  We show that the matrix-model action for noncommutative U(n) gauge theory
actually describes SU(n) gauge theory coupled to gravity. This is elaborated in
the 4-dimensional case. The SU(n) gauge fields as well as additional scalar
fields couple to an effective metric G_{ab}, which is determined by a dynamical
Poisson structure. The emergent gravity is intimately related to
noncommutativity, encoding those degrees of freedom which are usually
interpreted as U(1) gauge fields. This leads to a class of metrics which
contains the physical degrees of freedom of gravitational waves, and allows to
recover e.g. the Newtonian limit with arbitrary mass distribution. It also
suggests a consistent picture of UV/IR mixing in terms of an induced gravity
action. This should provide a suitable framework for quantizing gravity.
