Extended Theories of Electrodynamics in $f(R)$ Gravity

TL;DR

This paper extends $f(R)$ gravity by incorporating a function of the electromagnetic invariant, leading to generalized field equations that recover known models and could impact phenomenology in extreme astrophysical environments.

Contribution

Introduces a new coupling function $f( ext{F})$ in $f(R)$ gravity, generalizing electrodynamics and connecting to known models like Bopp-Podolsky.

Findings

Derives generalized field equations resembling a Klein-Gordon form.

Recovers Plebanski and Bopp-Podolsky models as special cases.

Suggests potential observable effects in extreme environments.

Abstract

Within the general framework of $f(R)$ gravity, we introduce a function of the electromagnetic curvature invariant $f(\mathbb{F})$ that couples minimally to gravitation to ensure a consistent treatment of curvature functions in these theories. We show that one of the solutions leads to field equations that are a generalization of the Klein-Gordon equation while the other leads to a typically non-linear massless solution. Focusing on flat spacetime, our formalism recovers the Plebanski family of models and Bopp-Podolsky electrodynamics as specific limits. These extensions may have phenomenological consequences in extreme environments, such as the early universe or near charged compact objects, where deviations from classical electrodynamics might be probed.