# Deforming black holes with even multipolar differential rotation boundary

## TL;DR

This paper constructs and analyzes black hole and soliton solutions in AdS$_4$ with quadrupolar differential rotation boundary conditions, revealing their stability and properties despite boundary-induced spacelike Killing vectors.

## Contribution

It introduces new quadrupolar differential rotation boundary conditions and numerically constructs associated deforming black hole and soliton solutions, extending previous dipolar studies.

## Key findings

- Solutions exist even when the Killing vector becomes spacelike.
- Quadrupolar deformation is smaller than dipolar at the same temperature.
- Solutions do not develop superradiant hair despite boundary conditions.

## Abstract

Motivated by the novel asymptotically global AdS$_4$ solutions with deforming horizon in [JHEP {\bf 1802}, 060 (2018)], we analyze the boundary metric with even multipolar differential rotation and numerically construct a family of deforming solutions with quadrupolar differential rotation boundary, including two classes of solutions: solitons and black holes. In contrast to solutions with dipolar differential rotation boundary, we find that even though the norm of Killing vector $\partial_t$ becomes spacelike for certain regions of polar angle $\theta$ when $\varepsilon>2$, solitons and black holes with quadrupolar differential rotation still exist and do not develop hair due to superradiance. Moreover, at the same temperature, the horizonal deformation of quadrupolar rotation is smaller than that of dipolar rotation. Furthermore, we also study the entropy and quasinormal modes of the solutions, which have the analogous properties to that of dipolar rotation.