The Einstein 3-form Gα and its Equivalent 1-form Lα in Riemann–Cartan Space

Abstract

The definition of the Einstein 3-form Gα is motivated by means of the contracted 2nd Bianchi identity. This definition involves at first the complete curvature 2-form. The 1-form Lα is defined via Gα = L β ∧ ⋆ (ϑ β ∧ ϑα) (here ⋆ is the Hodge-star, ϑα the coframe). It is equivalent to the Einstein 3-form and represents a certain contraction of the curvature 2-form. A variational formula of Salgado on quadratic invariants of the Lα 1-form is discussed, generalized, and put into proper perspective.

Schwarzschild and Kerr solutions of Einstein's field equation: An Introduction

“…( 94) to (98). How such a Reduce-Excalc program can be set up, is demonstrated in Stauffer et al [178] and in Socorro et al [176], for the Einstein 3-form, see Heinicke [82]:…”

Quadratic curvature gravity with second order trace and massive gravity models in three dimensions

“…Rθ α is the Schouten 1-form, defined in n ≥ 3 dimensions [24] and G α = R α − 1 2 Rθ α is the Einstein 1-form. Various properties of (32) such as the variational derivatives with respect to L α and G α have been studied in [25] in more general context of Riemann-Cartan geometry using the language of exterior differential forms. Finally, note that, with the addition of an appropriate cosmological constant to (32), the lagrangian density becomes the critical theory [12] which has recently been introduced for n = 4.…”