Evolution of Energy and Linear Momentum in Asymptotically Hyperboloidal Initial Data Sets

Abstract

I will discuss the evolution of geometric invariants in asymptotically hyperboloidal initial data sets, focusing on energy and linear momentum as defined by Michel. Using a hyperboloidal time function to describe the evolution of these initial data sets, I introduce the concept of E-P chargeability, a property of the initial data set, and show that it remains preserved under the Einstein evolution equations. This preservation ensures that energy and momentum charges are well-defined along the evolution, forming the foundation of our approach.

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. This provides an alternative perspective for understanding mass loss in asymptotically hyperboloidal initial data sets.