The Analysis to Quasi-Local Energy and Hamiltonian Constraint based on Variation

Abstract

In this paper, by arising condition in variation, from equal time to non-equal time, I reconsider how geometrodynamics equations allow to be derived from variational principle in general relativity and then find the variation of extrinsic curvature dependent only locally on its induced metric and unit normal. I thus try to attribute the quasi-local energy to the integrability of submanifold. At last I discuss the dynamical degrees of freedom on Hamiltonian constraint by analyzing non-equal time variation which also represents a global transformation.