Berwald spacetimes and very special relativity

In this work, we study Berwald spacetimes and their vacuum dynamics, where the latter are based on a Finsler generalization of Einstein's equations derived from an action on the unit tangent bundle. In particular, we consider a specific class of spacetimes that are nonflat generalizations of the very special relativity (VSR) line element, which we call "very general relativity" (VGR). We derive necessary and sufficient conditions for the VGR line element to be of Berwald type. We present two novel examples with the corresponding vacuum field equations: a Finslerian generalization of vanishing scalar invariant (VSI) spacetimes in Einstein's gravity as well as the most general homogeneous and isotropic VGR spacetime.