Geometric Constraints on Quantum Gravity-Inspired Dispersion Relations

TL;DR

This work develops a coordinate-invariant geometric framework to analyze modified dispersion relations (MDRs) arising in quantum gravity. By embedding each MDR as an off-shell surface in energy–momentum space and studying its Gaussian curvature, the authors diagnose hyperbolicity (K<0), potential instabilities (K>0), and thresholds via tangency conditions. Applying this method to non-polynomial MDRs (logarithmic, exponential, trigonometric) and to Loop Quantum Gravity (LQG)–motivated MDRs, they derive model-independent constraints and show that LQG-inspired MDRs remain strictly hyperbolic in the phenomenologically relevant regime. They also demonstrate that Doubly Special Relativity (DSR) realizations correspond to a reparametrization of SR with no new intrinsic curvature, while providing quantitative bounds on non-polynomial MDR parameters from high-energy observations. Overall, the geometric framework offers a robust bridge between quantum-gravity kinematics and empirical data, yielding universal stability criteria and thresholds beyond the reach of EFT.

Abstract

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.~\cite{GRP}, 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.

Figures (3)

• Figure 1: Gaussian curvature $K(E,p)$ of representative logarithmic, exponential, cubic, and trigonometric MDRs. Red regions indicate elliptic domains ($K>0$), associated with loss of hyperbolicity, while blue regions mark hyperbolic domains ($K<0$) corresponding to stable propagation.Each panel shows the onset of curvature sign changes and potential critical structures characteristic of each MDR class.
• Figure 2: Gaussian curvature $K(E,p)$ for the four principal LQG–motivated modified dispersion relations: (a) polymeric (sine) MDR; (b) holonomy–induced MDR in the energy; (c) inverse–triad MDR; and (d) semiclassical/DSR-like MDR. The solid white curve indicates the physical mass shell $f(E,p)=0$, while the dashed black curve marks the locus $K=0$ separating hyperbolic ($K<0$) from elliptic ($K>0$) regions. In all cases the entire phenomenologically relevant part of the mass shell lies strictly within the hyperbolic domain, demonstrating the kinematical robustness of LQG-inspired MDRs at sub-Planckian energies.
• Figure 3: Exclusion lines for logarithmic (a) and exponential (b) MDRs. For each messenger—photons ($E\!\lesssim\!100$--$300$ TeV), neutrinos ($E\!\lesssim\!1$ PeV), and UHECR ($E\!\lesssim\!1$ EeV)—the colored curve marks the threshold at which vacuum decay or related instabilities would already occur. Parameter values below each curve are excluded; the region above the curves is phenomenologically viable.