Stochastic description of near-horizon fluctuations in Rindler-AdS

We study quantum spacetime fluctuations near light-sheet horizons associated with a Rindler wedge in anti-de Sitter (AdS) spacetime, in the context of AdS=CFT. In particular, we solve the vacuum Einstein equation near the light-sheet horizon, augmented with the ansatz of a quantum source smeared out in a Planckian width along one of the light-cone directions. Such a source, whose physical interpretation is of gravitational shock waves created by vacuum energy fluctuations, alters the Einstein equation to a stochastic partial differential equation taking the form of a Langevin equation. By integrating fluctuations along the light sheet, we find an accumulated effect in the round-trip time of a photon to traverse the horizon of the Rindler wedge that depends on both the d-dimensional Newton constant God thorn N and the AdS curvature L, in agreement with previous literature utilizing different methods.