Static spherically symmetric solutions for conformal gravity in three dimensions

TL;DR

The problem addressed is the lack of propagating degrees of freedom for GR in three dimensions, which traditionally yields constant-curvature spacetimes, while conformal gravity with vanishing Cotton tensor admits richer structures. The authors exploit the conformal invariance of the field equations to generate nontrivial static solutions via improper conformal transformations that glue constant-curvature patches. They present a general static spherically symmetric metric $ds^2 = -(a r^2 + b r + c) dt^2 + (dr^2)/(a r^2 + b r + c) + r^2 dphi^2$, yielding spacetimes with asymptotic curvature $-a$ and, for suitable parameters, black holes with up to two horizons and a wormhole solution with neck radius $l_0$. These solutions illustrate that conformal matching across infinite boundaries, impossible in GR, is feasible in conformal gravity, connecting different constant-curvature regions (e.g., BTZ-related patches for the wormhole). The results broaden the landscape of exact solutions in 3D gravity and raise open questions about conserved charges and black hole thermodynamics, with potential extensions to rotating cases.

Abstract

Static spherically symmetric solutions for conformal gravity in three dimensions are found. Black holes and wormholes are included within this class. Asymptotically the black holes are spacetimes of arbitrary constant curvature, and they are conformally related to the matching of different solutions of constant curvature by means of an improper conformal transformation. The wormholes can be constructed from suitable identifications of a static universe of negative spatial curvature, and it is shown that they correspond to the conformal matching of two black hole solutions with the same mass.

Figures (1)

• Figure 1: The causal structure of the conformal gravity black holes (\ref{['final']}) and the wormhole (\ref{['wormholeansatz']}) is shown in (A-D).1 and E.1, respectively. These spacetimes can be constructed from the conformal matching of different patches of constant curvature spacetimes at the corresponding spatial infinities, as depicted in (A-E).2. Each square diagram appearing in Figs.(B-E).2 describes the causal structure of static BTZ black holes. In Fig. A.2 the square diagrams correspond to the causal structure of static solutions with negative mass and positive cosmological constant. In the figures A.2 and B.2, the region $\rho<\rho_{*}$ must be excised and for D.2 the excised region correspond to $\rho>\rho_{*}$. Here $r_{c}$ corresponds to the matching surface where the constant curvature spacetimes are smoothly glued; for the wormhole the matching surface is located at the neck ($z=0$).