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Complementarity Is Not Enough

Raphael Bousso

TL;DR

This paper analyzes the firewall paradox, arguing that causal patch complementarity cannot resolve the conflicting entanglement requirements of near-horizon modes, interior degrees of freedom, and early Hawking radiation. It outlines three non-mutually-consistent postulates—unitarity, no drama for infalling observers, and EFT validity outside the stretched horizon—and examines three potential resolutions: information preservation for outside observers, firewalls, or a breakdown of EFT. It then explores two EFT-breakdown strategies: identifying the interior with a subset of early radiation and the Horowitz–Maldacena final-state proposal, highlighting the conceptual and technical challenges each faces. The work emphasizes the need for radical new physics or refined consistency conditions to reconcile unitarity with the equivalence principle in black hole spacetimes.

Abstract

The near-horizon field B of an old black hole is maximally entangled with the early Hawking radiation R, by unitarity of the S-matrix. But B must be maximally entangled with the black hole interior A, by the equivalence principle. Causal patch complementarity fails to reconcile these conflicting requirements. The system B can be probed by a freely falling observer while there is still time to turn around and remain outside the black hole. Therefore, the entangled state of the BR system is dictated by unitarity even in the infalling patch. If, by monogamy of entanglement, B is not entangled with A, the horizon is replaced by a singularity or "firewall". To illustrate the radical nature of the ideas that are needed, I briefly discuss two approaches for avoiding a firewall: the identification of A with a subsystem of R; and a combination of patch complementarity with the Horowitz-Maldacena final-state proposal.

Complementarity Is Not Enough

TL;DR

This paper analyzes the firewall paradox, arguing that causal patch complementarity cannot resolve the conflicting entanglement requirements of near-horizon modes, interior degrees of freedom, and early Hawking radiation. It outlines three non-mutually-consistent postulates—unitarity, no drama for infalling observers, and EFT validity outside the stretched horizon—and examines three potential resolutions: information preservation for outside observers, firewalls, or a breakdown of EFT. It then explores two EFT-breakdown strategies: identifying the interior with a subset of early radiation and the Horowitz–Maldacena final-state proposal, highlighting the conceptual and technical challenges each faces. The work emphasizes the need for radical new physics or refined consistency conditions to reconcile unitarity with the equivalence principle in black hole spacetimes.

Abstract

The near-horizon field B of an old black hole is maximally entangled with the early Hawking radiation R, by unitarity of the S-matrix. But B must be maximally entangled with the black hole interior A, by the equivalence principle. Causal patch complementarity fails to reconcile these conflicting requirements. The system B can be probed by a freely falling observer while there is still time to turn around and remain outside the black hole. Therefore, the entangled state of the BR system is dictated by unitarity even in the infalling patch. If, by monogamy of entanglement, B is not entangled with A, the horizon is replaced by a singularity or "firewall". To illustrate the radical nature of the ideas that are needed, I briefly discuss two approaches for avoiding a firewall: the identification of A with a subsystem of R; and a combination of patch complementarity with the Horowitz-Maldacena final-state proposal.

Paper Structure

This paper contains 8 sections, 10 equations, 1 figure.

Table of Contents

  1. The Firewall Paradox
  2. Why Complementarity is Not Enough
  3. Three Choices
  4. Information Loss for the Outside Observer
  5. Firewalls
  6. Breakdown of Effective Quantum Field Theory
  7. Identification of Hilbert Spaces
  8. Final State Quantum Mechanics

Figures (1)

  • Figure 1: An example of the Horowitz-Maldacena final-state proposal as applied to an observer (shaded causal patch) who enters the black hole at the Page time. Entanglement is denoted by semicircles. By the infall time, the initial state has evolves into an entangled state of the Hawking radiation $R$ and the half-evaporated black hole before $X_1$. The initial state also imposes the infalling vacuum, so $A$ and $B$ are maximally entangled. If the final state maximally entangles $X_1$ with $A$, then the quantum information in $X_1$ is teleported to $B$, and $BR$ will be in an entangled pure state.