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Energy-Momentum Surfaces: A Differential Geometric Framework for Dispersion Relations

Gines R. Perez Teruel

TL;DR

The paper develops a differential-geometric framework in which dispersion relations are treated as energy–momentum surfaces in $\, ext{R}^3$, enabling the use of tangent spaces, first and second fundamental forms, and curvature to study kinematical constraints. The Newtonian relation yields a developable, flat surface with no critical points, while Special Relativity produces a hyperbolic-paraboloid embedding with a single saddle at the origin and globally negative Gaussian curvature, reflecting the universal light cone. Modified dispersion relations can introduce additional critical points and regions of positive Gaussian curvature, signaling new invariant scales or thresholds. This unifying approach provides a robust geometric diagnostic to distinguish factorizable MDRs (Gravity’s Rainbow-style) from genuinely new deformations and offers a transparent language for exploring Lorentz-invariance violations and Planck-scale phenomenology.

Abstract

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.

Energy-Momentum Surfaces: A Differential Geometric Framework for Dispersion Relations

TL;DR

The paper develops a differential-geometric framework in which dispersion relations are treated as energy–momentum surfaces in , enabling the use of tangent spaces, first and second fundamental forms, and curvature to study kinematical constraints. The Newtonian relation yields a developable, flat surface with no critical points, while Special Relativity produces a hyperbolic-paraboloid embedding with a single saddle at the origin and globally negative Gaussian curvature, reflecting the universal light cone. Modified dispersion relations can introduce additional critical points and regions of positive Gaussian curvature, signaling new invariant scales or thresholds. This unifying approach provides a robust geometric diagnostic to distinguish factorizable MDRs (Gravity’s Rainbow-style) from genuinely new deformations and offers a transparent language for exploring Lorentz-invariance violations and Planck-scale phenomenology.

Abstract

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.
Paper Structure (26 sections, 1 theorem, 98 equations, 4 figures)

This paper contains 26 sections, 1 theorem, 98 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Energy--momentum parametric surfaces
  3. The Newtonian energy--momentum surface
  4. Tangent and normal vectors
  5. First and second fundamental forms
  6. Curvatures
  7. Critical points and discussion
  8. The hyperboloid and its hyperbolic paraboloid embedding: Special Relativity as a parametric surface
  9. Tangent and normal vectors
  10. First and second fundamental forms
  11. Curvature and critical points
  12. Relation with Gravity's Rainbow
  13. A geometric mapping to the SR mass shell
  14. Pullback of the first fundamental form
  15. Leading-order expansion in $E/M_{Pl}$
  16. ...and 11 more sections

Key Result

Proposition 1

Let a modified dispersion relation be written as with $g\in C^2(\mathbb{R})$. The associated energy--momentum surface $\mathbf{r}(E,p)=(E,p,f(E,p))$ satisfies:

Figures (4)

  • Figure 1: Gaussian curvature of the Lorentzian paraboloid. The surface is everywhere negatively curved, with asymptotic directions along $E=\pm p$, corresponding to the massless dispersion relation.
  • Figure 2: Classification of modified dispersion relations (MDRs) in the embedding framework. Factorizable MDRs (e.g. Rainbow/DSR) are diffeomorphic to the SR hyperboloid and introduce no new geometry. Non-factorizable MDRs (e.g. polynomial, logarithmic, exponential) cannot be straightened by a diffeomorphism, leading to genuinely new geometric structures.
  • Figure 3: Gaussian curvature of the MDR surface for $n=3$, $\kappa_{A}>0$. Unlike the Lorentzian paraboloid, the curvature is not everywhere negative: it varies with momentum and can change sign, reflecting the richer kinematical structure implied by the MDR.
  • Figure 4: Gaussian curvature maps of the dispersion surfaces. (a) Newtonian case: flat with $K=0$ and no critical points. (b) Special Relativity: strictly negative curvature, with a unique saddle-type critical point at the origin. (c) Modified dispersion relation ($n=3$, $\kappa=-1$, $M_{\rm Pl}=1$): curvature of variable sign and an additional critical point, signaling a new kinematical threshold.

Theorems & Definitions (2)

  • Proposition 1: Critical points and curvature of general MDR surfaces
  • proof : Sketch of proof