Unveiling the electrodynamics of the first nonlinearly charged rotating black hole

After many years of efforts, the first nonlinearly charged rotating black hole has been finally reported by Garcia-Diaz in two recent works. This is an important result that was pending in General Relativity, since nonlinear generalizations of the Kerr-Newman solution were not yet known. Unfortunately, the Lagrangian supporting this configuration cannot be expressible in terms of the standard invariants using elementary functions. In the present work we circumvent this problem by using the formulations of nonlinear electrodynamics in terms of mixed electromagnetic eigenvalues, introduced by Salazar, Garcia-Diaz and Plebanski almost four decades ago. In doing so, we prove that the underlying theory becomes fully determined, and hence the new found nonlinearly charged stationary axisymmetric spacetimes correspond to exact solutions of a well-defined self-gravitating nonlinear electrodynamics whose fundamental structural functions are provided here.