First we briefly review our covariant Hamiltonian approach to quasi-local energy, noting that the Hamiltonian-boundary-term quasi-local energy expressions depend on the chosen boundary conditions and reference configuration. Then we present the quasi-local energy values resulting from the formalism applied to homogeneous Bianchi cosmologies. Finally we consider the quasi-local energies of the FRW cosmologies. Our results do not agree with certain widely accepted quasi-local criteria.

We have proposed a program for determining the reference for the quasi-local energy defined in the covariant Hamiltonian formalism. Our program has been tested by applying it to the spherically symmetric spacetimes. With respect to different observers we found that the quasi-local energy can be positive, zero, or even negative. The observer measuring the maximum energy was identified; the associated energy is positive for both the Schwarzschild and the Friedmann-Lemaître-RobertsonWalker spacetimes.PACS numbers: 04.20. Cv, 04.20.Fy, 98.80.Jk Introduction. An outstanding fundamental problem in general relativity is that there is no proper definition for the energy density of gravitating systems. (This can be understood as a consequence of the equivalence principle.) The modern concept is that gravitational energy should be non-local, more precisely quasi-local, i.e., it should be associated with a closed two-surface (for a comprehensive review see [1]). Here we consider one proposal based on the covariant Hamiltonian formalism [2] wherein the quasi-local energy is determined by the Hamiltonian boundary term. For a specific spacetime displacement vector field on the boundary of a region (which can be associated with the observer), the associated quasi-local energy depends not only on the dynamical values of the fields on the boundary but also on the choice of reference values for these fields. Thus a principal issue in this formalism is the proper choice of reference spacetime for a given observer.

Abstract. The Hamiltonian for dynamic geometry generates the evolution of a spatial region along a vector field. It includes a boundary term which determines both the value of the Hamiltonian and the boundary conditions. The value gives the quasi-local quantities: energy-momentum, angularmomentum/center-of-mass. The boundary term depends not only on the dynamical variables but also on their reference values; the latter determine the ground state (having vanishing quasi-local quantities). For our preferred boundary term for Einstein's GR we propose 4D isometric matching and extremizing the energy to determine the reference metric and connection values.

The specification of energy for gravitating systems has been an unsettled issue since Einstein proposed his pseudotensor. It is now understood that energy-momentum is quasi-local (associated with a closed 2-surface). Here we consider quasi-local proposals (including pseudotensors) in the Lagrangian-Noether-Hamiltonian formulations. There are two ambiguities: (i) there are many possible expressions, (ii) they depend on some non-dynamical structure, e.g., a reference frame. The Hamiltonian approach gives a handle on both problems. The Hamiltonian perspective helped us to make a remarkable discovery: with an isometric Minkowski reference a large class of expressions-namely all those that agree with the Einstein pseudotensor's Freud superpotential to linear order-give a common quasi-local energy value. Moreover, with a bestmatched reference on the boundary this is the Wang-Yau mass value.