Nonlinear relativity in position space

We propose a new method for obtaining the position space of nonlinear relativity, i.e., the dual of its usual momentum space formulation. We require that plane waves still be solutions to free field theory. This is equivalent to postulating the invariance of the linear contraction between position and momentum spaces, and dictates a set of energy-dependent space-time Lorentz transformations. In turn this leads to an energy-dependent metric. An alternative, more problematic approach, would allow for position and momentum space to acquire independent nonlinear representations of the Lorentz group ; this requires a nonlinear contraction between momentum and position spaces. We discuss a variety of physical implications of these approaches and show how they point to two rather distinct formulations of theories of gravity with an invariant energy and/or length scale.