Wind Finslerian Structures: From Zermelo’s Navigation to the Causality of Spacetimes
Erasmo Caponio, Miguel Ángel Javaloyes, Miguel Sánchez
Abstract
The notion of <italic>wind Finslerian structure</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma"> <mml:semantics> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:annotation encoding="application/x-tex">\Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <italic>wind Riemannian structures</italic> , they admit a double interpretation which provides: (a) a model for classical <italic>Zermelo’s navigation problem</italic> even when the trajectories of the moving objects (planes, ships) are influenced by <italic>strong</italic> winds or streams, and (b) a natural description of the <italic>causal structure</italic> of relativistic spacetimes endowed with a non-vanishing Killing vector field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ( <italic>SSTK splittings</italic> ), 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma"> <mml:semantics> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:annotation encoding="application/x-tex">\Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is described in terms of two (conic, pseudo) Finsler metrics, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript l"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>l</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">F_l</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the former with a convex indicatrix and the latter with a concave one. Notions such as balls and geodesics are extended to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma"> <mml:semantics> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:annotation encoding="application/x-tex">\Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Among the applications, we obtain the solution of Zermelo’s navigation with arbitrary time-independent wind, metric-type properties for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma"> <mml:semantics> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:annotation encoding="application/x-tex">\Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (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 <roman>S</roman> STK\xspace spacetimes and Zermelo’s navigation, are also provided.