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

Abstract

The notion of wind Finslerian structure is developed; this is a generalization of Finsler metrics 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 (SSTK splittings), in terms of Finslerian elements. These elements can be regarded as conformally invariant Killing initial data on a partial Cauchy hypersurface. The wind Finslerian structure is described in terms of two (conic, pseudo) Finsler metrics, one with a convex indicatrix and the other with a concave one. However, the spacetime viewpoint for the wind Riemannian case gives a useful unified viewpoint. A thorough study of the causal properties of such a spacetime is carried out in Finslerian terms. Randers-Kropina metrics appear as the Finslerian counterpart to the case of an SSTK when the Killing vector field is either timelike or lightlike. Among the applications, we obtain the solution of Zermelo's navigation with arbitrary stationary wind, metric-type properties (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.

Figures (14)

• Figure 1: Wind Minkowskian structures
• Figure 2: In the top figure, $\Sigma$ is a smooth hypersurface of $TM\equiv \mathds R^2$ consisting of two curves which intersect the vertical space at $(p_0,v_0)$ (depicted as a vertical line) non-transversely. So $\Sigma$ satisfies the property (a) in Definition \ref{['windStruct']} and (as the curves are symmetric with respect to the zero section of $TM$) it determines continuously a scalar product in the tangent space at each $p\in M\equiv\mathds R$. Nevertheless, the failure of (b) implies that this product does not vary smoothly with respect to $p$ and, so, $\Sigma$ does not determine a (smooth) Riemannian metric on $M$. In the second figure, changing the lower curve by a horizontal line, one obtains at each tangent space a wind Minkowskian structure varying continuously (but not smoothly) with the point. Moreover, the vector field determined by the centroids (the dashed curve) is not differentiable at $p_0$.
• Figure 3: A wind Minkowskian structure $\Sigma$ in $\mathds R^2\setminus\{Q\}$. The shaded regions represent the wind balls $B^+_{\Sigma}(0,1/5)$ and $B^+_{\Sigma}(0,1)$ which satisfy $\hat{B}^+_{\Sigma}(0,1/5)=\bar{B}^+_{\Sigma}(0,1/5)$ but $P\in \bar{B}^+_{\Sigma}(0,1)\setminus \hat{B}^+_{\Sigma}(0,1)$.
• Figure 4: A wind Finslerian cylinder $(S^1\times \mathds R, \Sigma)$. The shaded region represents the c-ball $\hat{B}_\Sigma^+((0,0),1/2)$
• Figure 5: The time cone in an $\rm SSTK$ splitting

Theorems & Definitions (232)

• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5
• proof
• Remark 2.6
• Definition 2.8
• Remark 2.9
• Proposition 2.10