Geodesic structure of the Euler-Heisenberg static black hole Daniel Amaro * Departamento de Física, Universidad Autónoma Metropolitana–Iztapalapa, Apdo. Postal 55–534, C.P. 09340, Ciudad de M´ exico, M´ exico Alfredo Macías † Physics Department, Universidad Autónoma Metropolitana–Iztapalapa, PO. Box 55–534, C.P. 09340, CDMX, M´ exico (Received 16 August 2020; accepted 28 October 2020; published 19 November 2020) We derive an electrically charged static black hole spacetime of the Einstein-Euler-Heisenberg theory, in terms of the Plebański dual variables. This solution is a nonlinear electromagnetic generalization of the Reissner-Nordström 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 α. We study all possible equatorial trajectories of test particles. Moreover, the orbits of photons are analyzed by means of the effective Plebański 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. DOI: 10.1103/PhysRevD.102.104054 I. INTRODUCTION The coupling of the Einstein theory to the class of nonlinear electrodynamics (NLED) proposed by Plebański [1] admits regular black hole solutions [2], i.e., black holes whose curvature invariants R, R μν R μν , and R μναβ R μναβ are nonsingular. Therefore, there exist nowadays a great revival of interest on it. The interest in nonlinear electrodynamics began in 1912 when Mie [3] put forward the first model for nonlinear electrodynamics. Between 1932 and 1935 Born and Infeld [4] proposed their nonlinear theory, which represents a classical generalization of the Maxwell-Lorentz theory for accommodating stable solutions for the description of electrons. Because of the nonlinearity of the electromag- netic theory, the field of a point charge turns out to be finite at r ¼ 0, in contrast to the well-known 1=r 2 singularity of the Coulomb field in Maxwell-Lorentz electrodynamics. Moreover, the characteristic surface, the light cone, depends on the field strength, and the superposition principle for the electromagnetic field does not hold any longer. Black hole solutions to the Born-Infeld (BI) nonlinear electrodynamics have been found first by Hoffmann [5] in 1935 and later by Salazar et al. [6] in 1987. Then, Plebański [1] postulated a more general nonlinear electrodynamics, which contains the Born-Infeld theory as special case. Parity violating terms could emerge in Plebański nonlinear electrodynamics [7]. A regular black hole solution to this theory has been obtained by Ayón–Beato et al. [2] in 1998. 1 Additionally, Bretón in 2002 [8] studied the trajectories of test particles in a geometry that is the Born-Infeld nonlinear electromagnetic generalization of the Reissner-Nordström solution. Currently, there exists a revival of interest in nonlinear electrodynamics since the effective theory arising from superstrings is an electrodynamics of the Born-Infeld type [9–12]. Besides, much attention has been deserved to the interpretation of the solutions to the Born-Infeld equations as states of D-branes [13]. Moreover, quantum electrodynamical vacuum correc- tions to the Maxwell-Lorentz theory can be accounted for by an effective nonlinear theory derived by Euler and Heisenberg [14,15]. The vacuum is treated as a specific type of medium, the polarizability and magnetizability properties of which are determined by the clouds of virtual charges surrounding the real currents and charges [16]. Recently, Brodin et al. [17] proposed a possible direct measurement of the Euler-Heisenberg effect. This theory is a valid physical theory [18], and it is the low field limit of the Born-Infeld one [19]. On the other hand, the concepts of a black hole shadow surrounded by a photon ring have played an important role since they are crucial for the interpretation of the obser- vations recently reported by the Event Horizon Telescope team from the supermassive black hole encountered at the * amaro@xanum.uam.mx † amac@xanum.uam.mx 1 The curvature invariants mentioned above are nonsingular. We do not know if its geodesic completeness has been studied. PHYSICAL REVIEW D 102, 104054 (2020) 2470-0010=2020=102(10)=104054(14) 104054-1 © 2020 American Physical Society