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Behind the geon horizon

Monica Guica, Simon F. Ross

TL;DR

This work analyzes the Papadodimas–Raju interior-reconstruction program in the explicit RP$^2$ geon, a simple one-sided AdS$_3$ black hole dual to a pure CFT state. It shows that, for late-time local observables, the PR mirror operator is a simple, state-dependent, early-time operator: $ ilde{\mathcal{O}}_g(t,\phi)=\mathcal{O}(-t,\phi+\pi)$, thereby providing a transparent interior picture in this setting. The geon state is constructed via a Euclidean path integral with a cross-cap, revealing maximal left-right entanglement and a Cardy-like high-energy spectrum $d_C(E) \sim e^{\pi\sqrt{cE/3}}$, with late-time thermality matching BTZ while early-time structure remains non-thermal. The paper also explores deformations of the geon state, including a special unitary rotation to compare with the $J$ quotient and infinitesimal mode rotations that smear the mirror, illustrating how interior geometry and mirror locality respond to state changes and offering concrete tests for the PR framework in a controlled holographic example.

Abstract

We explore the Papadodimas-Raju prescription for reconstructing the region behind the horizon of one-sided black holes in AdS/CFT in the case of the RP^2 geon - a simple, analytic example of a single-sided, asymptotically AdS_3 black hole, which corresponds to a pure CFT state that thermalises at late times. We show that in this specific example, the mirror operators involved in the reconstruction of the interior have a particularly simple form: the mirror of a single trace operator at late times is just the corresponding single trace operator at early times. We use some explicit examples to explore how changes in the state modify the geometry inside the horizon.

Behind the geon horizon

TL;DR

This work analyzes the Papadodimas–Raju interior-reconstruction program in the explicit RP geon, a simple one-sided AdS black hole dual to a pure CFT state. It shows that, for late-time local observables, the PR mirror operator is a simple, state-dependent, early-time operator: , thereby providing a transparent interior picture in this setting. The geon state is constructed via a Euclidean path integral with a cross-cap, revealing maximal left-right entanglement and a Cardy-like high-energy spectrum , with late-time thermality matching BTZ while early-time structure remains non-thermal. The paper also explores deformations of the geon state, including a special unitary rotation to compare with the quotient and infinitesimal mode rotations that smear the mirror, illustrating how interior geometry and mirror locality respond to state changes and offering concrete tests for the PR framework in a controlled holographic example.

Abstract

We explore the Papadodimas-Raju prescription for reconstructing the region behind the horizon of one-sided black holes in AdS/CFT in the case of the RP^2 geon - a simple, analytic example of a single-sided, asymptotically AdS_3 black hole, which corresponds to a pure CFT state that thermalises at late times. We show that in this specific example, the mirror operators involved in the reconstruction of the interior have a particularly simple form: the mirror of a single trace operator at late times is just the corresponding single trace operator at early times. We use some explicit examples to explore how changes in the state modify the geometry inside the horizon.

Paper Structure

This paper contains 18 sections, 92 equations, 10 figures.

Table of Contents

  1. Introduction and summary
  2. The $\mathbb{R}\mathrm{P}^2$ geon geometry
  3. Definition
  4. Thermality of the $\mathbb{R}\mathrm{P}^2$ geon
  5. Path integral construction of the geon state
  6. Entanglement structure
  7. Support of the geon state at high energies
  8. Other boundary states
  9. Mirror operators in the $\mathbb{R}\mathrm{P}^2$ geon geometry
  10. Local and smeared bulk operators in BTZ
  11. Local and smeared bulk operators in the geon geometry
  12. Identification of the mirror operators
  13. Modifications of the geon state
  14. A special unitary rotation
  15. Infinitesimal mode rotations
  16. ...and 3 more sections

Figures (10)

  • Figure 1: The Penrose diagram for the eternal BTZ spacetime, indicating the direction of $\chi = \partial_t$ in each region.
  • Figure 2: The construction of the geon spacetime as a quotient of BTZ. There is a time-translation symmetry that acts only in the exterior region, which is indicated by the arrow.
  • Figure 3: The geodesics contributing to the two-point function of two operators on the same boundary (red), and for operators on different boundaries (blue) in BTZ.
  • Figure 4: Contributing bulk diagrams to the boundary three-point function. The interaction vertex is to be integrated only in the region outside the horizon.
  • Figure 5: The Klein bottle can be thought of as the quotient of a rectangular torus by the $\mathbb Z_2$ action $(\tau, \phi) \sim (-\tau, \phi+\pi)$. There are two natural fundamental regions for this identifications. In (a), we have a representation as the propagation between two cross-caps. In (b), we have the alternative representation with an orientation-reversing identification of the two sides.
  • ...and 5 more figures