Quantum-Gravity Boundary Data from the Interior

A microscopic account of the black-hole interior need not reproduce a chosen Schwarzschild-time slice or a local field at an unanchored coordinate point. It must reproduce the invariant content that made the semiclassical description successful: asymptotic charges and fluxes, causal and relational observables, regularity or breakdown statements, exterior correlators, and a normalized map between complete initial and final data. This page identifies those inputs without presuming that classical spacetime or tensor factorization survives microscopically.

Helpful background. Semiclassical Gravity supplies renormalized fluxes and matching regimes; Quantum Information supplies algebras and channels; Quantum Fields in Curved Spacetime supplies exterior states and correlators; Black-Hole Information supplies the recovery question; Quantum Gravity supplies microscopic programmes; and Causality and Global Structure supplies invariant causal and extension language.

Boundary charges and fluxes survive the slicing test

The most robust data live at a physical boundary. ADM or Bondi energy, angular momentum, gauge charge, incoming state preparation, outgoing radiation algebra, and their flux-balance laws do not depend on how an interior foliation is drawn. A candidate evolution should reproduce, in its semiclassical regime,

$$Q_{\text{initial}}-Q_{\text{final}} =\int_{\mathscr I^+}\!\mathcal F_Q,$$

with the appropriate horizon or additional-boundary terms stated. The symbol $Q$ may denote energy, angular momentum, or a gauge charge; the measure and flux operator belong to the particular asymptotic formulation. A missing term is physical only after the complete boundary algebra has been declared.

Information adds a stronger requirement. If the complete asymptotic system is closed, its evolution should define a normalized linear channel, and a claim of unitarity should state the state domain and error norm. If baby-universe, remnant, or additional boundary sectors are included, they belong in the complete system before a purity verdict is assigned. This is invariant data; the statement that “information is at radius $r$ on this slice” generally is not.

Relational observables replace coordinate locations

Coordinates label a gauge choice. A field value $\phi(x)$ at a coordinate point is therefore not by itself a diffeomorphism-invariant interior observable. One can instead anchor an operator to an asymptotic frame, a worldline and its proper time, a matter clock, or a geometrically selected surface. Gravitational dressing then extends beyond a compact coordinate neighborhood and can alter commutators. Donnelly and Giddings construct perturbative dressings and display the obstruction to exactly local commuting subsystems in Donnelly and Giddings 2016, §§ 2–5.

The choice of anchor remains part of the observable. Two dressings can agree on the leading local excitation yet differ in their asymptotic gravitational field. A microscopic theory need not reproduce every dressing as a separate degree of freedom, but it must reproduce their gauge-invariant correlations and transformation laws within the claimed perturbative order. The operational cost of constructing such observables is developed separately in Gravitationally Dressed Observables.

Causal, regularity, and semiclassical matching data

A causal diagram encodes invariant relations only after its boundaries and extensions are specified. Useful inputs include the existence of trapped or antitrapped surfaces, causal accessibility between relationally anchored events, and whether the maximal semiclassical development is extendible in a declared regularity class. Event-horizon location on a finite slice is teleological; apparent-horizon position is foliation dependent. Neither should be promoted to microscopic data without qualification.

Classical inner-horizon results illustrate the needed precision. In spherical Einstein–Maxwell–scalar evolution, a continuous extension can coexist with mass and curvature blowup Dafermos 2005, main theorems, pp. 445–504. A completion must therefore say whether it matches the continuous metric, the divergent higher derivatives, the causal boundary, or a replacement quantum observable. “The Cauchy horizon survives” has no invariant content until that regularity and model are named.

Semiclassical matching should be quantitative. Outside a declared quantum region, compare renormalized stress tensors, detector responses, low-point correlators, and charge fluxes with effective field theory. State an expansion parameter $\epsilon$ and require, for a chosen observable family,

$$\bigl\lvert \langle\mathcal O\rangle_{\text{micro}} -\langle\mathcal O\rangle_{\text{semi}} \bigr\rvert \le C_{\mathcal O}\epsilon.$$

The constants, state family, time interval, and allowed operators must be specified. Agreement for simple exterior correlators does not establish a smooth interior for exponentially complex probes.

Entanglement data require a named algebra

Entropy is attached to a state restricted to an algebra or factor, not to a coordinate region alone. In gravity, centers, constraints, area operators, and dressing complicate factorization. A microscopic proposal should identify the radiation algebra, the code domain, and the map relating logical interior operators to asymptotic representatives. It should then report relative entropy, recovery error, or correlator agreement in an explicit norm.

