Geometric Constraints on Quantum Gravity–Inspired Dispersion Relations

Modified dispersion relations (MDRs) arise in many quantum-gravity approaches, often in non-polynomial or non-analytic form beyond the reach of effective field theory (EFT). Logarithmic, exponential and trigonometric MDRs appear in causal set theory, nonlocal gravity and κ-Poincaré models, while Loop Quantum Gravity (LQG) yields polymeric (sine), holonomy, inverse-triad and semiclassical corrections. Using the geometric framework of Ref. [16], we analyse the intrinsic curvature of the associated energy-momentum surfaces, where negative curvature ensures hyperbolic and stable propagation, and curvature sign changes or critical points indicate kinematical instabilities or new invariant scales. We apply this method exhaustively to all major MDRs derived in LQG and find that they remain strictly hyperbolic in the entire phenomenologically relevant regime, with no elliptic patches or critical branching. The same framework provides universal constraints on representative logarithmic, exponential and trigonometric MDRs beyond EFT. Thus, geometric criteria yield a unified and coordinate-independent assessment of stability, thresholds and invariant scales, and demonstrate the robustness of MDRs emerging from LQG.