Aspects of Warped AdS$_3$ geometries

TL;DR

The paper develops a group-manifold framework for warped AdS$_3$ geometries by viewing AdS$_3$ as the group $SL(2,\mathbb{R})$ and producing warped spaces through a controlled deformation of the bi-invariant metric via a right-invariant one-form. It provides a comprehensive construction of warped AdS$_3$ quotients with residual $\mathbb{R}\times SO(2)$ symmetry, classifies warpings (spacelike, timelike, lightlike), and analyzes their causal structure through projection diagrams. The work demonstrates that warped AdS$_3$ and their quotients are exact solutions to multiple three-dimensional gravity theories, including TM gravity, Einstein--Cartan, and Einstein--Maxwell--Chern--Simons, and it presents explicit geodesic solutions, Killing vectors, Killing spinors, and a coordinate-decryption algorithm. The results offer a cohesive, geometrical toolkit for understanding warped AdS$_3$ black holes and near-horizon geometries, with implications for holography, causality, and string-theoretic backgrounds. Overall, the paper unifies warped AdS$_3$ geometry, quotients, and their gravity-model realizations under a group-theoretic, coadjoint-orbit perspective and provides practical methods for analyzing their global and causal properties.

Abstract

We discuss the geometry of three-dimensional warped Anti-de Sitter spaces and quotients thereof, paying special attention to their underlying group manifold nature. We perform a systematic analysis of warped Anti-de Sitter geometries, focusing on their global properties and illustrating their occurrence as special solutions of various three-dimensional gravity theories.

Figures (9)

• Figure 1: Configurations without a horizon, encountered in types (a): $\boldsymbol I _a$, $\boldsymbol I \space\boldsymbol I _a$, $\boldsymbol I _b$, $\boldsymbol I _c^\pm, \boldsymbol I \space\boldsymbol I _b^\pm$, $\tilde{\boldsymbol I }_a$, $\tilde{\boldsymbol I \space\boldsymbol I }^\pm_b$, ${\boldsymbol I \space\boldsymbol I \space\boldsymbol I }^+$, ${\tilde{\boldsymbol I \space\boldsymbol I }}_a$, each only for $\lambda \neq 0$(b): $\tilde{\boldsymbol I }_a$, only for $\lambda \neq 0$(c): $\boldsymbol I \space\boldsymbol I _a$, $\boldsymbol I \space\boldsymbol I _b^\pm$, ${\boldsymbol I \space\boldsymbol I \space\boldsymbol I }^\pm$, each only for $\lambda \neq 0$(d): self-dual case (${L_{\text{\tiny {(L)}}}} = 0$), ${\epsilon_{\text{\tiny {(R)}}}} = 1$, ${\epsilon_{\text{\tiny {(L)}}}} = -1$(e): $\boldsymbol I _a$, $\boldsymbol I _c^-$, $\boldsymbol I \space\boldsymbol I _b^-$ each only for $\lambda = 0$(f): ${\boldsymbol I \space\boldsymbol I \space\boldsymbol I }^-$ only for $\lambda = 0$
• Figure 2: Configurations with one horizon, no singularity, encountered in types(a): ${\tilde{\boldsymbol I \space\boldsymbol I }}_a$, only for $\lambda \neq 0$(b): self-dual case (${L_{\text{\tiny {(L)}}}} = 0$), ${\epsilon_{\text{\tiny {(R)}}}} = 1$, ${\epsilon_{\text{\tiny {(L)}}}} =0$
• Figure 3: Configurations with one horizon, encountered in types(a): $\boldsymbol I \space\boldsymbol I _a$, ${\tilde{\boldsymbol I \space\boldsymbol I }}_a$ for $\lambda \neq 0$(b): $\boldsymbol I \space\boldsymbol I _a$ for $\lambda = 0$
• Figure 4: Configurations with one horizon(a): $\boldsymbol I _b$ for $\lambda \neq 0$(b): $\boldsymbol I _b$ for $\lambda = 0$
• Figure 5: Configurations with two horizons, no singularity, encountered in types(a): $\boldsymbol I _b$ for $\lambda \neq 0$(b): self-dual case (${L_{\text{\tiny {(L)}}}} = 0$), ${\epsilon_{\text{\tiny {(R)}}}} =1$, ${\epsilon_{\text{\tiny {(L)}}}} =1$