Stochastic Description of Near-Horizon Fluctuations in Rindler-AdS

We study quantum spacetime fluctuations near light-sheet horizons associated with a Rindler wedge in 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 shockwaves 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 $G_N^{(d)}$ and the AdS curvature $L$, in agreement with previous literature utilizing different methods.