Quasi-local energy and Oppenheimer-Snyder collapse

We consider a scenario of the gravitational collapse of the Oppenheimer-Snyder dust cloud with spatially constant matter density. Given a closed two-surface within the star, three versions of the quasi-local energy are investigated. We show that, during the collapse, (i) the Geroch energy of the surface is nonpositive and increases to zero; (ii) the Hawking energy of the surface is monotonically increasing and approaches to the Hayward energy of the initial surface; and (iii) the Hayward energy of the surface is conserved and remains positive. These results have no restrictions on the topology and the symmetry of the surface.