Geodesics, One Point Functions and Black Hole Perturbations
Parijat Dey, Arundhati Goldar, Nirmalya Kajuri
TL;DR
The authors probe the robustness of the holographic relation between a thermal boundary one-point function and the boundary-to-horizon geodesic length in Euclidean BTZ against infinitesimal radial metric perturbations. Using a first-order perturbation of $g_{rr}$, they compute the induced change in the bulk-boundary propagator via WKB and verify it against exact expressions, showing the leading effect is captured by $-m\mathcal{K}_0(r)\delta\ell(r)$. They then relate this to the boundary observable by saddle-point evaluation, finding $\delta\langle\mathcal{O}\rangle \sim e^{-m\ell_{\text{hor}}}\delta\ell_{\text{hor}}$ up to calculable prefactors, thereby yielding $\langle\mathcal{O}\rangle \to e^{-m(\ell_{\text{hor}}+\delta\ell_{\text{hor}})}$ to leading order. The work confirms that the geodesic-based exponential mapping remains valid for small radial deformations, reinforcing the picture that heavy boundary operators faithfully probe bulk geometry even away from exact backgrounds, and suggests extensions to Lorentzian setups and time-dependent perturbations.
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.