Ambiguities in the Seiberg-Witten map and emergent gravity

The θ 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.