Modified Palatini action that gives the Einstein-Maxwell theory

TL;DR

This paper introduces a modified Palatini action quadratic in derivatives that naturally yields Einstein-Maxwell and Einstein-Proca theories, linking geometric structures to electromagnetic phenomena.

Contribution

It proposes a novel quadratic Palatini action that reproduces Einstein-Maxwell and Einstein-Proca theories, expanding the scope of Palatini formulations.

Findings

Derives Einstein-Maxwell theory from a quadratic Palatini action.

Shows the antisymmetric Ricci tensor component acts as electromagnetic field.

Connects geometric derivatives to electromagnetic field strength.

Abstract

The actions for bosonic fields typically contain terms quadratic in the derivatives of the fields. This is not the case in the Palatini approach to general relativity. The action does not contain any derivatives of the metric and it only contains terms linear in the derivatives of the connection. In general relativity the covariant derivative of the metric vanishes, so it is not possible to include such terms in the action. However, in more general theories this is not the case. In this paper I consider an action which is quadratic in the derivatives of the metric and connection and show that it leads to the coupled Einstein-Maxwell theory or the coupled Einstein-Proca theory with the antisymmetric part of the Ricci tensor playing the role of the electromagnetic field strength.