Complementarity, Firewalls, Remnants, and Final-State Proposals

Complementarity, firewalls, remnants, and final-state boundary conditions evade the information conflict by modifying different assumptions. None is specified by its name alone: each needs an operator algebra, an evolution or endpoint rule, and tests for signaling, cloning, uncontrolled state density, or postselection-induced nonlinearity.

Required background. The Information Problem: Assumptions and Observables supplies the assumption set. Locality, Code Distance, and Causal Constraints supplies the consistency tests for overlapping descriptions.

Helpful background. Causal, Killing, Trapping, and Apparent Horizons: The QFT Interface fixes the geometry. QEC Evidence, Current Disputes, and Status supplies the modern code-subspace comparison.

The old-black-hole entanglement conflict

Let $R$ be early radiation, $B$ a late outgoing mode, and $A$ its interior partner. A smooth horizon requires $AB$ to be nearly pure, so $S_{AB}\simeq0$ and $S_A\simeq S_B$. Unitary evaporation after the Page time requires the late mode to reduce the radiation entropy, $S_{BR}<S_R$.

Weak monotonicity,

$$S_{AB}+S_{BR}\ge S_A+S_R,$$

then gives a contradiction: the left side is smaller than $S_B+S_R$, while the right side is approximately $S_B+S_R$. This is the AMPS formulation of the conflict, conditional on $A$, $B$, and $R$ being ordinary compatible subsystems and on the semiclassical horizon state Almheiri et al. 2013.

Four modifications and their costs

Proposal	Assumption modified	Required construction	Principal danger
Complementarity	Independent simultaneous $A$ and exterior descriptions	An observer-dependent or code-subspace algebra with no operational cloning	Contradictory measurements or hidden acausal signaling
Firewall	Smooth infalling vacuum at an old horizon	Order-one horizon-scale dynamics producing the altered state	Conflict with low-energy effective theory and equivalence expectations
Remnant	Complete evaporation into a finite exterior state space	Stable endpoint with a finite production law and enough internal states	Infinite species, uncontrolled pair production, or inaccessible information
Final state	Ordinary initial-value quantum mechanics	A specified postselected interior boundary condition	Nonlinear evolution, fine tuning, or signaling

Black-hole complementarity originally proposed that no single observer can compare duplicated descriptions Susskind, Thorlacius, and Uglum 1993. A modern code interpretation can make overlapping representatives consistent only within a declared code subspace and error tolerance; it does not permit two independent copies of the logical operator.

Horowitz and Maldacena proposed a maximally entangled final boundary condition at the singularity that teleports information into outgoing radiation Horowitz and Maldacena 2004. Generic interactions spoil exact unitarity unless the final state is correlated with them; postselection also demands careful causality analysis Gottesman and Preskill 2004.

Application: the common old-black-hole experiment

Test every proposal on the same setup: prepare an old black hole entangled with accessible $R$, let an infaller measure near-horizon correlations, and later decode $B$ from radiation if the proposal permits it.

• Complementarity must show that the decoded representative and the infaller’s $A$ cannot be jointly used to signal or verify cloning.
• A firewall predicts a detectable failure of the local vacuum correlations and must locate its onset.
• A remnant must give a finite state count and production amplitude compatible with the energy remaining.
• A final-state model must give a linear, normalized map for arbitrary entangled inputs, not only for a tuned product state.

Holding the asymptotic algebra and entropy convention fixed makes the differing physical predictions visible.

Adversarial consistency conditions

Enlarge the code subspace, couple the black hole to an external reference, and allow generic interior interactions. A proposal that works only for one known state is state preparation, not a universal evaporation rule. Check energy and global charges, positivity of probabilities, linearity on superpositions, and whether an observer can arrange a closed signaling loop. For remnants, sum production over internal species rather than examining one state.

Scope and handoff

These proposals are logical response classes, not established outcomes of black-hole evaporation. Island calculations modify the entropy accounting through a gravitational replica prescription and should be evaluated separately, beginning with Page Curves and Entropy Bookkeeping.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

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

References

• Almheiri, A., D. Marolf, J. Polchinski, and J. Sully. “Black Holes: Complementarity or Firewalls?” Journal of High Energy Physics 2013, 2 (2013): 062. DOI.
• Gottesman, D., and J. Preskill. “Comment on ‘The Black Hole Final State’.” Journal of High Energy Physics 2004, 3 (2004): 026. DOI.
• Horowitz, G. T., and J. Maldacena. “The Black Hole Final State.” Journal of High Energy Physics 2004, 2 (2004): 008. DOI.
• Susskind, L., L. Thorlacius, and J. Uglum. “The Stretched Horizon and Black Hole Complementarity.” Physical Review D 48 (1993): 3743–3761. DOI.