Rigorous Formulation of Duality in Gravitational Theories

TL;DR

This paper rigorously formulates duality in gravitational theories, establishing conditions under which Hodge duals of torsion and curvature forms can be interpreted as related geometric structures, and introduces a new Einstein equation involving dual Riemann tensors.

Contribution

It provides a rigorous mathematical framework for duality in gravitational theories, connecting dual torsion and curvature forms with Riemann-Cartan structures and proposing a novel Einstein equation with dual Riemann tensors.

Findings

Conditions for dual torsion and curvature forms as Riemann-Cartan structures

New Einstein equation involving dual Riemann tensor

Comparison with existing literature on duality in gravity

Abstract

In this paper we evince a rigorous formulation of duality in gravitational theories where an Einstein like equation is valid, by providing the conditions under which the Hodge duals (with respect to the metric tensor g) of T^a and R_b^a may be considered as the torsion and curvature 2-forms associated with a connection D', part of a Riemann-Cartan structure (M,g',D'), in the cases g = g' and g does not equal g', once T^a and R_b^a are the torsion and curvature 2-forms associated with a connection D part of a Riemann-Cartan structure (M,g,D). A new form for the Einstein equation involving the dual of the Riemann tensor of D is also provided, and the result is compared with others appearing in the literature.