Real-Time Claims, Horizons, and Validity Limits

Every real-time holographic conclusion carries a causal contract: state-preparation contour, operator ordering, horizon or interior condition, fluctuation channel, coupling expansion, and time window. A regular classical event horizon supports infalling linear response and quasinormal decay in its saddle. An apparent horizon, a transient trapped region, or a horizonless microstate does not inherit all of those conclusions automatically.

Required background. Schwinger–Keldysh Contours and Real-Time Bulk Geometries fixes the state and ordering, and Quasinormal Modes, Poles, and Spectral Response fixes saddle relaxation.

Helpful background. Semiclassical Breakdown Diagnostics and Causal, Killing, Trapping, and Apparent Horizons: The QFT Interface supply gravitational limits. Finite Size, Symmetry Sectors, and Scrambling False Positives and Scrambling Evidence and Claim-Status Matrix supply finite-system controls.

Horizon-dependent real-time statements

Claimed result	Required bulk input	Principal limitation
Retarded absorption	Future-horizon regularity in the prepared state	A horizonless interior requires a different boundary condition
Quasinormal decay	Stationary saddle and linearized channel	Intermediate-time saddle response, not exact finite-$N$ decay
Thermalization	Dynamical solution and observable comparison	Apparent-horizon formation is neither necessary nor sufficient for every probe
Hydrodynamization	Long-wavelength stress tensor and small residual	Does not require isotropy; depends on tolerance and frame-invariant observables
OTOC growth	Regulated contour, elastic eikonal regime, factorization	Ends before or at scrambling and is operator-dependent

The infalling retarded prescription is justified by future-horizon regularity in the selected thermal state Son and Starinets 2002. It is not a local rule that can be pasted onto any geometry containing a surface called a horizon.

A correlator across four time regimes

For a black-brane retarded correlator, separate:

1. Early transients: source shape and high-frequency response matter.

2. Quasinormal window: a few poles approximate the classical response,

   $$G_R(t,\mathbf k) \simeq-i\theta(t)\sum_nR_n(\mathbf k)e^{-i\omega_n t}.$$

3. Nonlinear or nonperturbative crossover: mode coupling, loops, or exponentially small saddles become comparable to the decayed leading term.

4. Finite-$N$ late time: discreteness, noise, and recurrences invalidate exact exponential decay.

The transition between 2 and 3 is observable-dependent. If the leading pole has damping rate $\Gamma$ and an omitted correction has size $e^{-S}$, equality occurs parametrically near

$$t_{\mathrm{cross}}\sim\frac{S}{\Gamma},$$

not at a universal geometric time. This estimate assumes the omitted contribution is approximately stationary; a growing secular term gives a different crossover.

Horizon-replacement adversary

Replace the stationary event horizon by a transient apparent horizon. The local geometry may still absorb over a finite interval, but stationarity and a global frequency-domain pole expansion are absent. Replace it instead by a smooth horizonless interior. The radial problem can have real normal modes or extremely narrow resonances rather than quasinormal absorption.

Each claim must be rederived:

• causal retarded response follows from the full contour and interior condition;
• decay follows only over the interval in which leakage or coarse graining is effective;
• thermality requires an observable comparison, not a trapped surface alone;
• exact unitarity cannot be inferred from a classical causal diagram.

The time-dependent numerical framework of Chesler and Yaffe 2014 illustrates why constraints, slicing, and boundary one-point functions must be tracked together.

What the result establishes

Literature cutoff: 25 July 2026. Classical horizons robustly organize leading retarded response, hydrodynamics, and chaos in controlled top-down or EFT saddles. They do not determine exact finite-$N$ late time or prove that a different interior with similar coarse exterior observables has the same microscopic response.

Curved-spacetime QFT develops horizon taxonomy and semiclassical validity; Thermal and Nonequilibrium QFT develops general real-time methods. The next page treats exact spectral discreteness.

The chapter overview compares these assumptions and limitations.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Chesler, Paul M., and Laurence G. Yaffe. “Numerical Solution of Gravitational Dynamics in Asymptotically Anti-de Sitter Spacetimes.” Journal of High Energy Physics 2014, 086 (2014). doi:10.1007/JHEP07(2014)086.
• 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.