New improved massive gravity and three-dimensional spacetimes of constant curvature and constant torsion

Abstract

We derive the field equations for topologically massive gravity coupled with the most general quadratic curvature terms using the language of exterior differential forms and a first order constrained variational principle. We find variational field equations both in the presence and absence of torsion. We then show that spaces of constant negative curvature (i.e. the anti de-Sitter space AdS 3 ) and constant torsion provide exact solutions.

New improved massive gravity

“…That is to say, we evaluate curvatures and their derivatives without differentiating any functions. The relevant differential geometric techniques are explained elsewhere [14]. Our starting point will be the structure equations satisfied by an orthonormal set of left-invariant basis 1-forms {e a } on AdS 3 : e 0 = cosh χdt + cos t sin t sinh χdχ + sin 2 t sinh 2 χdθ, e 1 = sinh χ cos θdt + (cos t sin t cosh χ cos θ + sin 2 t sin θ)dχ + sin 2 t sinh χ(cosh χ cos θ − cot t sin θ)dθ, e 2 = sinh χ sin θdt + (cos t sin t cosh χ sin θ − sin 2 t cos θ)dχ…”

“…The notion of constant torsion in 3 dimensions [13] was implicit in Cartan's work and it is called Cartan's spiral staircase in a recent review [11]. Details of the calculations that follow were presented elsewhere [14].…”

The Variational Field Equations of Cosmological Topologically Massive Supergravity

“…The variational field equations of these theories are derived by augmenting the Einstein-Hilbert action (possibly coupled to other terms as well) by a Lorentzian Chern-Simons term. Some examples of massive gravity theories are studied in references [1]- [8]. The presence of Chern-Simons term helps generate a propagating massive spin-2 field for the otherwise dynamically trivial Einstein's theory [3].…”

Weyl covariant theories of gravity in 3-dimensional Riemann–Cartan–Weyl space-times

“…QCG is defined via the action of 3D GR augmented with generic quadratic curvature invariants. This model is motivated by NMG and the new improved massive gravity (NIMG) [12,13] models. On the one hand, in NIMG we have studied a general 3D gravitational model that contains TMG, MMG and NMG as subcases and obtained exact background solutions.…”

“…In our earlier papers[12,13], using an identy we replaced one of the three quadratic curvature invariants in terms of the other two. Here, we do not do this and keep all three quadratic curvature invariants.…”