Einstein-Maxwell fields as solutions of Einstein gravity coupled to conformally invariant non-linear electrodynamics

TL;DR

This paper characterizes Einstein-Maxwell solutions that can be extended to conformally invariant non-linear electrodynamics, providing criteria and examples for such extendable configurations.

Contribution

It introduces a criterion for extendability of Einstein-Maxwell solutions to conformally invariant non-linear electrodynamics and demonstrates their applicability to various known solutions.

Findings

All static Einstein-Maxwell configurations are extendable.

Explicit examples include black holes, gravitational waves, and cosmological solutions.

Duality invariance allows for dyonic solutions in extended theories.

Abstract

We study Einstein-Maxwell (non-null) sourcefree configurations that can be extended to any conformally invariant non-linear electrodynamics (CINLE) by a constant rescaling of the electromagnetic field. We first obtain a criterion which characterizes such extendable solutions in terms either of the electromagnetic invariants, or (equivalently) of the canonical Newman-Penrose form of the self-dual Maxwell field. This is then used to argue that all static configurations are extendable (more generally, all configurations admitting a non-null twistfree Killing vector field). One can thus draw from the extensive literature to straightforwardly extend to CINLE various known exact solutions, whereby the duality invariance of the Einstein-Maxwell theory allows for dyonic solutions even in more general theories. This is illustrated by a few explicit examples, including the homogeneous $\Lambda<0$ universe of Ozsv\'ath, a black hole in the universe of Levi-Civita, Bertotti and Robinson, a generalization of the charged $C$-metric, and non-expanding gravitational waves in the Bonnor-Melvin background.