Black hole solutions in Euler-Heisenberg theory

We construct static and spherically symmetric black hole solutions in the Einstein-Euler-Heisenberg (EEH) system which is considered as an effective action of a superstring theory. We consider electrically charged, magnetically charged, and dyon solutions. We can solve analytically for the magnetically charged case. We find that they have some remarkable properties about causality and black hole thermodynamics depending on the coupling constant of the EH theory a and b, though they have a central singularity as in the Schwarzschild black hole. We restrict $a>\mathrm{}0\mathrm{}$ because it is natural if we think of EH theory as a low-energy limit of the Born-Infeld (BI) theory. (i) For the magnetically charged case, whether or not the extreme solution exists depends on the critical parameter ${a=a}_{\mathrm{crit}}.$ For $a<~a_{\mathrm{crit}},$ there is an extreme solution as in the Reissner-Nortström (RN) solution. The main difference from the RN solution is that there appear solutions below the horizon radius of the extreme solution and they exist till $r_H\to 0.$ Moreover, for $a>\mathrm{}{a}_{\mathrm{crit}},$ there is no extreme solution. For arbitrary a, the temperature diverges in the $r_H\to 0$ limit. (ii) For the electrically charged case, the inner horizon appears under some critical mass $M_0$ and the extreme solution always exists. The lower limit of the horizon radius decreases when the coupling constant a increases. (iii) For the dyon case, we expect a variety of properties because of the term $b({\epsilon }_{\mu \nu \rho \sigma }{F}^{\mu \nu }{F}^{\rho \sigma }{)}^2$ which is peculiar to the EH theory. But their properties are mainly decided by the combination of the parameters $a+8b.\mathrm{}$ We show that solutions have similar properties to the magnetically charged case in the $r_H\to 0$ limit for $a+8b<~0.$ For $a+8b>\mathrm{}0,$ it depends on the parameters $a,b.\mathrm{}$