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Capacity of Entanglement and Replica Backreaction in RST Gravity

Raúl Arias, Daniel Fondevila

Abstract

We compute the capacity of entanglement in two dimensional dilaton gravity in a setting where Hawking radiation, backreaction, and islands can be treated analytically. Our focus is the eternal black hole of the Russo Susskind Thorlacius model coupled to N conformal matter fields. Unlike previous gravitational computations, which were mostly carried out in JT gravity, the RST model forces one to deal with a genuinely dynamical conformal factor and with the global constraints of the replica construction. The main technical step is therefore to solve the replica deformation on the orbifold globally at first order near n=1, including the homogeneous sector fixed by single valuedness and by the requirement of a fixed microcanonical state. For a single interval we obtain a time independent generalized capacity, parallel to the generalized entropy. For two intervals, even in the late time factorization regime, the global solution generates an interaction term between replica fixed points; after Lorentzian continuation this produces a time dependent capacity on the two QES saddle, despite the corresponding entropy plateau. We discuss the regime of validity of the resulting expressions and explain how the large size of the two QES capacity implies a highly non uniform saddle competition near n=1, providing a concrete mechanism for sharp features of the capacity at the Page transition.

Capacity of Entanglement and Replica Backreaction in RST Gravity

Abstract

We compute the capacity of entanglement in two dimensional dilaton gravity in a setting where Hawking radiation, backreaction, and islands can be treated analytically. Our focus is the eternal black hole of the Russo Susskind Thorlacius model coupled to N conformal matter fields. Unlike previous gravitational computations, which were mostly carried out in JT gravity, the RST model forces one to deal with a genuinely dynamical conformal factor and with the global constraints of the replica construction. The main technical step is therefore to solve the replica deformation on the orbifold globally at first order near n=1, including the homogeneous sector fixed by single valuedness and by the requirement of a fixed microcanonical state. For a single interval we obtain a time independent generalized capacity, parallel to the generalized entropy. For two intervals, even in the late time factorization regime, the global solution generates an interaction term between replica fixed points; after Lorentzian continuation this produces a time dependent capacity on the two QES saddle, despite the corresponding entropy plateau. We discuss the regime of validity of the resulting expressions and explain how the large size of the two QES capacity implies a highly non uniform saddle competition near n=1, providing a concrete mechanism for sharp features of the capacity at the Page transition.
Paper Structure (16 sections, 116 equations, 4 figures, 1 table)

This paper contains 16 sections, 116 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review of RST gravity
  3. Eternal black hole solution
  4. Generalized entropy for 1 and 2 intervals
  5. Replica equations and Capacity of entanglement
  6. RST replica equations of motion - 1 interval
  7. RST replica equations of motion - 2 intervals
  8. Capacity of entanglement in Eternal Black Holes
  9. Single-interval capacity (one-QES)
  10. Two intervals capacity: early and late-time regime
  11. Discussion and outlook
  12. The $U(1)$--symmetric ansatz
  13. Deriving $C_{\text{matter}}^{(n)}$ for an $n$ dependent conformal factor
  14. Matter stress tensor and the origin of the time dependence in the two--QES capacity
  15. Saddle competition in the $(n,t)$ plane and the determination of the relevant transition time for the capacity
  16. ...and 1 more sections

Figures (4)

  • Figure 1: Penrose diagram of the eternal RST black hole. The wavy boundaries $\Omega=1/4$ represent the singularity. The dashed lines indicate the horizons. Kruskal type coordinates and right wedge Minkowski coordinates are also shown.
  • Figure 2: Single-interval configuration in the eternal RST black hole. The radiation region is anchored at $P_O$ on the asymptotic boundary (red curve). In the island saddle the generalized entropy (and capacity) is evaluated using a single quantum extremal point $P_Q$, with the exterior and island segment shown in red and the complement segment in blue.
  • Figure 3: Two-interval configuration in the eternal RST black hole. Left: the radiation region is anchored at two boundary points $P_{\bar{O}}, P_O$ (one on each asymptotic boundary), no-island configuration. Right: in the island saddle the entanglement wedge includes a symmetric island bounded by two quantum extremal points $[P_{\bar{Q}},P_Q]$; the generalized entropy is evaluated on the red segments (exterior + island), which gives the same as for the blue segments (assuming global pure state and Haag duality).
  • Figure 4: Qualitative comparison of generalized entropy and generalized capacity for two intervals as a function of $t_O$. Before $t_{\rm Page}$ both follow the same linear branch; after $t_{\rm Page}$ the entropy saturates while the capacity exhibits a jump and grows rapidly.