Poincar\'{e} gauge theory of gravity

TL;DR

This paper develops a (2+1)-dimensional Poincaré gauge theory of gravity using Riemann-Cartan space-time, deriving field equations, exploring solutions for weak fields, and connecting to Einstein and teleparallel theories.

Contribution

It introduces a comprehensive Poincaré gauge theory of gravity in 2+1 dimensions with a general quadratic Lagrangian, linking it to Einstein and teleparallel theories.

Findings

Solutions of vacuum Einstein equations satisfy the theory's vacuum equations.

Torsion can be 'frozen' at sources with no spin, aligning with Einstein equations.

A Newtonian limit exists under specific parameter conditions.

Abstract

A Poincar\'{e} gauge theory of (2+1)-dimensional gravity is developed. Fundamental gravitational field variables are dreibein fields and Lorentz gauge potentials, and the theory is underlain with the Riemann-Cartan space-time. The most general gravitational Lagrangian density, which is at most quadratic in curvature and torsion tensors and invariant under local Lorentz transformations and under general coordinate transformations, is given. Gravitational field equations are studied in detail, and solutions of the equations for weak gravitational fields are examined for the case with a static, \lq \lq spin"less point like source. We find, among other things, the following: (1)Solutions of the vacuum Einstein equation satisfy gravitational field equations in the vacuum in this theory. (2)For a class of the parameters in the gravitational Lagrangian density, the torsion is \lq \lq frozen" at the place where \lq \lq spin" density of the source field is not vanishing. In this case, the field equation actually agrees with the Einstein equation, when the source field is \lq \lq spin"less. (3)A teleparallel theory developed in a previous paper is \lq \lq included as a solution" in a limiting case. (4)A Newtonian limit is obtainable, if the parameters in the Lagrangian density satisfy certain conditions.