Overview Introduction Finslerian and spacetime viewpoints Background: Riemann-Finsler, Lorentz Applications: Fermat and Zermelo Fermat vs Zermelo Lorentzian geodesics Geodesics: � g ( γ ′ ( s ) , γ ′ ( s )) ds Critical points of E ( γ ) = (1 / 2) � Euler-Lagrange equation in terms of Levi-Civita ∇ g ( γ ′ , γ ′ ) constant: timelike, lightlike, spacelike Local maximization properties only for causal (timelike or lightlike) M. S´ anchez Generalized Fermat and Zermelo
Overview Introduction Finslerian and spacetime viewpoints Background: Riemann-Finsler, Lorentz Applications: Fermat and Zermelo Fermat vs Zermelo Lorentzian geodesics Geodesics: � g ( γ ′ ( s ) , γ ′ ( s )) ds Critical points of E ( γ ) = (1 / 2) � Euler-Lagrange equation in terms of Levi-Civita ∇ g ( γ ′ , γ ′ ) constant: timelike, lightlike, spacelike Local maximization properties only for causal (timelike or lightlike) Interpretations f-d timelike (unit) curves ≡ observers f-d lightlike geod. ≡ light rays Fermat principle ≡ light arrives fastest/critical M. S´ anchez Generalized Fermat and Zermelo
Overview Introduction Finslerian and spacetime viewpoints Background: Riemann-Finsler, Lorentz Applications: Fermat and Zermelo Fermat vs Zermelo Fermat principle Classical relativistic Fermat principle (Kovner ’90, Perlick ’90): Point p ∈ M (event), observer α : I ⊂ R → M Among lightlike curves from p to α : pregeodesics are critical curves for the arrival time t ∈ I at α (parameter of α ) � in particular, first arriving (minima) are pregedesics M. S´ anchez Generalized Fermat and Zermelo
Overview Introduction Finslerian and spacetime viewpoints Background: Riemann-Finsler, Lorentz Applications: Fermat and Zermelo Fermat vs Zermelo Fermat principle Classical relativistic Fermat principle (Kovner ’90, Perlick ’90): Point p ∈ M (event), observer α : I ⊂ R → M Among lightlike curves from p to α : pregeodesics are critical curves for the arrival time t ∈ I at α (parameter of α ) � in particular, first arriving (minima) are pregedesics � Existence of lightlike geodesics, multiplicity, Morse relations: Existence of critical points: Fortunato, Giannoni, Masiello ’95, etc. M. S´ anchez Generalized Fermat and Zermelo
Overview Introduction Finslerian and spacetime viewpoints Background: Riemann-Finsler, Lorentz Applications: Fermat and Zermelo Fermat vs Zermelo Link Fermat/Zermelo Start with Zermelo on M and represent graphs of curves adding a coordinate “time” as a dimension more M. S´ anchez Generalized Fermat and Zermelo
Overview Introduction Finslerian and spacetime viewpoints Background: Riemann-Finsler, Lorentz Applications: Fermat and Zermelo Fermat vs Zermelo Link Fermat/Zermelo Maximum velocities: add a “unit of time” to all the indicatrices � cone structure compatible with a ( conformal class of ) Lorentz g (independent of t ) M. S´ anchez Generalized Fermat and Zermelo
Overview Introduction Finslerian and spacetime viewpoints Background: Riemann-Finsler, Lorentz Applications: Fermat and Zermelo Fermat vs Zermelo Link classical Fermat/Zermelo Now, for mild wind, let x , y ∈ M : “Vertical” lines R × { x } , R × { y } are timelike � observers Connecting Z -unit curve c : [0 , T ] → M ⇐ ⇒ g -lightlike curve γ ( t ) = ( t , c ( t )) on R × M from (0 , x ) to ( T , y ) ∈ R × { y } c unit Z -geodesic (critical for length) ⇐ ⇒ γ = ( t , c ( t )) a lightlike g -pregeodesic ⇐ ⇒ γ Fermat critical curve from p to the observer α y ( s ) = ( s , y ) ∈ R × { y } M. S´ anchez Generalized Fermat and Zermelo
Overview Introduction Finslerian and spacetime viewpoints Background: Riemann-Finsler, Lorentz Applications: Fermat and Zermelo Fermat vs Zermelo Generalized Fermat and Zermelo What about if the wind is not mild? Arrival vertical curve (observer?) R × { y } non-timelike M. S´ anchez Generalized Fermat and Zermelo
Overview Introduction Finslerian and spacetime viewpoints Background: Riemann-Finsler, Lorentz Applications: Fermat and Zermelo Fermat vs Zermelo Generalized Fermat and Zermelo What about if the wind is not mild? Arrival vertical curve (observer?) R × { y } non-timelike Goal However, there is still a Fermat principle M. S´ anchez Generalized Fermat and Zermelo
Overview Introduction Finslerian and spacetime viewpoints Background: Riemann-Finsler, Lorentz Applications: Fermat and Zermelo Fermat vs Zermelo Generalized Fermat and Zermelo What about if the wind is not mild? Arrival vertical curve (observer?) R × { y } non-timelike Goal However, there is still a Fermat principle No Zermelo (Finsler) metric but a wind Riemann. st. Goal wind Riemm./ Finslerian st. admit a notion of geodesic The relation with spacetimes holds M. S´ anchez Generalized Fermat and Zermelo
