Spinor formulation of topologically massive gravity

TL;DR

This paper develops a real spinor and differential form formalism for topologically massive gravity in three dimensions, providing new geometric insights and solutions including black holes.

Contribution

It introduces a real spinor formulation of TMG using differential forms, extending previous triad approaches and clarifying geometric identities and solutions.

Findings

Derived basic geometrical identities for TMG

Presented exact Bianchi Type VIII - IX solutions

Identified black hole solutions via re-identifications

Abstract

In the framework of real 2-component spinors in three dimensional space-time we present a description of topologically massive gravity (TMG) in terms of differential forms with triad scalar coefficients. This is essentially a real version of the Newman-Penrose formalism in general relativity. A triad formulation of TMG was considered earlier by Hall, Morgan and Perjes, however, due to an unfortunate choice of signature some of the spinors underlying the Hall-Morgan-Perjes formalism are real, while others are pure imaginary. We obtain the basic geometrical identities as well as the TMG field equations including a cosmological constant for the appropriate signature. As an application of this formalism we discuss the Bianchi Type $VIII - IX$ exact solutions of TMG and point out that they are parallelizable manifolds. We also consider various re-identifications of these homogeneous spaces that result in black hole solutions of TMG.