The Einstein 3-form G_a and its equivalent 1-form L_a in Riemann-Cartan space

TL;DR

This paper explores the Einstein 3-form G_a and its equivalent 1-form L_a in Riemann-Cartan space, providing definitions, relationships, and variational formulas related to curvature and quadratic invariants.

Contribution

It introduces a new perspective on the Einstein 3-form and its 1-form L_a, including their definitions, equivalence, and a generalized variational formula for quadratic invariants.

Findings

G_a defined via contracted Bianchi identity involving curvature 2-form

L_a shown to be equivalent to G_a and a contraction of curvature

A generalized variational formula for quadratic invariants of L_a

Abstract

The definition of the Einstein 3-form G_a is motivated by means of the contracted 2nd Bianchi identity. This definition involves at first the complete curvature 2-form. The 1-form L_a is defined via G_a = L^b \wedge #(o_b \wedge o_a). Here # denotes the Hodge-star, o_a the coframe, and \wedge the exterior product. The L_a 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 L_a 1-form is discussed, generalized, and put into proper perspective.