Non-isometric code models demonstrate why the domain matters. Akers and collaborators obtain approximate preservation and interior reconstruction for restricted state families while sufficiently complex or outside-domain operations reveal failure Akers et al. 2022, §§ 4–8. The result supplies a possible interface principle, not a unique microscopic geometry. A smooth code-subspace experience and global unitarity can coexist without an independent interior tensor factor, but only inside the stated approximation.

The programme-specific construction of microscopic black holes belongs to Evaporation and Information Interfaces in Quantum Gravity. The present page supplies the black- hole-information observables against which those programmes are compared; it does not select among them.

Results and open questions through 2026-08-19

This discussion reflects results available through 2026-08-19. A conservative position demands only asymptotic charges, a unitary S-matrix, and controlled exterior effective field theory, regarding the interior as nonfundamental. A geometric position also requires relational observables that reproduce smooth infall and the expected causal structure. A code-based position asks for approximate algebraic reconstruction on a state domain rather than one global interior geometry. These competing positions differ in formulation; none licenses coordinate- dependent data as microscopic observables.

Evidence or requirement	Result	Scope or limitation
Perturbative dressing in Donnelly and Giddings 2016, §§ 2–5	Relational or asymptotically anchored operators are gauge invariant but not exactly compactly local.	Perturbative gravity with chosen dressing and boundary conditions; not a full interior completion.
Charged spherical interior in Dafermos 2005, main theorems, pp. 445–504	Causal extension and higher-regularity blowup are distinct boundary data.	Classical Einstein–Maxwell–scalar symmetry class, not quantum or rotating evolution.
Restricted non-isometric encoding in Akers et al. 2022, §§ 4–8	Interior correlators and reconstruction can be approximately preserved on a controlled state family.	Random-model, code-domain, complexity, and norm assumptions; failure outside the domain is essential.
Foliation-change adverse check	Boundary charges and relational correlations remain fixed while apparent-horizon location and interior coordinate labels can move.	Removes gauge artifacts but does not decide which quantum observables replace the classical interior.

The controlled evidence identifies data that a completion should match; it does not establish that one currently known framework matches every row in a generic evaporating four-dimensional spacetime. That synthesis remains unresolved.

A portable comparison procedure

First declare the complete boundary system, conserved charges, state class, and asymptotic channel. Next list relational observables with their anchors and commutators. State causal and regularity claims without coordinate labels. Specify the semiclassical region, expansion parameter, error norm, and time interval. Finally test a different foliation, dressing, and state just outside the code domain. Any leading prediction that changes under a pure gauge choice is not acceptable boundary data.

The conclusions could change after a microscopic programme derives all listed asymptotic and relational observables in one controlled evaporating model, or a theorem shows that a proposed code domain reproduces smooth-infall correlators and unitary radiation with quantified errors, or an adverse gauge, dressing, or outside-domain test changes a claimed invariant conclusion.

Exercises

On two foliations of the same spacetime, an apparent horizon occurs at different coordinate radii, while Bondi energy flux and a detector’s proper-time response agree. Which quantities are suitable microscopic boundary data, and what extra information is needed to use the horizon statement?

Solution

The Bondi flux and relational detector response are suitable because their boundary or worldline anchors survive the foliation change. The coordinate radius is not. The apparent-horizon statement becomes meaningful only after the two-surface, null expansions, foliation, and mapping between descriptions are specified; its existence may be useful, but its coordinate position is not fundamental data. One should then test whether any proposed microscopic observable reproduces the invariant expansion and causal relations rather than the original label.

References

• Akers, Chris, Netta Engelhardt, Daniel Harlow, Geoff Penington, and Shreya Vardhan. 2022. “The Black Hole Interior from Non-Isometric Codes and Complexity.” arXiv:2207.06536. arXiv.
• Dafermos, Mihalis. 2005. “The Interior of Charged Black Holes and the Problem of Uniqueness in General Relativity.” Communications on Pure and Applied Mathematics 58, 445–504. arXiv:gr-qc/0307013.
• Donnelly, William, and Steven B. Giddings. 2016. “Observables, Gravitational Dressing, and Obstructions to Locality and Subsystems.” Physical Review D 94, 104038. DOI. arXiv:1607.01025.