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

TL;DR

This work provides a gauge-invariant, unitary description of scalar field dynamics across the horizon of a two-sided AdS black hole by formulating the bulk theory in the maximal slicing gauge and expressing bulk operators through boundary CFTs on both AdS boundaries. The authors construct a time-dependent Hermitian Hamiltonian $\hat{H}(\bar{t})$ acting on the Hartle–Hawking state $|\mathrm{HH}\rangle$ and show that horizon-crossing excitations evolve unitarily, with a bulk reconstruction kernel relating bulk operators to dual CFT operators on the two boundaries. A crucial technical step is smoothing Hartle–Hawking modes via Hermite-smearing to obtain horizon-differentiable bulk modes $\hat{g}_{nq}$, enabling a well-defined quadratic Hamiltonian and a Bogoliubov-diagonalizable evolution that remains normalizable. The paper also derives a boundary-relevant order parameter signaling horizon crossing and computes bulk Wightman functions, recovering exterior Hawking thermodynamics only after relating wormhole coordinates to BTZ coordinates, while revealing unitary interior correlations in the exact (large-$N$) limit. Overall, the results illuminate how horizon-crossing physics can be described unitarily within the bulk effective field theory and encoded in the dual two-boundary CFT data, with clear implications for the black hole information issue in AdS/CFT contexts.

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.

Figures (13)

• Figure 1: The Kruskal (left) and Penrose (right) diagrams for the two-sided BTZ black hole solution. The two solid black lines are the asymptotic AdS boundaries, the diagonal gray lines are the event horizons and the wiggly red lines are the singularities. The horizons divide the two-sided black hole into four regions labelled I, II, F and P, each of which have a static Schwarzschild-like coordinate system which is displayed in the Penrose diagram.
• Figure 2: The region $R_0$ (shaded gray) covered by the maximal slicing solution is demarcated by the orange lines in the BTZ Kruskal diagram (left) and Penrose diagram right) of the fully extended BTZ black hole. The three regions $R_0$, $R_1$ and $R_{-1}$ cover the full Penrose diagram. The regions $R_1$ and $R_{-1}$ (unshaded) are covered by a different family of maximal slices as is explained in the main text.
• Figure 3: Plots of constant $\bar{t} = 0, \pm 0.5, \pm 1,\pm 1.5, \pm 2$ slices, i.e., maximal slices (solid blue lines) in the BTZ Kruskal diagram (left) and Penrose diagram (right) of the fully extended BTZ black hole. As $\bar{t}\to\infty$, the final slice approaches $R_\infty = \ell\sqrt{M/4\pi}$ (orange curve) from below.
• Figure 4: The action of the large diffeomorphism corresponding $c = 1$, $\tilde{c} = 0$, which translations points on the right boundary but fixes points on the left boundary. The blue curves are the slices of the symmetric foliation and the green curves attached to the blue curves on the left boundary are their images under the large diffeomorphism.
• Figure 5: The function \ref{['w1-ac']} for timelike separated insertions is plotted as a function of time $\bar{t}$ with $x'=0.5 R_h$ held fixed. The imaginary part (blue) has the UV divergence at small times but decays at large times. On the other hand the real part (orange) saturates to a finite non-zero value at late times, showing that information is not lost even as $\bar{t}\to\infty$. The two insertions are also shown schematically in the Kruskal-Szekeres diagram in the top figure.