Energy–momentum surfaces: A differential geometric framework for dispersion relations

We propose a geometric framework where dispersion relations are viewed as parametric surfaces in energy–momentum space. Within this picture, the presence and type of critical points of the surface emerge as clear geometric signatures of kinematical restrictions. The Newtonian relation corresponds to a developable surface with no critical points, reflecting the absence of invariant limits. Special Relativity generates a saddle point and globally negative curvature, encoding the universal light cone. Modified dispersion relations may introduce additional critical points, signaling new invariant energy scales or thresholds. This unifying approach not only recasts known results in a transparent geometric language but also provides a simple diagnostic tool for exploring departures from Lorentz invariance and their physical implications.