Geodesic structure of the Euler-Heisenberg static black hole

Abstract

We derive an electrically charged static black hole spacetime of the Einstein-Euler-Heisenberg theory, in terms of the Pleba\ifmmode \acute{n}\else \'{n}\fi{}ski dual variables. This solution is a nonlinear electromagnetic generalization of the Reissner-Nordstr\om solution, and it is characterized by the mass $M$, the electric charge $Q$ of the black hole, and the Euler-Heisenberg nonlinear constant, which includes the fine structure constant $\ensuremath{\alpha}$. We study all possible equatorial trajectories of test particles. Moreover, the orbits of photons are analyzed by means of the effective Pleba\ifmmode \acute{n}\else \'{n}\fi{}ski pseudometric related to the geometrical metric and to the electromagnetic energy-momentum tensor. The shape of the shadow of the black hole is also presented and discussed.