Horizon and Effective-Field-Theory Requirements

Saying that a proposal “modifies the horizon” is not a physical specification. One must say where the modification acts, when it turns on, which state and frequencies it affects, how large it is in a controlled norm, and what a freely falling or exterior detector measures. These questions distinguish a hard firewall from soft atmosphere-scale couplings, a state-dependent interior map, a smooth microstate geometry, and a replica calculation that changes an entropy saddle without changing local Lorentzian effective field theory.

Required background. Quantum Information for Black-Hole Physics supplies entanglement, trace distance, channels, and the small-correction language used to test horizon correlations.

Helpful background. Black-Hole Thermodynamics and Semiclassical Gravity supplies stress tensors and backreaction; Quantum Fields in Curved Spacetime supplies local horizon states; Black-Hole Information supplies the AMPS and island questions; Quantum Gravity supplies candidate departures from local spacetime physics.

State the effective theory and its domain

For a large black hole of curvature radius $r_s$, local effective field theory asserts that observables with wavelength $\ell$ in a freely falling laboratory obey the low-energy QFT prediction when

$$\ell_{\mathrm P}\ll \ell\ll r_s, \qquad \frac{\ell_{\mathrm P}^2}{\ell^2}\ll1, \qquad \frac{\ell^2}{r_s^2}\ll1.$$

The claim also requires a state: near a regular future horizon the short- distance two-point function has the Hadamard vacuum singularity, while the global Unruh state contains outgoing Hawking flux. A proposal must identify whether it changes the state, operator algebra, Hamiltonian, interaction, factorization, or spacetime itself. It must also state the cutoff and the first operator omitted from its description.

The equivalence principle does not demand that every finite laboratory see empty Minkowski space. Curvature, acceleration, long-time correlations, and state-dependent fluxes remain measurable. The relevant smoothness test is that renormalized local observables and detector response stay within the claimed semiclassical error for a freely falling trajectory, not that coordinates erase all physics.

Quantify the entanglement change

Let $b$ be a late exterior mode and $\widetilde b$ its near-horizon partner. Smoothness requires their state to be close, on the selected mode algebra, to the local vacuum entanglement. If

$$\frac12\left\lVert \rho_{b\widetilde b}-\rho^{\mathrm{vac}}_{b\widetilde b} \right\rVert_1\le\varepsilon,$$

continuity bounds restrict how much their entropy can differ in a regulated finite-dimensional mode space. Trace distance alone does not bound an unbounded energy or stress observable: a tiny high-energy tail can carry a large mean energy. A cutoff or independent energy-moment bound is needed for that inference. A recovery proposal must show whether its cumulative correction over many emissions remains $O(\varepsilon)$, grows with the number of modes, or becomes order one near the Page time.

Mathur’s small-corrections argument makes one such sequential-emission assumption set explicit; see Mathur 2009, §§4–6. Its conclusion is not that every imaginable nonlocal or nonfactorizing theory fails. It is that uniformly small corrections satisfying the sequential pair-state hypotheses cannot reverse the leading entanglement growth. A proposal evades the bound only by identifying which hypothesis—factorization, pair-state closeness, state space, or correction size—changes.

Compare modifications on common axes

Horizon scenario	Location and scale	Required detector or stress-tensor check
Hard firewall	Planck-distance region with order-one occupation of infalling modes	Compute freely falling excitation energy and renormalized stress tensor, including backreaction.
Soft atmosphere transfer	Region of order $r_s$ with weak, state-dependent couplings integrated over long times	Close information and energy fluxes while bounding exterior correlator and image deviations.
Fuzzball-like structure	State-dependent horizon-scale microstructure replacing the conventional interior	State which known microstates are represented and predict infalling and asymptotic observables.
State-dependent reconstruction	Semiclassical code subspace with interior operators depending on the equilibrium state	Test overlaps, superpositions, commutators, and Born-rule consistency on the declared domain.
Island/QES prescription	Change of dominant generalized-entropy saddle in a bath calculation	Do not infer a local Lorentzian stress-tensor modification unless a separate dynamics supplies one.

The AMPS argument shows why unitarity, a smooth horizon, semiclassical exterior factorization, and old-hole entanglement cannot all retain their naive forms; the precise mode and mining qualifications appear in Almheiri et al. 2013, §§2–3 and Appendices B–C. It does not select one row of the table. Each row has a different observable cost and therefore must be tested separately.

