Dynamics of Geometric Invariants in the Asymptotically Hyperboloidal Setting: Energy and Linear Momentum

Abstract: We investigate the evolution of geometric invariants, as defined by Michel \cite{Michel}, in the context of asymptotically hyperboloidal initial data sets. Our focus lies on the charges of energy and linear momentum, and we study their behavior under the Einstein evolution equations. We construct foliations describing the evolution of asymptotically hyperboloidal initial data sets using hyperboloidal time function. We define E-P chargeability as a property of the initial data set, and we show that it is preserved under the evolution for our choice of time function. This ensures that the charges are well-defined along the evolution, which is crucial for our approach. Along such foliations, we recover the same energy-loss and linear momentum-loss formulae as those derived by Bondi, Sachs, and Metzner \cite{Bondi-vanderBurg-Metzner} while operating under weaker asymptotic assumptions. Our approach is distinct from previous work as we do not utilize conformal compactifications and work directly at the level of the initial data set.