The Finsler spacetime condition for (α,β)-metrics and their isometries

For the general class of pseudo-Finsler spaces with $(\alpha ,\beta )$-metrics, we establish necessary and sufficient conditions such that these admit a Finsler spacetime structure. This means that the fundamental tensor has Lorentzian signature on a conic subbundle of the tangent bundle and thus the existence of a cone of future pointing timelike vectors is ensured. The identified $(\alpha ,\beta )$-Finsler spacetimes are candidates for applications in gravitational physics. Moreover, we completely determine the relation between the isometries of an $(\alpha ,\beta )$-metric and the isometries of the underlying pseudo-Riemannian metric $a$; in particular, we list all $(\alpha ,\beta )$-metrics which admit isometries that are not isometries of $a$.