Cauchy Slice Holography and the Semiclassical Approximation

Abstract

We investigate a new approach to holography in asymptotically AdS spacetimes, in which time rather than space is the emergent dimension. By making a sufficiently large $T^2$-deformation of a Euclidean CFT, we define a holographic theory that lives on Cauchy slices of the Lorentzian bulk. (More generally, for an arbitrary Hamiltonian constraint equation that closes, we show how to obtain it by an irrelevant deformation from a CFT with suitable anomalies.) The partition function of this theory defines a natural map between the bulk canonical quantum gravity theory Hilbert space, and the Hilbert space of the usual (undeformed) boundary CFT. We argue for the equivalence of the ADM and CFT Hamiltonians. We also explain how bulk unitarity emerges naturally, even though the boundary theory is not reflection-positive. This allows us to reformulate the holographic principle in the language of Wheeler-DeWitt canonical quantum gravity. Along the way, we outline a procedure for obtaining a bulk Hilbert space from the gravitational path integral with Dirichlet boundary conditions. Following previous conjectures, we postulate that this finite-cutoff gravitational path integral agrees with the $T^2$-deformed theory living on an arbitrary boundary manifold---at least near the semiclassical regime. However, the $T^2$-deformed theory may be easier to UV complete, in which case it would be natural to take it as the definition of nonperturbative quantum gravity. We then explore the semiclassical approximation to canonical quantum gravity and how a classical background emerges from the Wheeler-DeWitt (WDW) states. By employing the Wigner functional analysis, we derive the backreacted Einstein-Hamilton-Jacobi equation as an approximation to the WDW equation, along with the requisite validity conditions. We then apply this understanding to both AdS/CFT and dS/CFT correspondences, to explain how the bulk is encoded in the correlation functions of the $T^2$-deformed theory. We then explain an appropriate description for scenarios in which gravity behaves quantum mechanically in certain regions of spacetime and explain its relation to subregion holography. We derive the validity conditions for gravity to be semiclassical near any co-dimension 1 time-like surface and employ these conditions to explore the black hole information paradox. Our analysis suggests that for evaporating black holes, there might be a violation of semiclassical gravity in the near-horizon region close to the Page time, although this is contingent upon certain assumptions. This also provides insights into the fate of information trapped within evaporating black holes. We then explore this issue from the perspectives of both external and infalling observers. We then explain how to employ this new approach to study the retrieval of information from evaporating black holes, presenting a comprehensive approach to tackle this complex issue in quantum gravity.