AdS black holes in two-dimensional dilaton gravity and holography

TL;DR

This work constructs two analytic AdS$_2$ black holes in a 1+1 dimensional dilaton gravity model with two non-minimally coupled scalars. The solutions feature two integration constants in the blackening function, enabling extremality and yielding constant negative curvature; their global structure is analyzed via Kruskal extensions and Penrose diagrams, and a Hamilton–Jacobi counter-term provides a finite renormalized action for a semi-classical partition function. A consistent canonical ensemble thermodynamics is developed, including an extremal limit with vanishing temperature, and the boundary effective theory for Solution I is shown to be a Schwarzian action augmented by a mass term tied to the bulk integration constants. Holographically, the boundary dynamics reduce to Schwarzian behavior with mass deformations, while Solution II corresponds to a trivial boundary theory due to a constant dilaton; these results link two-dimensional dilaton gravity with Jackiw–Teitelboim-like descriptions and generalized conformal structures, offering paths to higher-dimensional generalizations and Lifshitz-like extensions.

Abstract

In this paper, we present two novel analytic AdS black hole solutions in a two-dimensional dilaton gravity theory with two scalar fields non-minimally coupled to gravity. Our solutions contain two arbitrary integration constants in the blackening factor $f(r)$, allowing for an extremal configuration. Solution I reproduces a previously reported AdS black hole when one of the integration constants in $f(r)$ vanishes. For our black hole configurations, the scalar curvature is constant and negative, corresponding to the $AdS_2$ spacetime. In order to elucidate their black hole nature, we explore the causal structure of these solutions with the aid of suitable Kruskal-like coordinates and Penrose diagrams. By employing the Hamilton-Jacobi method, we construct a boundary counter-term that renders a renormalized action with a vanishing variation. We use this finite action for the partition function in the semi-classical approximation. We establish a consistent Thermodynamics, verified by the first law, for our black hole solutions, including the extremal case. Finally, we perform a holographic analysis of the effective theory at the boundary of the black hole solution I. This theory is characterized by a Schwarzian action supplemented by a black hole mass term determined by the two integration constants in $f(r)$. We also examine the holographic implications of the boundary counter-term.

Figures (4)

• Figure 1: Graphical representation of an example of the dilaton field $X(r) \equiv e^{ \phi_1+\phi_2}=r-\frac{c_1}{2}$ in the solution I. We employ the following particular values: $c_3=c_5=0$, $c_1=4$, $c_2=2$. We observe a finite value of the dilaton field $X(r)$ at the horizon $r_+=\sqrt{2}+2$, represented here with the dashed vertical line, and its asymptotically singular behavior.
• Figure 2: Kruskal diagram in coordinates ($T_+$, $R_+$).
• Figure 3: Kruskal diagram for coordinates ($T_-$, $R_-$).
• Figure 4: Conformal diagram for our $AdS_2$ black holes. The spacetime boundary $r \rightarrow \infty$ is represented by the dashed vertical straight lines and the singularity $r=0$ is depicted by the zigzag vertical ones. Here we illustrate a particular motion through a timelike path.