  A homogeneous part of the Seiberg-Witten gauge equivalence relation for gauge
fields can lead to solutions that involve matter fields in such a way that the
gauge equivalence and the dimensional and index structures are preserved. In
particular, we consider scalar fields coupled to U(1) gauge fields. The matter
fields appear non-linearly in the map. As an application, we analyze the
implication of this ambiguity to emergent gravity at the first order in
noncommutative parameter and show that the new ambiguity restores the
possibility of conformal coupling of real scalar density field that is coupled
non-minimally to the emergent gravity induced by gauge fields -- a possibility
that is strictly not allowed if we consider only the already known ambiguity in
the Seiberg-Witten map.
