  We study the properties of the energy-momentum tensor of gauge fields coupled
to matter in non-commutative (Moyal) space. In general, the non-commutativity
affects the usual conservation law of the tensor as well as its transformation
properties (gauge covariance instead of gauge invariance). It is known that the
conservation of the energy-momentum tensor can be achieved by a redefinition
involving another star-product. Furthermore, for a pure gauge theory it is
always possible to define a gauge invariant energy-momentum tensor by means of
a Wilson line. We show that the latter two procedures are incompatible with
each other if couplings of gauge fields to matter fields (scalars or fermions)
are considered: The gauge invariant tensor (constructed via Wilson line) does
not allow for a redefinition assuring its conservation, and vice-versa the
introduction of another star-product does not allow for gauge invariance by
means of a Wilson line.
