  The theta expansion of the Seiberg-Witten map has ambiguities which can be
removed by a gauge transformation and/or a field redefinition. In the context
of emergent gravity such a field redefinition changes the emerging metric and
requires the presence of non-minimal gravitational couplings. It also requires
that a real scalar field becomes a scalar density and allows the introduction
of a potential. We also find that the potential can have only one term and that
a quartic interaction is not allowed. Even though the metric depends on the
ambiguity we show that the dispersion relation does not present any sign of it.
A proposal for an exact Seiberg-Witten map is used to derive the full metric
going beyond the linearized limit.
