Abstract
We study the free motion of a massive particle moving in the background of a Finslerian deformation of a plane gravitational 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 certain deformation of Cohen and Glashow's very special relativity group $\mathrm{ISIM}(2)$. 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 gravitational wave for any value of the Bogoslovsky-Finsler parameter $b$. We then replace the underlying plane gravitational wave with a homogeneous $pp$-wave solution of the Einstein-Maxwell equations. We conclude by extending the theory to the Finsler-Friedmann-Lema\^{\i}tre model.
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