Area inequalities for stable marginally outer trapped surfaces in Einstein-Maxwell-dilaton theory

Abstract

We prove area inequalities for stable marginally outer trapped surfaces in Einstein-Maxwell-dilaton theory. Our inspiration comes on the one hand from a corresponding upper bound for the area in terms of the charges obtained recently by Dain, Jaramillo and Reiris in the pure Einstein-Maxwell case without symmetries, and on the other hand from Yazadjiev's inequality in the axially symmetric Einstein-Maxwell-dilaton case. The common issue in these proofs and in the present one is a functional $\mathcal{W}$ of the matter fields for which the stability condition readily yields an upper bound. On the other hand, the step which crucially depends on whether or not a dilaton field is present is to obtain a lower bound for $\mathcal{W}$ as well. We obtain the latter by first setting up a variational principle for $\mathcal{W}$ with respect to the dilaton field $\phi$, then by proving existence of a minimizer $\psi$ as solution of the corresponding Euler-Lagrange equations and finally by estimating $\mathcal{W} (\psi)$. In the special case that the normal components of the electric and magnetic fields are proportional we obtain the area bound $A \geq 8\pi PQ$ in terms of the electric and magnetic charges. In the generic case our results are less explicit but imply rigorous 'perturbation' results for the above inequality. All our inequalities are saturated for a 2-parameter family of static, extreme solutions found by Gibbons. Via the Bekenstein-Hawking relation $A = 4S$ our results give positive lower bounds for the entropy $S$ which are particularly interesting in the Einstein-Maxwell-dilaton case.