Bounds on M/R for charged objects with positive cosmological constant

We consider charged spherically symmetric static solutions of the Einstein–Maxwell equations with a positive cosmological constant Λ. If r denotes the area radius, m g and q the gravitational mass and charge of a sphere with area radius r respectively, we find that for any solution which satisfies the condition p + 2 p ⊥ ⩽ ρ, where p ⩾ 0 and p ⊥ are the radial and tangential pressures respectively, ρ ⩾ 0 is the energy density, and for which the inequality holds. We also investigate the issue of sharpness, and we show that the inequality is sharp in a few cases but generally this question is open.