Wind Finslerian structures: from Zermelo's navigation to the causality of spacetimes

Abstract

The notion of wind Finslerian structure Σ is developed; this is a generalization of Finsler metrics (and Kropina ones) where the indicatrices at the tangent spaces may not contain the zero vector. In the particular case that these indicatrices are ellipsoids, called here wind Riemannian structures, they admit a double interpretation which provides: (a) a model for classical Zermelo's navigation problem even when the trajectories of the moving objects (planes, ships) are influenced by strong winds or streams, and (b) a natural description of the causal structure of relativistic spacetimes endowed with a non-vanishing Killing vector field K (SSTK splittings), in terms of Finslerian elements. These elements can be regarded as conformally invariant Killing initial data on a partial Cauchy hypersurface. The links between both interpretations as well as the possibility to improve the results on one of them using the other viewpoint are stressed.The wind Finslerian structure Σ is described in terms of two (conic, pseudo) Finsler metrics, F and F l , the former with a convex indicatrix and the latter with a concave one. Notions such as balls and geodesics are extended to Σ. Among the applications, we obtain the solution of Zermelo's navigation with arbitrary time-independent wind, metric-type properties for Σ (distance-type arrival function, completeness, existence of minimizing, maximizing or closed geodesics), as well as description of spacetime elements (Cauchy developments, black hole horizons) in terms of Finslerian elements in Killing initial data. A general Fermat's principle of independent interest for arbitrary spacetimes, as well as its applications to SSTK spacetimes and Zermelo's navigation, are also provided.

Connecting and closed geodesics of a Kropina metric

“…This correspondence has been introduced in [8] extending that one between standard stationary Lorentzian metrics and Finsler metrics of Randers type ( [6,7]). Actually in [8], the class of spacetimes S × R considered is larger and the Killing vector field ∂ t can also be spacelike in some points (but in this case, the associated Finsler geometry is not simply of Kropina or Randers type, see [8] for details).…”

“…A second model in which Kropina spaces appear is related to the Zermelo's navigation problem which consists in finding the paths between two points x 0 and x 1 that minimize the travel time of a ship or an airship moving in a wind in a Riemannian landscape (S, g 0 ) (see [32,9,28,31]). If the wind is time-independent then it can be represented by a vector field W on S. When g 0 (W, W ) = 1, called critical wind in [8], the solutions of the problem (if they exist) are the pregeodesics of the Kropina metric K(v) := − g0 (v,v) 2g0(W,v) associated to the Zermelo's navigation data g 0 and W which are minimizer of the length functional associated to K see [8,]. This result and more general ones contained in [8] are strictly connected to the causality properties of the spacetime S ×R which is also associated to Zermelo navigation data ( [8,Theorem 6.15]).…”

“…If the wind is time-independent then it can be represented by a vector field W on S. When g 0 (W, W ) = 1, called critical wind in [8], the solutions of the problem (if they exist) are the pregeodesics of the Kropina metric K(v) := − g0 (v,v) 2g0(W,v) associated to the Zermelo's navigation data g 0 and W which are minimizer of the length functional associated to K see [8,]. This result and more general ones contained in [8] are strictly connected to the causality properties of the spacetime S ×R which is also associated to Zermelo navigation data ( [8,Theorem 6.15]).…”

“…Thus, K is not extendible by continuity at 0. We point out that {0} ∪ I x is a compact strongly convex hypersurface in T x S (it is an ellipsoid, see [8,Prop. 2.57]).…”

“…Since ω doesn't vanish on S, we have that g is a Lorentzian metric, t is a temporal function and −∇t is timelike (see [8,Proposition 3.3]); hence (S × R, g) is timeoriented by −∇t; moreover ∂t is a lightlike Killing vector field. Observe now that a vector (v, τ ) ∈ T x S × R is future pointing and lightlike if and only if v ∈ A x and τ = K(v), K in (1).…”