Gauge invariant Hamiltonian evolution across the black hole horizon in asymptotically AdS spacetimes
Phys. Rev. D 113, 106019 – Published 18 May, 2026
DOI: https://doi.org/10.1103/gz2s-mqk5
Abstract
We study the quantum dynamics of a probe scalar field in the background of a black hole in asymptotically AdS 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, within effective field theory, 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 and 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 Bañados-Teitelboim-Zanelli black hole. We recover Hawking’s thermodynamic results in the exterior region when expressed in terms of Bañados-Teitelboim-Zanelli coordinates that are related to the wormhole coordinates by a transformation which is singular at the horizon. We compute the two-point function with one insertion in the future or past interior and the other in the exterior. Both are related by a time reflection symmetry and asymptote to a nonzero constant as the coordinate time between the two points becomes large.
