a geometric scrambling calculation.
5. Holographic Out-of-Time-Order Correlators fixes contour, normalization, factorization window, $\lambda_L$, and $v_B$.
6. Finite-Coupling, Higher-Spin, and String Corrections to Chaos separates Einstein control from Regge intercepts, extra exchanges, and finite-$N$ limitations.

The route deliberately goes from causal linear response to a genuinely out-of-time-order observable. A pole is evidence only in its channel; the same background may contain nonhydrodynamic poles that never control operator growth.

Results and limitations through 2026-08-20

The formulation axes are state, operator channel, Fourier convention, horizon boundary condition, momentum domain, time contour, large-$N$ window, and bulk coupling expansion.

Evidence or position	Controlled result	Limitation or adverse discriminator
Retarded prescription, Son and Starinets 2002, §§2–4, pp. 3–14	Ingoing bulk solutions yield causal retarded correlators whose poles are quasinormal frequencies	Pole locations alone do not determine OTOCs, many-body chaos, or nonlinear saturation
Pole-skipping in Einstein–matter gravity, Blake, Davison, Grozdanov, and Liu 2018, §§II–IV, pp. 7–31	A horizon equation degenerates at the maximal-chaos frequency and butterfly momentum in the energy channel	Nonuniqueness needs the full boundary map; other channels, higher derivatives, and nonmaximal chaos can modify the relation
Shockwave calculation, Shenker and Stanford 2014, §§3–4, pp. 6–17	Exponential horizon blueshift produces $t_*\sim\lambda_L^{-1}\log(1/\varepsilon_{\rm sh})$, with a dimensionless initial shock amplitude, in controlled AdS black holes	String spreading, Planck effects, state choice, and late-time nonlinearities bound the calculation
Chaos bound, Maldacena, Shenker, and Stanford 2016, §§2–4, pp. 5–15	Analyticity, boundedness, and factorization assumptions give $\lambda_L\leq2\pi T$	The bound does not assert exponential growth for every operator or prove that a pole-skipping point saturates it

Controlled holographic Einstein regimes connect the four diagnostics. Competing positions differ on whether pole-skipping is a universal chaos measurement or a horizon constraint that needs additional boundary evidence. The latter caution is essential: outside the controlled channel and approximation, the identification remains model-dependent, and a general converse theorem is unresolved.

Coherence and independent checks

Every page uses the same retarded Fourier sign, so decaying quasinormal poles have $\operatorname{Im}\omega\lt0$, whereas the chaos-related pole-skipping point lies at positive imaginary frequency and imaginary momentum. Retarded analyticity in the upper frequency half-plane holds at real physical momentum for a stable state; it does not forbid this exceptional point after momentum is also continued to a complex value. Numerically, residues and convergence with radial resolution must accompany pole locations; hydrodynamic dispersion must recover the appropriate susceptibility and conservation law.

For chaos, one must state the factorization window $t_d\ll t\ll t_*$ and verify that the connected OTOC is initially small. The Lyapunov exponent, butterfly velocity, and scrambling time answer different questions. Finite-coupling or string corrections can shift an effective Regge intercept and the spatial profile before Planck-scale physics matters. Higher-spin exchanges may also invalidate a pure Einstein interpretation without violating the general analytic bound.

The chapter’s strongest check is triangular: compute $(\lambda_L,v_B)$ from an OTOC or shockwave, locate the energy-channel pole-skipping point, and verify that the full retarded solution has the predicted slope structure. Agreement is compelling within one model; disagreement identifies which assumption or channel failed.

What could change these conclusions

The conclusions could change if a theorem proves or disproves a general converse between pole-skipping and OTOC growth, if an independent calculation changes the finite-coupling Regge correction, or if a counterexample displays the same horizon degeneracy without the inferred chaos data. The conclusions also depend on whether new boundary conditions, operator channels, higher-spin spectra, numerical replications, or finite-$N$ results alter the factorization window.

Review the chapter

A calculation finds a nonhydrodynamic quasinormal damping rate $2\pi T$ and a horizon degeneracy at $\omega=i2\pi T$. Determine whether maximal chaos follows and list the missing checks.

Solution

Maximal chaos does not follow from the damping rate because a lower-half-plane retarded pole and an upper-half-plane horizon degeneracy answer different analytic questions. One must solve the energy-channel boundary problem near the special point, verify simultaneous zero and pole behavior with the expected slope dependence, and independently calculate an OTOC or shockwave in a factorization window. The operator normalization, state, contour, and butterfly momentum must agree. Only then does $\lambda_L=2\pi T$ have the controlled many-body interpretation; finite coupling or additional exchanges can still restrict its domain.

References

• Blake, Mike, Richard A. Davison, Sašo Grozdanov, and Hong Liu. “Many-Body Chaos and Energy Dynamics in Holography.” Journal of High Energy Physics 10 (2018) 035. arXiv:1809.01169.
• Maldacena, Juan, Stephen H. Shenker, and Douglas Stanford. “A Bound on Chaos.” Journal of High Energy Physics 08 (2016) 106. arXiv:1503.01409.
• Shenker, Stephen H., and Douglas Stanford. “Black Holes and the Butterfly Effect.” Journal of High Energy Physics 03 (2014) 067. arXiv:1306.0622.
• Son, Dam T., and Andrei O. Starinets. “Minkowski-Space Correlators in AdS/CFT Correspondence: Recipe and Applications.” Journal of High Energy Physics 09 (2002) 042. arXiv:hep-th/0205051.