I will present joint work with Saradha Senthil Velu, in which we study the evolution of geometric invariants in asymptotically hyperboloidal initial data sets, focusing on the notions of energy and linear momentum. The evolution of such initial data sets is described using a hyperboloidal time function. A key element of our approach is the introduction of the concept of E–P chargeability—a property of the initial data set that ensures the energy and momentum charges are well defined—and our proof that this property is preserved along the chosen evolution. A central result of this work is the derivation of energy-loss and linear momentum-loss formulae, recovering the well-known results of Bondi, Sachs, and Metzner—but under weaker asymptotic assumptions. Our approach differs from those using conformal compactifications, as we work directly at the level of the initial data set. If time permits, I will conclude with a discussion of further developments and current work in progress.