Exact vacuum solution of a (1+2)-dimensional Poincare gauge theory: BTZ solution with torsion

TL;DR

This paper derives an exact vacuum solution in a (1+2)-dimensional Poincaré gauge gravity model with torsion, generalizing the BTZ black hole and exploring conformally flat solutions and Cartan's spiral staircase.

Contribution

The paper presents the first exact vacuum solution of the Mielke-Baekler model with torsion, extending the BTZ solution to include torsion effects and analyzing its properties.

Findings

Recovered BTZ solution when torsion vanishes

Derived the general conformally flat vacuum solution with torsion

Connected Cartan's spiral staircase to solutions with constant pressure and torque

Abstract

In (1+2)-dimensional Poincar\'e gauge gravity, we start from a Lagrangian depending on torsion and curvature which includes additionally {\em translational} and {\em Lorentzian} Chern-Simons terms. Limiting ourselves to to a specific subcase, the Mielke-Baekler (MB) model, we derive the corresponding field equations (of Einstein-Cartan-Chern-Simons type) and find the general vacuum solution. We determine the properties of this solution, in particular its mass and its angular momentum. For vanishing torsion, we recover the BTZ-solution. We also derive the general conformally flat vacuum solution with torsion. In this framework, we discuss {\em Cartan's} (3-dimensional) {\em spiral staircase} and find that it is not only a special case of our new vacuum solution, but can alternatively be understood as a solution of the 3-dimensional Einstein-Cartan theory with matter of constant pressure and constant torque.