Rotating black holes in the Einstein–Euler–Heisenberg theory

We apply the Newman–Janis algorithm and its Azreg–Aïnou formulation to obtain the rotating models from a seed static Einstein–Euler–Heisenberg electrically charged black hole solution. Both algorithms generate solutions whose geometric part do not correspond to the energy–momentum tensor of the Euler–Heisenberg electromagnetic field. Only for slow rotations one has a Lense–Thirring-like slow rotating approximate solution to the field equations. According to the Azreg–Aïnou method, the obtained rotating metric is a solution with a kind of ad hoc constructed electromagnetic imperfect fluid as a source. This explanation is artificial, since one is not working with the source at all, instead one is working only with the geometry via Einstein tensor. Both methods lead to the rotating Gürses and Gürsey metric, that we consider as a geometrical model. The corresponding energy conditions are analyzed, its event horizons, ergoregions and test particle circular orbits are studied, and its shadow is shown as well. (paper)