The definition of an energy for the gravitational field remains a challenging problem in general relativity. In this talk, the Hawking energy associated with the past lightcone of a point in a cosmological spacetime is used to assign an energy to the observable universe, and its positivity and monotonicity in the weak gravitational lensing regime is shown. In the strong lensing regime, when the lightcone exhibits self-intersections and wave-front singularities, it remains well-defined for swallow-tail singularities, but in general has no positivity or monotonicity property anymore. The Hawking energy can be used to compare a spacetime domain in an inhomogeneous universe with a homogeneous one and may be used to arrive at bounds on the cosmic fluid density and equation-of-state parameters. This talk is based on ArXiv 2003.13583 & 2010.07896

Title: Multipole moments of stationary space-times

Abstract:
Multipole moments are defined for stationary, asymptotically flat, source-free solutions of Einstein's equation. There arise two sets of multipole moments, the mass moments and the angular momentummoments. These quantities emerge as tensors at a point A "at spatial infinity." They may be expressed as certain combinations of the derivatives at A of the norm and twist of the timelike Killing vector. In the Newtonian limit, the moments reduce to the usual multipole moments of the Newtonian potential. Some properties of these moments are derived, and, as an example, the multipole moments of the Kerr solution are discussed.