Inside Out: Meet The Operators Inside The Horizon

TL;DR

This work develops a Lorentzian framework for reconstructing bulk operators behind horizons in time-dependent AdS spacetimes by combining HMPS bulk–boundary constraints with geodesic gravitational dressing. It introduces an extremal-geodesic, RT-surface anchored foliation that yields gauge-invariant, state-sensitive bulk operators whose boundary representations can be related through boundary time evolution and modular flow, including resummation across shockwaves. The authors provide explicit AdS$_2$ and AdS$_3$ examples (notably Vaidya-type geometries) where HKLL-like reconstructions, modular flow, and resummation procedures are demonstrated and compared, highlighting the necessity of resumming $O(G_N)$ corrections and the subtleties of background dependence. The results illuminate how entanglement wedge reconstruction can be organized in dynamical settings, with implications for locality, gauge invariance, and the role of the RT surface in encoding behind-horizon physics.

Abstract

Based on the work of Heemskerk, Marolf, Polchinski and Sully (HMPS), we study the reconstruction of operators behind causal horizons in time dependent geometries obtained by acting with shockwaves on pure states or thermal states. These geometries admit a natural basis of gauge invariant operators, namely those geodesically dressed to the boundary along geodesics which emanate from the bifurcate horizon at constant Rindler time. We outline a procedure for obtaining operators behind the causal horizon but inside the entanglement wedge by exploiting the equality between bulk and boundary time evolution, as well as the freedom to consider the operators evolved by distinct Hamiltonians. This requires we carefully keep track of how the operators are gravitationally dressed and that we address issues regarding background dependence. We compare this procedure to reconstruction using modular flow, and illustrate some formal points in simple cases such as AdS$_2$ and AdS$_3$.

Figures (14)

• Figure 1: A bulk local operator expressed using a retarded(left)/spacelike(right) Green's function. The green line denotes the gravitational dressing of the operator. This operator, along with its smearing can be represented in terms of initial data for the matter part (red smearing) which itself is graviationally dressed (blue).
• Figure 2: Depiction of our definition of "same point," $(X_B^t,t)\sim(X_\mathbbmsl{H},\mathbbmsl{t})$. Note that the dressings reach the same point in the old geometry as defined by their destance to the original black hole horizon or RT surface. By backwards evolving using the respective Hamiltonians should give equivalent expressions for the operators as described in equation \ref{['eq:newvsoldv2']}.
• Figure 3: Illustration of step \ref{['step:remove']}, whereby we remove all shocks that do not cross $\Sigma_{t_0}$, the Cauchy surface our operator lies in.
• Figure 4: Illustration of step \ref{['step:move']}, whereby we simplify the operator by evolving it to a Cauchy slice that crosses fewer shocks.
• Figure 5: Operators in Cauchy slices that either cross a shock or do not. If the operator's dressing crosses no shocks---but the full Cauchy slice does---we can not use step \ref{['step:remove']} to remove said shock without first using step \ref{['step:move']} to move the operator to a different Cauchy slice which does not cross the shock.