Next, we study initial data sets $(M,g,k)$ which are used in General Relativity to describe isolated gravitational systems. We introduce spacetime harmonic functions, i.e. functions solving the PDE $\Delta u=-\text{tr}_gk|\nabla u|$, to give a new lower bound for the mass of $(M,g,k)$. This lower bound in particular implies the spacetime positive mass theorem including the case of equality.

Finally, we discuss recent progress towards the spacetime Penrose conjecture. We demonstrate how the famous monotonicity formula for the Hawking energy under inverse mean curvature flow can be generalized to initial data sets. This leads to a new notion of spacetime inverse mean curvature flow which is based on double null foliations.

Several of the above results have been obtained in collaboration with Simon Brendle, Florian Johne, Demetre Kazaras, Marcus Khuri and Yiyue Zhang.