Order-one coefficient in stretched-horizon entropy-current fluctuations

Determine the order-one numerical coefficient in the fluctuation relation for the boundary entropy-current two-point function S_vv on the stretched horizon of a causal diamond within the Carlip–Solodukhin horizon conformal field theory framework, after regulating the short-distance singularity by the momentum cutoff; specifically, fix the O(1) prefactor multiplying the near-coincident contribution to 〈S_vv(z) S_vv(w)〉 proportional to A/4G_N times (z−w)^{-4} that dominates the integral over the stretched horizon.

Background

The paper adopts the Carlip–Solodukhin ansatz that near-horizon fluctuations are captured by a 1+1 dimensional conformal field theory on the stretched horizon with central charge proportional to the area in Planck units. This yields a specific form for the entropy-current correlator 〈S_vv(z) S_vv(w)〉 scaling as (z−w)^{-4}, with the singular short-distance behavior regulated by a momentum cutoff of the horizon CFT.

In the nested causal diamond setup, only near-coincident points contribute appreciably to integrals over the stretched horizon due to localization and large area growth. The authors note that, in this regime, the precise order-one prefactor in the fluctuation relation cannot be fixed with confidence, creating an explicit uncertainty that affects quantitative predictions of the induced non-local terms in the effective hydrodynamic action.