When null energy condition meets ADM mass

We give a conjecture on the lower bound of the ADM mass $M$ by using the null energy condition. The conjecture includes a Penrose-like inequality $3M\geqκ\mathcal{A}/(4π)+\sqrt{\mathcal{A}/4π}$ and the Penrose inequality $2M\geq\sqrt{\mathcal{A}/{4π}}$ with $\mathcal{A}$ the event horizon area and $κ$ the surface gravity. Both the conjecture in the static spherically symmetric case and the Penrose inequality for a dynamical spacetime with spherical symmetry are proved by imposing the null energy condition. We then generalize the conjecture to a general dynamical spacetime. Our results raise a new challenge for the famous unsettled question in general relativity: in what general case can the null energy condition replace other energy conditions to ensure the Penrose inequality?

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