  The possible role of gravity in a noncommutative geometry is investigated.
Due to the Moyal *-product of fields in noncommutative geometry, it is
necessary to complexify the metric tensor of gravity. We first consider the
possibility of a complex Hermitian, nonsymmetric $g_{\mu\nu}$ and discuss the
problems associated with such a theory. We then introduce a complex symmetric
(non-Hermitian) metric, with the associated complex connection and curvature,
as the basis of a noncommutative spacetime geometry. The spacetime coordinates
are in general complex and the group of local gauge transformations is
associated with the complex group of Lorentz transformations CSO(3,1). A real
action is chosen to obtain a consistent set of field equations. A Weyl
quantization of the metric associated with the algebra of noncommuting
coordinates is employed.
