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We show that the classical equations of motion for a particle on threedimensional fuzzy space and on the fuzzy sphere are underpinned by a naturalLorentz geometry. From this geometric perspective, the equations of motiongenerally correspond to forced geodesic motion, but for an appropriate choiceof noncommutative dynamics, the force is purely noncommutative in origin andthe underpinning Lorentz geometry some standard space-time with, in general,non-commutatuve corrections to the metric. For these choices of thenoncommutative dynamics the commutative limit therefore corresponds to geodesicmotion on this standard space-time. We identify these Lorentz geometries to bea Minkowski metric on $\mathbb{R}^4$ and $\mathbb{R} \times S ^2$ in the casesof a free particle on three dimensional fuzzy space ($\mathbb{R}^3_\star$) andthe fuzzy sphere ($S^2_\star$), respectively. We also demonstrate theequivalence of the on-shell dynamics of $S^2_\star$ and a relativistic chargedparticle on the commutative sphere coupled to the background magnetic field ofa Dirac monopole.

Fuzzy Classical Dynamics as a Paradigm for Emerging Lorentz Geometries