A new formal scheme is presented in which Einstein's classical theory of
General Relativity appears as the common, invariant sector of a one-parameter
family of different theories. This is achieved by replacing the Poincare` group
of the ordinary tetrad formalism with a q-deformed Poincare` group, the usual
theory being recovered at q=1. Although written in terms of noncommuting
vierbein and spin-connection fields, each theory has the same metric sector
leading to the ordinary Einstein-Hilbert action and to the corresponding
equations of motion. The Christoffel symbols and the components of the Riemann
tensor are ordinary commuting numbers and have the usual form in terms of a
metric tensor built as an appropriate bilinear in the vierbeins. Furthermore we
exhibit a one-parameter family of Hamiltonian formalisms for general
relativity, by showing that a canonical formalism a` la Ashtekar can be built
for any value of q. The constraints are still polynomial, but the Poisson
brackets are not skewsymmetric for q different from 1.Comment: LaTex file, 21 pages, no figure
