Growth conditions, Riemannian completeness and Lorentzian causality

Can We Make a Finsler Metric Complete by a Trivial Projective Change?

“…As it was observed and actively studied in [4,5,8,10], this Finsler metric and the initial Lorentz metric G are closely related. In particular, for every light-like geodesic γ(τ ) = (t(τ ), x 1 (τ ), x 2 (τ ), x 3 (τ )) of G, its "projection" to S, i.e., the curve τ → (x 1 (τ ), x 2 (τ ), x 3 (τ )) on S is a (probably, reparameterized) geodesic of the Finsler metric (1.6).…”

On the Completeness of Trajectories for Some Mechanical Systems

“…In Theorem 2, F is the norm of the Riemannian metric but, obviously, functional (14) makes sense for any Finsler metric and, under some the conditions as above, its Euler-Lagrange equation can be written as in (14). We say that γ : I ⊂ R → M is a trajectory for the potential V if its restriction to any compact subinterval [a, b] of I is a critical point of the action functional (21). Proof.…”

“…These improvements can be also extended to other contexts, as the completeness of certain Finler metrics in[21].…”

Almost isometries of non-reversible metrics with applications to stationary spacetimes

“…One of the simplest examples of nonreversible Finsler metrics are Randers metrics, which are defined as the sum of the square root of a Riemannian metric and a one-form having norm less than one at every point. Recently, a relation between Randers metrics and standard stationary spacetimes has been explored [4,7,8,9,10,11,16,19,21,22]. The core of this relation lies in the fact that, as a consequence of the Fermat principle, lightlike geodesics project up to parametrization into geodesics of a Randers metric that we call Fermat metric.…”