ADM mass and surface-gravity inequality for stationary black holes

Let $M$ be the ADM mass, $\kappa$ the surface gravity, and $\mathcal{A}$ the event-horizon area of a 4-dimensional asymptotically flat, static or axisymmetric stationary black hole with “$t-\phi$” reflection isometry. Assume that the Einstein equation and the null energy condition hold, and that the cross-section of the event horizon has $S^2$ topology. ADM mass and surface-gravity conjecture. The two independent inequalities

$$3M\geq\frac{\kappa\mathcal{A}}{4\pi}+\sqrt{\frac{\mathcal{A}}{4\pi}}$$

and

$$2M\geq\sqrt{\frac{\mathcal{A}}{4\pi}}$$

hold. Equality is achieved only when the exterior is Schwarzschild. Moreover, a regular stationary singularity-free spacetime satisfies $M\geq0$, with equality only for Minkowski spacetime. The conjecture proposes a lower bound on ADM mass using only the null energy condition and incorporates the Penrose inequality; its validity is supported by the Kerr–Newman solution and a proof in the static spherically symmetric case, but the general stationary case remains open.