Lorentzian, Nonequilibrium, and Chaotic Holography

Real-time holography connects boundary response and nonequilibrium evolution to Lorentzian bulk problems. The contour, initial state, and horizon conditions specify which correlator or evolution is being calculated.

Enter this chapter

Begin with Lorentzian Holographic Correlators and Infalling Conditions to derive retarded response from the bulk boundary-value problem.

Helpful background. Closed-Time-Path Generating Functionals in Practice supplies real-time contours; Out-of-Time-Order Correlators and Contour Regularization supplies OTOCs.

Topics

1. Lorentzian Holographic Correlators and Infalling Conditions — Derives the source–response ratio and spectral flux for a retarded black-brane correlator.
2. Schwinger–Keldysh Contours and Real-Time Bulk Geometries — Glues Euclidean and doubled Lorentzian saddles and tests normalization and KMS.
3. Holographic Initial States and Euclidean Caps — Converts cap sources into normalizable Lorentzian initial data and rejects singular preparations.
4. Holographic Influence Functionals and Open-Sector Dynamics — Extracts causal dissipation and symmetrized noise before testing the Markovian limit.
5. Quasinormal Modes, Poles, and Spectral Response — Solves the infalling, source-free eigenproblem and separates poles from finite-$N$ levels.
6. Bulk Quasinormal Modes and Boundary Hydrodynamic Poles — Derives shear, sound, and diffusion dispersions in invariant channels.
7. Time-Dependent Geometries and Holographic Thermalization — Compares local, geodesic, and extremal-surface probes in collapse.
8. Holographic Quenches and Entanglement Growth — Extracts growth and saturation while bounding entanglement-velocity claims.
9. Holographic Hydrodynamization, Attractors, and Gradient Asymptotics — Separates asymptotic hydrodynamics from quasinormal transients and initial-state dependence.
10. Shockwaves, OTOCs, and Scrambling — Derives the eikonal growth, butterfly front, and scrambling window for a regulated OTOC.
11. Real-Time Claims, Horizons, and Validity Limits — Identifies which response statements survive changes of horizon type or time regime.
12. Finite-N Spectra, Recurrences, and the Late-Time Plateau — Compares smooth saddle decay with discrete spectra, averaged ramps, plateaus, and recurrences.

How the ideas fit together

For retarded response around an appropriate black-hole background, infalling horizon conditions select causal propagation. Quasinormal frequencies then describe poles governing linear relaxation Son and Starinets 2002, § 3. A general real-time state or higher ordering requires the fuller contour construction, rather than just a horizon rule.

Nonlinear relaxation, entanglement growth, and shockwave diagnostics extend this framework in different directions. A classical dissipative horizon captures controlled large-N behavior, but it cannot determine exact finite-N recurrences or a late-time plateau from the same approximation.

The comparison below separates the assumptions and tests needed for these different conclusions.

Claim-domain comparison

Representative claim domains and validity boundaries for Lorentzian, Nonequilibrium, and Chaotic Holography

Claim object	State, ensemble, and conventions	Approximation, status, and evidence timing	Uncertainty and counterevidence	Falsifier	Failure condition	Licensed conclusion
retarded response	Declare source convention and infalling condition; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: real-time contour and state → Lorentzian bulk geometry → initial and horizon conditions → poles, response, and chaos tests → time-window conclusion. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “flux and Kramers-Kronig checks” check is counterevidence to the promoted claim.	The proposed response is nonzero before the source, or fails the appropriately subtracted dispersion relation where its analyticity and growth hypotheses hold.	Inferring all Schwinger–Keldysh correlators from one retarded Green function.	a causal Green function
quasinormal mode	Declare background, channel, and boundary condition; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: real-time contour and state → Lorentzian bulk geometry → initial and horizon conditions → poles, response, and chaos tests → time-window conclusion. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “pole and convergence test” check is counterevidence to the promoted claim.	The proposed physical mode fails the source-free boundary or horizon condition, or its frequency does not converge under the stated numerical refinement.	Identifying a linear relaxation scale with a nonlinear thermalization time.	linear relaxation scale
scrambling or plateau	Declare operator ordering, N, and time limits; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: real-time contour and state → Lorentzian bulk geometry → initial and horizon conditions → poles, response, and chaos tests → time-window conclusion. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “OTOC or spectral benchmark” check is counterevidence to the promoted claim.	With operator ordering, averaging, and limit order fixed, the proposed growth or plateau disagrees with the applicable benchmark inside its stated time window.	Promoting time-window behavior to exact finite-N irreversibility.	behavior in the stated window

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Where to go next

Continue to Holographic Matter, Transport, and Model Building to identify charge density and chemical potential in a black-brane solution.

References

• Son, Dam T., and Andrei O. Starinets. “Minkowski-Space Correlators in AdS/CFT Correspondence: Recipe and Applications.” Journal of High Energy Physics 2002, 042 (2002). doi:10.1088/1126-6708/2002/09/042.