A Hamiltonian formulation of the Einstein-Cartan-Sciama-Kibble theory of gravity

Abstract

A B S T R ACT. We postulate the energy-momentum function E for the ECSK theory of gravity and formulate the functional Hamiltonian equation in terms of the energy-momentum function E and the symplectic 2-form g2. The system of partial differential equations which follows from the functional Hamilton equation is equivalent to the system of variational equations of the ECSK theory. The Hamiltonian method gives rise to a natural division of these equations into i0 constraint equations and the set of dynamical equations. We discuss the geometric sense of the constraint equations and their relations to the initial value problem.

An SU(2)-covariant dynamical formulation of the Einstein-Cartan-Dirac theory

“…As was shown by Szczyrba (1982) there exists a unique boost transformation B'"'(P) such that the tetrad (3.4) ;(a) = B-'(")…”

“…It can be shown, however, that equations (4.18) follow directly from (4.17). The 39d,-differentiation of the relations (4.17b) and appropriate contractions lead to (4.18) (cf Frqckiewicz and Szczyrba 1982). Therefore, equations (4.13)-(4.16) together with relations (4.17) constitute a complete system of field equations proper for the discussion of the Cauchy-Kowalewska initial problem.…”

“…The occurrence of the mAB terms in the formula for the symplectic 2-form is a consequence of the 'caret' transformation we have performed to pass from (4.1) to (4.4). Such terms are typical for the (3 + 1) representation of symplectic forms in gauge theories of gravity (Szczyrba 1982). The complicated formula (A3.4) for mAB does not lead to any calculation problems because for solutions of field equations mAB = 0.…”

“…where 3* is the Hodge operator on the surface U (cf appendix 1). In local coordinates we may write The quantities eSA& represent a field of SU(2) bases built from vector densities on U, K / B represent the second fundamental form of the embedding U + M (Szczyrba 1982). Let us emphasise that the gravitational symplectic variables (4.6) diagonalise the gravitational part of the symplectic 2-form and are kinematically independent.…”