From BTZ Perturbations to Schwarzian Modes: A Geometrical and Perturbative Analysis

TL;DR

This paper derives Schwarzian modes in BTZ black hole geometry at finite temperature, clarifies their emergence conditions, and connects geometric and perturbative approaches, including a link to the double copy framework.

Contribution

It provides a detailed derivation of Schwarzian modes from the full BTZ geometry and introduces a geometric Kerr-Schild construction that relates to perturbative methods.

Findings

Schwarzian modes are derived explicitly from BTZ geometry.

A geometric Kerr-Schild approach reproduces the same modes.

The work clarifies the absence of rotational modes in the full geometry.

Abstract

We provide a detailed derivation of the Schwarzian modes in the full geometry of the Ba\~nados-Teitelboim-Zanelli (BTZ) black hole at finite temperature, establishing the precise conditions under which they emerge from the general solution, thereby clarifying the absence of rotational modes in the full geometry. In addition, we demonstrate that the same modes can be recovered through a purely geometric Kerr-Schild construction. This equivalent approach offers a new geometric understanding of the Schwarzian sector and highlights the correspondence between perturbative and pure geometric approaches, additionally it provides a connection with double copy.