Thermal one point functions, large $d$ and interior geometry of black holes

TL;DR

This paper investigates thermal one point functions of massive scalars in AdS black holes, revealing their exponentiation properties and how they encode interior black hole geometry, especially in large dimension limits and with various couplings.

Contribution

It demonstrates the exponentiation of one point functions in large d limits and shows they encode detailed interior black hole geometry, including horizons and singularities.

Findings

One point functions exponentiate in large d limit.

They encode proper time between horizons and proper length to singularity.

Gauss-Bonnet coupling affects one point functions as predicted by theory.

Abstract

We study thermal one point functions of massive scalars in $AdS_{d+1}$ black holes. These are induced by coupling the scalar to either the Weyl tensor squared or the Gauss-Bonnet term. Grinberg and Maldacena argued that the one point functions sourced by the Weyl tensor exponentiate in the limit of large scalar masses and they contain information of the black hole geometry behind the horizon. We observe that the one point functions behave identically in this limit for either of the couplings mentioned earlier. We show that in an appropriate large $d$ limit, the one point function for the charged black hole in $AdS_{d+1}$ can be obtained exactly. These black holes in general contain an inner horizon. We show that the one point function exponentiates and contains the information of both the proper time between the outer horizon to the inner horizon as well as the proper length from the inner horizon to the singularity. We also show that Gauss-Bonnet coupling induced one point functions in $AdS_{d+1}$ black holes with hyperbolic horizons behave as anticipated by Grinberg-Maldacena. Finally, we study the one point functions in the background of rotating BTZ black holes induced by the cubic coupling of scalars.