Geodesic motion in Bogoslovsky-Finsler Plane Gravitational Waves

We study the free motion of a massive particle moving in the background of a Finslerian deformation of a plane gravitional wave in Einstein's General Relativity. The deformation is a curved version of a one-parameter family of Relativistic Finsler structures introduced by Bogoslovsky, which are invariant under a deformation of Cohen and Glashow's Very Special Relativity group $ISIM(2)$. The partially broken Carroll Symmetry we derive using Baldwin-Jeffery-Rosen coordinates allows us to integrate the geodesics equations. The transverse coordinates of timelike Finsler-geodesics are identical to those of the underlying plane gavitational wave for any value of the Bogoslovsky-Finsler parameter $b$. We conclude by replacing the underlying plane gravitational wave by a homogenous pp-wave solution of the Einstein-Maxwell equations. The theory is extended to the Finsler-Friedmann-Lemaitre model.