On the definition and examples of cones and Finsler spacetimes

A systematic study of (smooth, strong) cone structures C and Lorentz-Finsler metrics L is carried out. As a link between both notions, cone triples (omega,T,F), where omega (resp. T) is a 1-form (resp. vector field) with omega(T)equivalent to 1 and F, a Finsler metric on ker(omega), are introduced. Explicit descriptions of all the Finsler spacetimes are given, paying special attention to stationary and static ones, as well as to issues related to differentiability. In particular, cone structures C are bijectively associated with classes of anisotropically conformal metrics L, and the notion of cone geodesic is introduced consistently with both structures. As a non-relativistic application, the time-dependent Zermelo navigation problem is posed rigorously, and its general solution is provided.