DOI: 10.36890/iejg.1949878 ISSN: 1307-5624

Energy Momentum Relations and Geometric Representations

Melek Erdoğdu, Ayşe Yavuz
Show PDF Cite
his study investigates energy-momentum dispersion relations as intrinsic geometric objects by employing the mathematical framework of Monge surfaces within Euclidean space. Traditionally treated as algebraic constraints, these relations are here reinterpreted through differential geometry to reveal the structural and properties in their surface representations. The research provides a comprehensive comparative analysis between the Newtonian energy momentum surface and the relativistic dispersion relation, modeled as a hyperbolic paraboloid. Using information such as the Weingarten map, Christoffel symbols, and Gaussian and mean curvatures, the local and global geometric characterizations of these surfaces are explicitly derived. Furthermore, the study characterizes the parameter curves (E-curves and p-curves) on these surfaces through the Darboux frame, calculating their normal curvature, geodesic curvature, and geodesic torsion to demonstrate how kinematic information is encoded within the curve geometry. The results emphasize that the developability, rigidity, and curvature properties of these surfaces provide a natural classification for extended kinematic models, offering a deeper understanding of the necessary connection between physical dispersion relations and surface geometry.

More from our Archive