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

Alberto A. Garcia, Friedrich W. Hehl, Christian Heinicke, Alfredo Macias

TL;DR

The paper analyzes (1+2)D gravity in the Poincaré gauge framework using the Mielke–Baekler model, which includes torsion and curvature Chern–Simons terms. It derives the exact vacuum field equations and constructs a BTZ-like black hole solution with nonzero torsion, detailing its mass, angular momentum, and the role of an effective cosmological constant $\Lambda_{\rm eff}$. It then establishes a broader conformally flat vacuum sector with torsion and connects Cartan’s spiral staircase to both MB vacuum solutions and 3D Einstein–Cartan theory with constant mechanical currents. Together these results illuminate how topological terms and torsion modify global charges and spacetime structure in 3D gravity and bridge gravitational and condensed-mmatter analogies.

Abstract

In (1+2)-dimensional Poincaré 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.

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

TL;DR

The paper analyzes (1+2)D gravity in the Poincaré gauge framework using the Mielke–Baekler model, which includes torsion and curvature Chern–Simons terms. It derives the exact vacuum field equations and constructs a BTZ-like black hole solution with nonzero torsion, detailing its mass, angular momentum, and the role of an effective cosmological constant . It then establishes a broader conformally flat vacuum sector with torsion and connects Cartan’s spiral staircase to both MB vacuum solutions and 3D Einstein–Cartan theory with constant mechanical currents. Together these results illuminate how topological terms and torsion modify global charges and spacetime structure in 3D gravity and bridge gravitational and condensed-mmatter analogies.

Abstract

In (1+2)-dimensional Poincaré 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.

Paper Structure

This paper contains 8 sections, 32 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Mielke-Baekler model and its BTZ-like exact solution
  3. Properties of our solution
  4. Autoparallels and extremals
  5. Killing vectors
  6. Quasilocal conserved quantities
  7. General conformally flat vacuum solution with torsion
  8. É. Cartan's spiral staircase

Figures (1)

  • Figure 1: Cartan's spiral staircase. Cartan's rules cartan22 for the introduction of a non-Euclidean connection in a 3D Euclidean space are as follows: (i) A vector which is parallelly transported along itself does not change (cf. a vector directed and transported in $x$-direction). (ii) A vector that is orthogonal to the direction of transport rotates with a prescribed constant 'velocity" (cf. a vector in $y$--direction transported in $x$--direction). The winding sense around the three coordinate axes is always positive.