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 alpha and b, though they have a central singularity as in the Schwarzschild black hole. We restrict alpha > 0 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 alpha = alpha (crit). For alpha less than or equal to alpha (crit), there is an extreme solution as in the Reissner-Nortstrom (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) --> 0. Moreover, for alpha > alpha (crit), there is no extreme solution. For arbitrary alpha, the temperature diverges in the r(H) --> 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 alpha increases. (iii) For the dyon case, we expect a variety of properties because of the term b(epsilon (mu nu rho sigmaFFrho sigma)-F-mu nu)(2) which is peculiar to the EH theory. But their properties are mainly decided by the combination of the parameters alpha + 8b. We show that solutions have similar properties to the magnetically charged case in the r(H) --> 0 limit for alpha + 8b less than or equal to 0. For alpha + 8b > 0, it depends on the parameters alpha ,b.