Stephenson-Kilmister-Yang theory of gravity and its dynamics.

Abstract

We investigate the Stephenson-Kilmister-Yang (SKY) gravitational Lagrangian in the framework of SO(3,1) gauge theories. It is proved that by an appropriate choice of dynamical variables this theory can be cast in a Hamiltonian form. Thirty-six dynamical variables ${X}^{k}$${\mathrm{}}_{(\ensuremath{\alpha})(\ensuremath{\beta})}$,${Y}^{k(\ensuremath{\alpha})(\ensuremath{\beta})}$ resemble the electric field and the magnetic induction in Yang-Mills theories of internal symmetries. Their evolution is governed by a Maxwell-type system of differential equations. There are 16 constraints for the initial values of the dynamical variables, of which 10 are first class and 6 are second class. The full gauge group of the theory is parametrized by 13 ``functions'' on spacetime and is essentially larger than the 10-parameter full gauge group of generic SO(3,1) theories of gravity. Additional 3-parameter gauge transformations in the set of field variables are generated by a nonstandard action of Lorentz boosts. There are 3 gauge variables related to these transformations. Only 10 of the 13 gauge transformations act independently in the set of dynamical variables. Therefore the theory has 36-(16+10)=10 independent degrees of freedom in the phase space. It is also shown that the SKY gravity naturally couples to matter Yang-Mills fields maintaining all of its features. A brief discussion of a conceivable coupling with vector matter fields is presented.