Variational approach to gravitational theories with two independent connections

TL;DR

This paper introduces a novel variational method for gravity theories involving two independent connections, leading to new field equations that differ from traditional approaches and depend on matter coupling.

Contribution

It proposes a new variational framework with two independent affine connections for gravity theories, extending beyond metric and Palatini formalisms.

Findings

Recovers Einstein equations for Einstein-Hilbert action.

Derives modified field equations for $f(R)$ and Scalar-Tensor theories.

Shows dependence of equations on matter action's connection dependence.

Abstract

A new variational approach for general relativity and modified theories of gravity is presented. In addition to the metric tensor, two independent affine connections enter the action as dynamical variables. In the matter action the dependence upon one of the connections is left completely unspecified. When the variation is applied to the Einstein-Hilbert action the Einstein field equations are recovered. However when applied to $f(R)$ and Scalar-Tensor theories, it yields gravitational field equations which differ from their equivalents obtained with a metric or Palatini variation and reduce to the former ones only when no connections appear in the matter action.