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Maxwell's equations are invariant under both duality rotations and conformaltransformations. Recently Bandos, Lechner, Sorokin, and Townsend have found anonlinear generalisation of electrodynamics which possesses both of thesesymmetries. We couple this theory to General Relativity and exploreself-gravitating configurations with a clear physical interpretation. We findcharged black holes and exact gravitational waves. The black hole geometry isReissner-Nordstr\"om, however, the non-linearity parameter acts as a screeningfactor allowing extremal configurations where the mass of the black hole issmaller than its charge. Furthermore, we also discuss the phenomenon ofbirefringence and determine the optical metrics associated to the propagationof photons.

Black holes and gravitational waves sourced by non-linear duality rotation-invariant conformal electromagnetic matter