# Study of Semiclassical Instability of the Schwarzschild AdS Black Hole in the Large $D$ Limit

## TL;DR

This paper investigates the semiclassical stability of Schwarzschild AdS black holes in the large D limit, identifying conditions for stability and instability through analytical methods and eigenvalue analysis.

## Contribution

It provides a simplified form of scalar perturbation equations and demonstrates stability for some classes while identifying unstable modes for small black holes in the large D limit.

## Key findings

- Stable scalar perturbations for certain classes via S-deformation method
- Unstable mode identified for small black holes at large D
- Eigenvalue expression matches previous numerical bounds

## Abstract

We analyze the semiclassical stability of the Schwarzschild AdS black hole in the Euclidean partition function approach. We perform this computation in the large $D$ limit and focus on scalar perturbations. We obtain the equations for non-spherically symmetric scalar perturbations in a simple form. For a class of perturbations stability is demonstrated by the S-deformation method. For some other classes we rule out unstable modes of $\mathcal{O}(D^2)$. We also analyze the spherically symmetric perturbations and demonstrate the appearance of an unstable mode for small black holes in the large $D$ limit. We obtain an expression for the eigenvalue corresponding to the unstable mode to next to leading order in a $1/D$ expansion. This result agrees with a previously obtained numerical bound on this eigenvalue. For cosmological constant zero, our answer matches a previous result obtained for the corresponding eigenvalue for the $D$ dimensional Schwarzschild-Tangherlini black hole to next to leading order in a $1/D$ expansion.