Teleparallel theory of (2+1)-dimensional gravity.

Abstract

A theory of (2 + 1)-dimensional gravity is developed on the basis of the Weitzenb\ock space-time characterized by the metricity condition and by the vanishing curvature tensor. The fundamental gravitational field variables are dreibein fields and the gravity is attributed to the torsion. The most general gravitational Lagrangian density quadratic in the torsion tensor is given by ${L}_{G}=\ensuremath{\alpha}{t}^{\mathrm{klm}}{t}_{\mathrm{klm}}+\ensuremath{\beta}{v}^{k}{v}_{k}+\ensuremath{\gamma}{a}^{\mathrm{klm}}{a}_{\mathrm{klm}}$. Here, ${t}_{\mathrm{klm}}$, ${v}_{k}$ and ${a}_{\mathrm{klm}}$ are irreducible components of the torsion tensor, and $\ensuremath{\alpha}$, $\ensuremath{\beta}$, and $\ensuremath{\gamma}$ are real parameters. A condition is imposed on $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ by the requirement that the theory has a correct Newtonian limit. A static circularly symmetric exact solution of the gravitational field equation in the vacuum is given. It gives space-times quite different from each other, according to the signature of $\ensuremath{\alpha}\ensuremath{\beta}$. These space-times have event horizons, if and only if $\ensuremath{\alpha}(3\ensuremath{\alpha}+4\ensuremath{\beta})l0$. Singularity structures of these space-times are also examined.