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

Abstract

We provide a detailed derivation of the Schwarzian modes in the full geometry of the Bañados-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.

Figures (3)

• Figure 1: Norm density of the Schwarzian modes, $||h||^2_{\sf den}=\sqrt{g}\,h_{\mu\nu}^{(n)}h^{\mu\nu\,(-n)}$, as a function of the proper radial coordinate $\log(r/r_+)$. Left: Fixed mode $n=3$ for different ratios $r_-/r_+$, with values matching those used in JOACO. The modes become increasingly localized near the horizon as extremality is approached ($r_- \to r_+$), eventually concentrating at the horizon. Away from extremality, the modes spread out and eventually become non-normalizable for higher $n$, as predicted by \ref{['schwarzian_spectrum']}, which manifests as a negative norm density as can be seen in the right plot. Right: Norm density for several modes at fixed $r_-=0.7\,r_+$. In agreement with \ref{['schwarzian_spectrum']}, only a finite number of Schwarzian modes are normalizable; in this case, modes with $n \geq 10$ become non-normalizable, while the lowest modes are increasingly localized near the horizon.
• Figure 2: Norm density of the rotational modes, $||h||^2_{\sf den}=\sqrt{g}\,h_{\mu\nu}^{(n)}h^{\mu\nu\,(-n)}$, as a function of the proper radial coordinate $\log(r/r_+)$. Left: Fixed mode $n=3$ for different ratios $r_-/r_+$, with values matching those used in Fig. \ref{['fig:Schwarzians']}. The modes become increasingly localized near the horizon as extremality is approached ($r_- \to r_+$), eventually concentrating at the horizon. These modes exhibit negative density near the horizon, but the norm remains well-defined (the integral remains positive). Away from extremality, the modes spread out and eventually become non-normalizable for higher $n$, as predicted by \ref{['rotational_spectrum']}, which manifests as a negative norm as seen for $r_-=0.6\,r_+$ (black line). Right: Norm density for several modes at fixed $r_-=0.8\,r_+$. In agreement with \ref{['rotational_spectrum']}, only a finite number of rotational modes are normalizable; in this case, modes with $n \geq 5$ become non-normalizable, while the lowest modes are increasingly localized near the horizon.
• Figure 3: Norm density of the vector modes, $||A||^2_{\sf den}=\sqrt{g}\,A_{\mu}^{(n)}A^{\mu\,(-n)}$, as a function of the proper radial coordinate $\log(r/r_+)$. Left: Fixed mode $n=3$ for different ratios $r_-/r_+$. The modes become increasingly localized near the horizon as extremality is approached ($r_- \to r_+$), but they do not become zero modes. Away from extremality, the modes spread out and eventually become non-normalizable, as predicted by \ref{['vector_norm2']}, which manifests as a negative norm density. Right: Norm density for several modes at fixed $r_-=0.9\,r_+$. In agreement with \ref{['vector_norm2']}, only a finite number of modes are normalizable; in this case, modes with $n \geq 11$ become non-normalizable, while the lowest modes are increasingly localized near the horizon.