Area, Entanglement Entropy and Supertranslations at Null Infinity

The area of a cross-sectional cut $\Sigma$ of future null infinity ($\mathcal{I}^+$) is infinite. We define a finite, renormalized area by subtracting the area of the same cut in any one of the infinite number of BMS-degenerate classical vacua. The renormalized area acquires an anomalous dependence on the choice of vacuum. We relate it to the modular energy, including a soft graviton contribution, of the region of $\mathcal{I}^+$ to the future of $\Sigma$. Under supertranslations, the renormalized area shifts by the supertranslation charge of $\Sigma$. In quantum gravity, we conjecture a bound relating the renormalized area to the entanglement entropy across $\Sigma$ of the outgoing quantum state on $\mathcal{I}^+$.