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Microcanonical Path Integrals and the Holography of small Black Hole Interiors

Donald Marolf

TL;DR

This paper develops a Lorentzian microcanonical path integral, inspired by Brown and York, to extend the AdS/CFT dictionary to the interiors of small AdS black holes. By fixing energy with a narrow window around $E_0$, the authors show that bulk saddles maximize a generalized entropy ${\cal S}= (\beta+it)E - I_{bulk}$, and that the dominant geometry is the entropically largest two-sided black hole (or its HRT counterpart) at energy $E_0$. The results imply that microcanonical thermofield-double states correspond to Kruskal-like bulk wormholes for which RT/HRT entropies govern the entanglement structure, while sufficiently small energy widths introduce bulk quantum fluctuations that can require superpositions of geometries rather than a single semiclassical spacetime. Altogether, the work broadens the holographic map to include interiors of small black holes and highlights the role of energy-constrained saddles in determining bulk topology and entanglement, with implications for quantum gravity and information-processing tasks in holographic theories.

Abstract

We use a microcanonical path integral closely related to that introduced by Brown and York in 1992 to add new entries to the AdS/CFT dictionary concerning the interiors of small black holes. Stationary points of such path integrals are also stationary points of more standard canonical-type path integrals with fixed boundary metric, but the condition for dominance is now maximizing Hubeny-Rangamani-Takayanagi entropy at fixed energy. As a result, such path integrals can bring to the fore saddles that fail to dominate in more familiar contexts. We use this feature to argue that the standard Kruskal-like two-sided extension of small AdS black holes with energy $E_0$ is dual to a microcanonical version of the thermofield double state for AdS black holes that maximize the microcanonical bulk entropy at this energy. We also comment on entanglement in such states and on quantum effects that become large when the energy-width of the microcanonical ensemble is sufficiently small.

Microcanonical Path Integrals and the Holography of small Black Hole Interiors

TL;DR

This paper develops a Lorentzian microcanonical path integral, inspired by Brown and York, to extend the AdS/CFT dictionary to the interiors of small AdS black holes. By fixing energy with a narrow window around , the authors show that bulk saddles maximize a generalized entropy , and that the dominant geometry is the entropically largest two-sided black hole (or its HRT counterpart) at energy . The results imply that microcanonical thermofield-double states correspond to Kruskal-like bulk wormholes for which RT/HRT entropies govern the entanglement structure, while sufficiently small energy widths introduce bulk quantum fluctuations that can require superpositions of geometries rather than a single semiclassical spacetime. Altogether, the work broadens the holographic map to include interiors of small black holes and highlights the role of energy-constrained saddles in determining bulk topology and entanglement, with implications for quantum gravity and information-processing tasks in holographic theories.

Abstract

We use a microcanonical path integral closely related to that introduced by Brown and York in 1992 to add new entries to the AdS/CFT dictionary concerning the interiors of small black holes. Stationary points of such path integrals are also stationary points of more standard canonical-type path integrals with fixed boundary metric, but the condition for dominance is now maximizing Hubeny-Rangamani-Takayanagi entropy at fixed energy. As a result, such path integrals can bring to the fore saddles that fail to dominate in more familiar contexts. We use this feature to argue that the standard Kruskal-like two-sided extension of small AdS black holes with energy is dual to a microcanonical version of the thermofield double state for AdS black holes that maximize the microcanonical bulk entropy at this energy. We also comment on entanglement in such states and on quantum effects that become large when the energy-width of the microcanonical ensemble is sufficiently small.

Paper Structure

This paper contains 9 sections, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Microcanonical Path Integrals
  3. Bulk microcanonical path integral and dominant saddles
  4. More general boundaries
  5. Physics in the Microcanonical Bulk
  6. Entanglement in the microcanonical ensemble
  7. Finite Width and Bulk Quantum Corrections
  8. Discussion
  9. Building horizon-free bulk states from the vacuum

Figures (1)

  • Figure 1: A rough conformal diagram of a 2-sided AdS-Schwarzschild black hole. The vertical (black) lines are the two disconnected components of the asymptotically-AdS conformal boundary. The heavy horizontal (red) lines are the singularity. In an exact conformal diagram these lines would not meet the AdS boundaries orthogonally (see explicit discussion in Fidkowski:2003nf and implicit discussion surrounding figure 7 of Klosch:1995qv). The diagonal (blue) lines are the horizons. The associated wormhole is almost traversable in the sense that the past event horizon of one boundary coincides with the future event horizon of the other, so that an infinitesimal perturbation of the geometry (violating the averaged null energy condition as in Gao:2016bin) could render the wormhole traversable.