Quasi-local energy in General Relativity

We discuss properties of the Wang--Yau quasi-local mass of a surface. The mass is defined by minimizing certain Hamiltonian energy among isometric embeddings of the given surface into the Minkowski space. We evaluate this quasi-local mass in several cases, including the large sphere limit at null infinity, the small sphere limit around a point and on apparent horizons.;We start by analyzing the Euler--Lagrange equation for this Hamiltonian energy. Under the compatibility condition of mean curvature, we are able to find a local minimum which is close to the embedding into R3 . Then we evaluate the quasi-local energy with respect to this embedding. We show that the limit satisfies the properties one expected.;We also study the Christodoulou memory effect at null infinity for spacetimes satisfying the Einstein--Maxwell equation. The non-linear effect relates the Bondi mass loss formula to the change of the distance between time-like geodesics, which could be observed by a gravitational wave experiment.