In this talk, we begin with an introduction to the ADM mass in general relativity, a quantity derived with a hamiltonian approach that represents the total energy of an isolated system. Such systems are modeled by asymptotically flat manifolds, which describe gravitating systems where all matter is confined in a compact region. We will then present a formula for this mass as the limit of the total mean curvature plus the total defect of dihedral angle of the boundary of large polyhedra. We will then discuss some geometric implications.
Next, we turn to a question motivated by cosmology: real astrophysical objects are not truly isolated but exist within an expanding universe. To address this, we introduce a definition of energy suited to an expanding cosmological setting, using the flat expanding de Sitter model as the background spacetime instead of the traditional Minkowski space. This shift necessitates a quasi-local definition of energy due to the presence of a cosmological horizon.
Parts of this talk are based on joint work with Pengzi Miao, and parts on joint work with Eric Ling and Rodrigo Avalos.
