Hamiltonian expression of curvature tensors in the York canonical basis: (I) Riemann tensor and Ricci scalars

Abstract

By using the York canonical basis of ADM tetrad gravity, in a formulation using radar 4-coordinates for the parametrization of the 3+1 splitting of the space-time, it is possible to write the 4-Riemann tensor of a globally hyperbolic, asymptotically Minkowskian space-time as a Hamiltonian tensor, whose components are 4-scalars with respect to the ordinary world 4-coordinates, plus terms vanishing due to Einstein's equations. Therefore, "on-shell" we find the expression of the Hamiltonian 4-Riemann tensor. Moreover, the 3+1 splitting of the space-time used to define the phase space allows us to introduce a Hamiltonian set of null tetrads and to find the Hamiltonian expression of the 4-Ricci scalars of the Newman–Penrose formalism. This material will be used in the second paper to study the 4-Weyl tensor, the 4-Weyl scalars and the 4-Weyl eigenvalues and to clarify the notions of Dirac and Bergmann observables.