Observer complementarity for black holes and holography

TL;DR

The paper argues that a form of black hole complementarity remains essential and can be realized by an observer-dependent formalism grounded in a quantum-to-classical rule. It develops an emergent-spacetime framework with non-isometric holographic codes and applies it to both the Antonini-Sasieta-Swingle-Rath geometry and evaporating black holes, deriving explicit observer-dependent bulk interpretations and information-theoretic task costs. By analyzing encoding maps, entanglement structures, and distillation/reconstruction complexities, it shows how interior spacetime can emerge consistently across scenarios while preserving unitarity and aligning with semiclassical expectations where appropriate. The resulting general principles offer a scalable, observer-aware approach to holographic emergence in closed universes and black hole evaporation, highlighting the roles of observer entropy and backreaction.

Abstract

We present a mathematical formulation of black hole complementarity based on recent rules for including the observer in quantum cosmology. We argue that this provides a self-consistent treatment of the interior of an evaporating black hole throughout its history, as well as the Antonini-Sasieta-Swingle-Rath configuration where a closed universe is entangled with a pair of AdS universes.

Figures (8)

• Figure 1: A holographic code $V:\mathcal{H}_\ell\otimes \mathcal{H}_r\otimes \mathcal{H}_R\to \mathcal{H}_B\otimes \mathcal{H}_R$ for a partially evaporated black hole on a "nice" time slice, shaded purple. The left and right interior modes $\ell$ and $r$ are acted upon with a unitary $U$, with some fixed auxiliary state $|\psi_0\rangle_f$ included to restrict the set of possible states. The unitary is followed by a projection $\langle 0|_P$ on a subset of the degrees of freedom and a rescaling by $\sqrt{|P|}$, where $|P|={\rm dim}\mathcal{H}_P$. The remaining system $B$ is the microstate Hilbert space of the black hole. The encoding map acts trivially outside of the black hole, e.g. on the Hawking radiation $R$.
• Figure 2: A holographic code for a completely evaporated black hole, acting on a state created from an infalling shell (orange line). $|\chi_{Hawk}\rangle$ is Hawking's state representing entanglement between interior and exterior outgoing modes (shaded light blue and red). The baby universe part of the slice, shaded green, is subject to a rank-one projection, so the circuit on the right sends pure states of $\ell$ to pure states of $R$.
• Figure 3: A holographic encoding map for the ASSR geometry. Left: the geometry. The shaded region is the Euclidean preparation, the solid lines are asymptotic AdS boundaries, and the dotted lines are surfaces where the suppressed sphere shrinks to zero size. The long blue line is the matter shell created by the boundary operator $\mathcal{O}$ that sources the closed universe, for which we have indicated a Cauchy slice in green. The purple slices are Cauchy slices of the two AdS regions. Effective degrees of freedom in the AdS regions are denoted $a$, those in the closed universe are denoted $b$, and the boundary CFT degrees of freedom are denoted $A$. The map is a rank-one projection on $b$ and the HKLL isometry on $a$, which maps the entangled state $|\psi_1\rangle_{ab}$ to a pure state of $A$.
• Figure 4: Observer codes for the ASSR geometry (suppressing $|\psi_0\rangle_f$). Cloning the observer out of the system to implement the quantum-to-classical channel creates an entangled state $|\omega\rangle$ of the observer, their environment, and their clone.
• Figure 5: Computing the expectation value of the encoded swap operator $\widetilde{\mathcal{S}}_{AdS}$ in the $\beta$ code.