Attractive gravity probe surface in the Einstein-Maxwell system

We derive areal inequalities for five types of attractive gravity probe surfaces, which were proposed by us in order to characterize the strength of gravity in different ways including weak gravity region, taking into account of contributions of electric and magnetic charges, angular momentum, gravitational waves, and matters. These inequalities are generalizations of the Riemannian Penrose inequality for minimal surfaces and lead to the concept of extremality for a given surface whose condition is given in terms of the gravitational mass and the electromagnetic charges. This means that the extremality is a characteristic property not only of black hole horizons or minimal surfaces but also of surfaces in weak gravity region. We also derive areal inequalities and extremality conditions for surfaces in asymptotically locally anti-de Sitter spacetimes.