Three-dimensional black holes, gravitational solitons, kinks and wormholes for BHT massive gravity

TL;DR

This work analyzes the three-dimensional Bergshoeff–Hohm–Townsend (BHT) massive gravity at the special point $m^{2}=\lambda$, showing that relaxing the standard Brown–Henneaux AdS boundary conditions admits a rich set of exact solutions—black holes, gravitational solitons, kinks, and a vacuum wormhole—whose properties are captured by Deser–Tekin charges and a central charge $c=\frac{3l}{G}$. The authors derive thermodynamics for the negative-$\Lambda$ black holes, construct rotating extensions, and explore the positive-$\Lambda$ and flat cases, including Euclidean actions that vanish for the instanton configurations. A double Wick rotation generates regular solitons and kinks, while identifications yield a vacuum wormhole; these solutions illustrate how hair-like deformations and relaxed asymptotics expand the solution space beyond standard GR in 3D. The results have implications for AdS$_3$/CFT$_2$ physics and highlight the interplay between higher-curvature terms, boundary conditions, and conserved charges in three-dimensional gravity.

Abstract

The theory of massive gravity in three dimensions recently proposed by Bergshoeff, Hohm and Townsend (BHT) is considered. At the special case when the theory admits a unique maximally symmetric solution, a conformally flat space that contains black holes and gravitational solitons for any value of the cosmological constant is found. For negative cosmological constant, the black hole is characterized in terms of the mass and the "gravitational hair" parameter, providing a lower bound for the mass. For negative mass parameter, the black hole acquires an inner horizon, and the entropy vanishes at the extremal case. Gravitational solitons and kinks, being regular everywhere, are obtained from a double Wick rotation of the black hole. A wormhole solution in vacuum that interpolates between two static universes of negative spatial curvature is obtained as a limiting case of the gravitational soliton with a suitable identification. The black hole and the gravitational soliton fit within a set of relaxed asymptotically AdS conditions as compared with the ones of Brown and Henneaux. In the case of positive cosmological constant the black hole possesses an event and a cosmological horizon, whose mass is bounded from above. Remarkably, the temperatures of the event and the cosmological horizons coincide, and at the extremal case one obtains the analogue of the Nariai solution, $dS_{2}\times S^{1}$. A gravitational soliton is also obtained through a double Wick rotation of the black hole. The Euclidean continuation of these solutions describes instantons with vanishing Euclidean action. For vanishing cosmological constant the black hole and the gravitational soliton are asymptotically locally flat spacetimes. The rotating solutions can be obtained by boosting the previous ones in the $t-φ$ plane.

Figures (6)

• Figure 1: Causal structure of the asymptotically AdS black holes with negative $b$.
• Figure 2: Penrose diagrams of the black holes with negative cosmological constant and positive $b$
• Figure 3: Causal structure of the wormhole. The dotted line corresponds to the location of the neck.
• Figure 4: Causal structure of the kink, that interpolates between $R\times H_{2}$ ($z\rightarrow-\infty$) and AdS ($z\rightarrow+\infty$)
• Figure 5: The causal structure of the black holes with positive cosmological constant