Causal ladder of Finsler spacetimes with a cone Killing vector field
Erasmo Caponio, Miguel Angel javaloyes
TL;DR
This work generalizes causal analysis from stationary and wind-driven spacetimes to the broader setting of cone structures equipped with cone Killing fields. It develops a CSTK splitting framework and a corresponding wind Finslerian description on a spacelike section, enabling causality to be read from a Finsler–Kropina type metric via the $d_F$ separation. The results establish a ladder of equivalences linking causal properties (continuity, simplicity, hyperbolicity) to concrete geometric and completeness conditions on the associated wind Finslerian metric, including Hopf–Rinow type phenomena. By treating arbitrary cone Killing fields, the paper provides a unified methodology to analyze causality in generalized spacetimes and connects cone geometry with wind Finslerian tools for practical causal classification.
Abstract
The correspondence between wind Riemannian structures and spacetimes endowed with a Killing vector field is deepened by considering a cone structure endowed with a vector field that preserve the structure (termed "cone Killing vector field") and a wind Finslerian structure. Causality properties of the former are characterized by using metric-type properties of the latter. A particular attention is posed to the case of a cone structure associated with a Finsler-Kropina type metric, i.e. a field of compact and strongly convex indicatrices that enclose the zero vector in the closure of its bounded interior at each tangent space of the manifold.