Geodesics, One Point Functions and Black Hole Perturbations
arXiv:2601.09397
The paper investigates the relationship between thermal one‑point functions in holographic black holes and bulk geodesic lengths, and tests its stability under small horizon‑preserving perturbations of an Euclidean BTZ black hole using WKB and saddle‑point techniques.
Abstract
Holographic black holes exhibit a striking relation between thermal boundary one-point functions and bulk geodesic lengths. In the large conformal-dimension limit, the one-point function of a primary operator is given by the exponential of the geodesic length from its boundary insertion point to the horizon. We test the robustness of this relation under perturbations by considering a class of deformations of an Euclidean BTZ black hole and working to first order in the perturbation.We find that, at leading order in the large conformal-dimension limit and to first order in the radial horizon-preserving perturbation, the logarithmic variation of the one-point function is governed by the variation of the renormalized boundary-to-horizon geodesic length. The result is established using WKB and saddle-point methods, and WKB expressions at large conformal dimension are checked against the exact Green function and bulk-boundary propagator.
20 pages, 1 figure. Revised version. Accepted for publication in Physical Review D