Comments on the non-commutative description of classical gravity

We find a one-parameter family of Lagrangian descriptions for classical gen eral relativity in terms of tetrads which are not c-numbers. Rather, they o bey exotic commutation relations. These noncommutative properties drop out in the metric sector of the theory, where the Christoffel symbols and the R iemann tensor are ordinary commuting objects and they are given by the usua l expression in terms of the metric tensor. Although the metric tensor is n ot a c-number, we argue that all measurements one can make in this theory a re 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 t orsion) is physically equivalent to Einstein's general relativity. (C) 1998 Published by Elsevier Science B.V. All rights reserved.