Geometry of physical dispersion relations

To serve as a dispersion relation, a cotangent bundle function must satisfy three simple algebraic properties. These conditions are derived from the inescapable physical requirements that local matter field dynamics must be predictive and allow for an observer-independent notion of positive energy. Possible modifications of the standard relativistic dispersion relation are thereby severely restricted. For instance, the dispersion relations associated with popular deformations of Maxwell theory by Gambini-Pullin or Myers-Pospelov are not admissible. Dispersion relations passing the simple algebraic checks derived here correspond to physically admissible Finslerian refinements of Lorentzian geometry.

Figure 1

Homogeneity of $P$ in its cotangent fiber gives rise to a cone of $P$-null covectors in each cotangent space. Three prototypical examples of such cones are shown, the first one being the familiar Lorentzian metric cone. Only for the second example do we have $N_x\ne N_x^{\mathrm{smooth}}$, with the difference set being constituted by the covectors lying in the intersection of the cones.

Figure 2

Hyperbolicity cones for two prototypical polynomials. On the left the familiar second-degree Lorentzian cone; on the right a fourth-degree cone defined, for simplicity, by a product of two Lorentzian metrics.

Figure 3

Gauss map sending the zero locus of a polynomial to the zero locus of a dual polynomial.

Figure 4

Example of a hyperbolic polynomial with nonhyperbolic dual polynomial.

Figure 5

An energy-distinguishing bihyperbolic dispersion relation.

Figure 6

Mass shells defined by bihyperbolic energy-distinguishing cotangent bundle functions $P$. On the left the familiar second-degree Lorentzian case; on the right a fourth-degree case defined by a product of two Lorentzian metrics.

Figure 7

Mass shell and Legendre map of massive momenta to tangent space.

Figure 8

Purely spatial directions with respect to $e_0$ are those annihilated by $L^{-1}(e_0)$. For the Lorentzian metric case on the left, this coincides with the space of vectors $g$-orthogonal to $e_0$.

Figure 9

Stability cone: If and only if an observer can ride on a particle, the particle cannot lose energy by a vacuum Cerenkov process.