Noncommutative dyonic black holes sourced by nonlinear electromagnetic fields

TL;DR

This work develops first-order noncommutative corrections to Einstein gravity coupled to nonlinear electrodynamics by employing a Killing Drinfel'd twist and the Seiberg-Witten map. The authors show that the NC effects introduce universal off-diagonal metric corrections, while leaving the mass and charges unchanged, and demonstrate explicit NC modifications for ModMax, Born-Infeld, and Euler–Heisenberg dyonic black holes. The approach resolves ordering ambiguities via Palais' symmetric criticality and reveals symmetry breaking (duality and conformal invariance) in the NC-corrected solutions. The results bridge noncommutative geometry with nonlinear electrodynamics in curved spacetime, offering a tractable perturbative framework and clear pathways for extension to higher orders, non-Killing twists, and modified gravity theories.

Abstract

We introduce the first-order noncommutative (NC) corrections to the general nonlinear electrodynamics (NLE) Lagrangian depending on two electromagnetic invariants. The NC deformation of Einstein-NLE theory is implemented using the $\partial_t\wedge\partial_\varphi$ Drinfel'd twist and the NC effects are encoded in the matter sector through the Seiberg-Witten map. The resulting equations of motion reflect two distinct sources of nonlinearity in this framework; one arising from replacing Maxwell's electrodynamics with its nonlinear modifications and another from the NC deformations. Assuming a general form of static, spherically symmetric dyonic black hole as a seed solution in the commutative limit, we solve the equations of motion perturbatively to the first order in the NC parameter $a$. Finally, we evaluate the obtained corrections to the metric tensor and gauge potential for several prominent NLE theories.