Geodesics, One Point Functions and Black Hole Perturbations

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 an arbitrary radial deformation of an Euclidean BTZ black hole and working to first order in the perturbation. We find that the relation remains robust: the corrected one-point function at large conformal dimension is still governed by an exponent proportional to the modified boundary-to-horizon geodesic length. The result is established using WKB and saddle-point methods, with the validity of the WKB approximation justified by exact analyses.