Investigation of Finsler geometry as a generalization to curved spacetime of Planck-scale-deformed relativity in the de Sitter case

Over the last few years, Planck-scale modifications to the dispersion relation of particles have been deeply studied for the possibility to formulate some phenomenology of Planckian effects in astrophysical and cosmological frameworks. There are some indications [F. Girelli, S. Liberati, and L. Sindoni, Phys. Rev. D 75, 064015 (2007)] that Finsler geometry can provide some generalization of Riemannian geometry which may allow us to account for the nontrivial (Planckian) structure of the relativistic particles’ configuration space. We investigate the possibility to formalize Planck-scale deformations to relativistic models in curved spacetime, within the framework of Finsler geometry. We take into account the general strategy of analysis of dispersion relations modifications in curved spacetimes proposed in [G. Rosati, G. Amelino-Camelia, A. Marciano, and M. Matassa, Phys. Rev. D 92, 124042 (2015)], generalizing to the de Sitter case the results obtained in [G. Amelino-Camelia, L. Barcaroli, G. Gubitosi, S. Liberati, and N. Loret, Phys. Rev. D 90, 125030 (2014)] for deformed relativistic particle kinematics in flat spacetime using Finsler formalism.

Figure 1

Two massless particles, a high energetic one (blue lines) and a low energetic one (red line), are emitted simultaneously at the origin. We can observe that the high energetic particle anticipates (dashed line) or follows (dotted line) the infrared one, whether the sum of parameters $\beta$ and $\gamma$ is, respectively, $\beta +\gamma <0$ or $\beta +\gamma >0$.

Figure 2

Particles with different energies experience a different cosmological redshift. The dashed-blue line shows the deformation effect for $\beta +\gamma >0$, while the dotted-blue one for $\beta +\gamma <0$. The red line represents the classical undeformed cosmological redshift effect, for a low-energetic particle.