Geodesics, One Point Functions and Black Hole Perturbations

TL;DR

This paper demonstrates that the relation between boundary one-point functions and bulk geodesic lengths in holographic black holes remains valid under first-order radial perturbations, confirming the robustness of this holographic correspondence.

Contribution

It shows that the exponential relation between one-point functions and geodesic lengths persists under arbitrary radial deformations of Euclidean BTZ black holes at first order.

Findings

The one-point function remains governed by the modified geodesic length under perturbations.

The relation is validated using WKB and saddle-point methods.

Exact analysis justifies the WKB approximation.

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 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.