Real-time gravitational replicas: formalism and a variational principle

Abstract

This work is the first step in a two-part investigation of real-time replica wormholes. Here we study the associated real-time gravitational path integral and construct the variational principle that will define its saddle-points. We also describe the general structure of the resulting real-time replica wormhole saddles, setting the stage for construction of explicit examples. These saddles necessarily involve complex metrics, and thus are accessed by deforming the original real contour of integration. However, the construction of these saddles need not rely on analytic continuation, and our formulation can be used even in the presence of non-analytic boundary-sources. Furthermore, at least for replica- and CPT-symmetric saddles we show that the metrics may be taken to be real in regions spacelike separated from a so-called ‘splitting surface’. This feature is an important hallmark of unitarity in a field theory dual.

Real-time gravitational replicas: low dimensional examples

“…where we have forward evolution for the 'kets' (k) and backward evolution for the 'bras' (b), resulting in the relative sign above. As argued in [1] (and earlier in [15,17]), the on-shell action I n in the Lorentzian context is real, and is given by…”

“…In addition, connections to the black hole information problem and baby universes have also been discussed recently in [17]. Our goal in this current paper is to exemplify the formal discussion in [1] with some concrete examples. For technical reasons our examples will rely on gravitational dynamics in low dimensions, especially in 2 and 3 spacetime dimensions, where one can write down explicit geometries that provide the appropriate stationary points.…”

“…Motivated by these developments, and by earlier efforts [15] to derive the covariant holographic entanglement entropy prescription of [16], in a companion paper [1] we outlined the general formalism for understanding the stationary phase approximation of the real-time gravitational functional integral. In addition, connections to the black hole information problem and baby universes have also been discussed recently in [17].…”

“…This boundary geometry provides the asymptotic boundary conditions for our AdS gravity dual, which we seek to determine. In what follows we will adhere to the terminology of [1] referring to the JHEP08(2021)171…”

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We continue the study of real-time replica wormholes initiated in [1]. Previously, we had discussed the general principles and had outlined a variational principle for obtaining stationary points of the real-time gravitational path integral.

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