  We find a one-parameter family of Lagrangian descriptions for classical
general relativity in terms of tetrads which are not c-numbers. Rather, they
obey exotic commutation relations. These noncommutative properties drop out in
the metric sector of the theory, where the Christoffel symbols and the Riemann
tensor are ordinary commuting objects and they are given by the usual
expression in terms of the metric tensor. Although the metric tensor is not a
c-number, we argue that all measurements one can make in this theory are
associated with c-numbers, and thus that the common invariant sector of our
one--parameter family of deformed gauge theories (for the case of zero torsion)
is physically equivalent to Einstein's general relativity.
