GEOMETRODYNAMICS VS CONNECTION DYNAMICS

The purpose of this review is to describe in some detail the mathemati cal relationship between geometrodynamics and connection dynamics in t he context of the classical theories of 2+1 and 3+1 gravity. We analyz e the standard Einstein-Hilbert theory (in any spacetime dimension), t he Palatini and Chem-Simons theories in 2+1 dimensions, and the Palati ni and self-dual theories in 3+1 dimensions. We also couple various ma tter fields to these theories and briefly describe a pure spin-connect ion formulation of 3+1 gravity. We derive the Euler-Lagrange equations of motion from an action principle and perform a Legendre transform t o obtain a Hamiltonian formulation of each theory. Since constraints a re present in all these theories, we construct constraint functions an d analyze their Poisson bracket algebra. We demonstrate, whenever poss ible, equivalences between the theories.