Spinor formulation of topologically massive gravity

Abstract

In the framework of real 2-component spinors in three dimensional spacetime 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 V III − 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.

Equivalence of three-dimensional spacetimes

“…Note that Φ AB has only five independent components, since we have the identity Φ 11 ≡ Φ 02 due to S = S a a = −2S ab n a k b + S ab m a m b = 0 (this point is missing in [29]). When n a and k a are swapped, the index 1 in Φ AB remains unchanged, while the rest flip 0 ↔ 2.…”

Properties of Solutions in 2 + 1 Dimensions

“…One may use the Newman-Penrose formalism for 2+1 dimensions as given Aliev and Nutku [14]. Similar formalism is also given in references [15].…”

Impulsive gravitational waves in general massive 3D gravity

“…The geometrical setting for the seminull coframe basis with varying conventions has been introduced previously [20][21][22][23] in the literature. The notation and most of the conventions used below follow those of Aliev and Nutku introduced in a spinor formulation of TMG.…”