Information flux and backreaction

Information transfer cannot be assessed from a correlation statement alone. If an interaction

$$\Delta H(t)=\sum_A \lambda_A(t)\, \mathcal O_A^{\mathrm{BH}}\mathcal O_A^{\mathrm{ext}}$$

acts outside the horizon, the functions $\lambda_A$, the operator bandwidth, state dependence, spatial support, and energy flux must be given. Enough independent channel capacity must accumulate to purify the radiation, but the same coupling must not produce an unacceptable stress tensor, violate the declared locality scale, or spoil known exterior physics. Giddings develops one explicit atmosphere-scale parametrization in Giddings 2013, §§2–5; it is a model class, not an established recovery mechanism.

Conversely, a proposal that preserves local EFT can move the burden to factorization, state dependence, or nonperturbative reconstruction. That is a legitimate logical option only if the new algebra is defined and agrees with ordinary QFT observables within its error. Generic fixed-background QFT and programme-specific string constructions are treated elsewhere; this page compares only the horizon assumptions they export to the information problem.

Results and open questions through 2026-08-19

Evidence or proposal	Assumptions	Controlled result	Limitation or open requirement
Local QFT near a large regular horizon, Harlow 2016, §§3.2–3.5 and 4.1–4.4	Hadamard-like state, low curvature, wavelengths above the cutoff, and a localized detector or operator algebra	Gives the standard short-distance entanglement, near-horizon mode structure, and semiclassical radiation baseline	Does not determine the exact late-time fine-grained state
Small-corrections and AMPS arguments, Mathur 2009, §§4–6 and Almheiri et al. 2013, §§2–3	Sequential emission or old-hole setup, declared subsystem factorization, smoothness, and unitarity assumptions	Gives sharp entropy-growth or monogamy incompatibilities within those assumption sets	Establishes an incompatibility, not a unique horizon replacement
Nonviolent nonlocality models, Giddings 2013, §§2–5	Effective atmosphere-scale couplings with specified support, bandwidth, state, and transfer rate	Parameterizes correlation transfer without imposing a Planckian wall	Microscopic origin, generic state dependence, causality, stress tensor, and distinctive observations remain unresolved
State-dependent interiors, Papadodimas and Raju 2014, §§2–5	Equilibrium state, small algebra, restricted excitations, and state-labelled mirror map	Reproduces selected smooth-horizon correlators on the declared domain	Overlap consistency, linearity on enlarged domains, Born-rule tests, and nonperturbative accuracy remain disputed
Microstate and fuzzball programs, Bena et al. 2022, §§2–8	Protected string-theory sectors and explicitly constructed state families or geometries	Supplies protected counting results and many nontrivial horizon-scale solutions	Representation of typical states and generic collapse or radiation dynamics is not settled

The competing positions are thus not simply “drama” versus “no drama.” They place the required order-one change in different objects: local state, Hamiltonian, algebra, factorization, or microscopic geometry. The conclusions could change if a proposal derives that change from a complete theory with quantified infalling and exterior errors, if a general obstruction rules out an entire class under weaker assumptions, or if a distinctive horizon observable is calculated and survives backreaction.

Check your understanding

A model changes each emitted pair by trace distance $10^{-6}$ and emits $10^{80}$ modes. Is the small number alone enough to conclude that the horizon remains semiclassically smooth and that the radiation purifies?

Solution

No. The trace distance must refer to a declared pair algebra and regulator, and one must control how errors compose across correlated emissions. Small corrections satisfying the sequential theorem leave the entropy increasing even when earlier emissions are correlated; statistical independence is not a required loophole. A nonfactorizing model changes the subsystem assumptions and needs a different analysis. Smoothness additionally requires local detector and stress-tensor bounds, and purification requires a complete reference-system channel test. The number $10^{-6}$ without a composition theorem, dynamics, or observable norm settles neither question.

References

• Almheiri, Ahmed, Donald Marolf, Joseph Polchinski, and James Sully, “Black Holes: Complementarity or Firewalls?” Journal of High Energy Physics 02 (2013) 062. arXiv
• Bena, Iosif, Emil J. Martinec, Samir D. Mathur, and Nicholas P. Warner, “Fuzzballs and Microstate Geometries: Black-Hole Structure in String Theory,” arXiv:2204.13113 (2022). Open manuscript
• Giddings, Steven B., “Nonviolent Nonlocality,” Physical Review D 88 (2013) 064023. arXiv
• Harlow, Daniel, “Jerusalem Lectures on Black Holes and Quantum Information,” Reviews of Modern Physics 88 (2016) 015002. DOI. Open manuscript
• Mathur, Samir D., “The Information Paradox: A Pedagogical Introduction,” Classical and Quantum Gravity 26 (2009) 224001. arXiv
• Papadodimas, Kyriakos and Suvrat Raju, “State-Dependent Bulk-Boundary Maps and Black Hole Complementarity,” Physical Review D 89 (2014) 086010. DOI. Open manuscript