A gauge invariant Hamiltonian evolution across the black hole horizon in asymptotically AdS spacetimes

TL;DR

This paper develops a gauge-invariant Hamiltonian framework for analyzing quantum scalar fields across black hole horizons in asymptotically AdS spacetimes, enabling a unitary description of horizon crossing phenomena.

Contribution

It introduces a well-defined, time-dependent Hamiltonian in the product space of dual CFTs that describes scalar field evolution across the horizon in a gauge-invariant manner.

Findings

Derived a Hamiltonian that acts on the Hartle-Hawking state.

Reconstructed bulk operators signaling horizon crossing.

Calculated two-point functions consistent with Hawking radiation.

Abstract

We study the quantum dynamics of a probe scalar field in the background of a black hole in AAdS spacetime in the Hamiltonian formulation of general relativity in the maximal slicing gauge. The black hole solution in this gauge is expressed in terms of wormhole coordinates, a smooth coordinate system with constant time slices that cut across the horizon, and asymptote to the Killing time slices at the boundaries. The quantum scalar field is expanded in terms of normalized solutions of the Klein-Gordon equation, that are valid at all points in spacetime. The operators that appear in the expansion are in the product space of the CFTs on the two spacetime boundaries, which are by definition gauge invariant under small bulk diffeomorphisms. The entangled Hartle-Hawking (HH) state arises naturally from this construction. One of our main results is a well defined formula for the time dependent Hermitian Hamiltonian of the probe scalar in the product space of the two CFTs, which describes the time development of operators/states along the maximal slices. This Hamiltonian acting on the HH state creates a state of finite norm. Consequently there is a unitary description of horizon crossing scalar field excitations on top of the HH state. We also present a bulk reconstruction formula that evaluates an order parameter that signals horizon crossing in the boundary theory. We calculate various bulk Wightman two-point functions on the two-sided BTZ black hole. We recover Hawking's thermodynamic results in the exterior region when expressed in terms of BTZ coordinates that are related to the wormhole coordinates by a singular transformation. We compute the two-point function with one insertion in the future/past interior and the other in the exterior. Both are related by a time reflection symmetry and asymptote to a non-zero constant as the coordinate time between the two points becomes large.