Overview Introduction Finslerian and spacetime viewpoints Background: Riemann-Finsler, Lorentz Applications: Fermat and Zermelo Fermat vs Zermelo Generalized Fermat and Zermelo What about if the wind is not mild? Arrival vertical curve (observer?) R × { y } non-timelike Goal However, there is still a Fermat principle No Zermelo (Finsler) metric but a wind Riemann. st. Goal wind Riemm./ Finslerian st. admit a notion of geodesic The relation with spacetimes holds � Fermat principle solves Zermelo problem M. S´ anchez Generalized Fermat and Zermelo
Overview Introduction Finslerian and spacetime viewpoints Background: Riemann-Finsler, Lorentz Applications: Fermat and Zermelo Fermat vs Zermelo Generalized Fermat and Zermelo What about if the wind is not mild? Arrival vertical curve (observer?) R × { y } non-timelike Goal However, there is still a Fermat principle No Zermelo (Finsler) metric but a wind Riemann. st. Goal wind Riemm./ Finslerian st. admit a notion of geodesic The relation with spacetimes holds � Fermat principle solves Zermelo problem Overall goal basics on wind Finslerian, spacetimes and Finsler/Lorentz correspondence (including Randers/stationary spacetimes) M. S´ anchez Generalized Fermat and Zermelo
Overview Introduction Finslerian and spacetime viewpoints Background: Riemann-Finsler, Lorentz Applications: Fermat and Zermelo Fermat vs Zermelo Plan General wind Finslerian structures + Spacetime viewpoint Applications: generalized Fermat and Zermelo (and more) M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Notion of wind Finslerian structure Definition For a vector space V : — Wind Minkowskian structure: Compact strongly convex smooth hypersurface Σ V embedded in V —Unit ball B Bounded open domain B enclosed by Σ V —Conic domain A : region determined half lines from 0 to B . 0 ∈ B ⇒ A = V 0 ∈ Σ V ⇒ A = half space 0 �∈ ¯ B ⇒ properly conic A M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Notion of wind Finslerian structure Definition For a manifold M : → TM : — Wind Finslerian str.: smooth hypersurface Σ ֒ Σ p = Σ ∩ T p M is wind Minkowski in T p M ( +transversality ) — Ball at p : B p ⊂ T p M ( � A p ) — Conic domain A := ∪ p A p ∈ ¯ — Region of strong wind: M l := { p ∈ M : 0 / B p } — Properly conic domain: A l := Σ ∩ π − 1 ( M l ) M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Notion of wind Finslerian structure Proposition Any Σ is the displacement of the indicatrix of Finsler metric F 0 along some vector field W : � v � Z ( v ) − W F 0 = 1 , (v ∈ Σ ⇐ ⇒ Z ( v ) is a solution) — Uniqueness if 0 p is required to be the barycentre of each F p — Wind Riemannian: displacement of F 0 = √ g R (ellipsoids) M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Notion of wind Finslerian structure Proposition Any Σ determines two “conic” pseudo-Finsler metrics: (i) F : A → [0 , + ∞ ) conic Finsler metric on all M, (ii) F l : A l → [0 , + ∞ ) F l is a Lorentz-Finsler metric in the region M l of strong wind with F < F l . Moreover, a cone structure appears M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Notion of wind Finslerian structure Cone structure on M l : Limit region F = F l : Cone ∪ A : “Σ-admissible vectors” it characterizes accessibility from x 0 to x 1 ( x 0 ≺ x 1 ) For wind Riemannian, associated to a Lorentzian metric Curvatures for F and F l are computable [JV] M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Balls and geodesics for wind Finsler No “distance” d F for Σ � redefinitions of balls and geodesics for any wind Finsler M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Balls and geodesics for wind Finsler No “distance” d F for Σ � redefinitions of balls and geodesics for any wind Finsler Σ admissible γ from x 0 to x : γ ′ in a closure of A ( ⊃ A l ). (Forward/backwards) wind balls [mild wind: usual open balls] B + Σ ( x 0 , r ) = { x ∈ M | ∃ γ Σ-a. x 0 to x : ℓ F ( γ ) < r < ℓ F l ( γ ) } , B − Σ ( x 0 , r ) = { x ∈ M | ∃ γ Σ-a. x to x 0 : ℓ F ( γ ) < r < ℓ F l ( γ ) } . M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Balls and geodesics for wind Finsler No “distance” d F for Σ � redefinitions of balls and geodesics for any wind Finsler Σ admissible γ from x 0 to x : γ ′ in a closure of A ( ⊃ A l ). (Forward/backwards) wind balls [mild wind: usual open balls] B + Σ ( x 0 , r ) = { x ∈ M | ∃ γ Σ-a. x 0 to x : ℓ F ( γ ) < r < ℓ F l ( γ ) } , B − Σ ( x 0 , r ) = { x ∈ M | ∃ γ Σ-a. x to x 0 : ℓ F ( γ ) < r < ℓ F l ( γ ) } . Wind c-balls: B + ˆ Σ ( x 0 , r ) = { x ∈ M | ∃ γ Σ-a. x 0 to x : ℓ F ( γ ) ≤ r ≤ ℓ F l ( γ ) } , ˆ B − Σ ( x 0 , r ) = { x ∈ M | ∃ γ Σ-a. x to x 0 : ℓ F ( γ ) ≤ r ≤ ℓ F l ( γ ) } . Closed balls: (usual closures) ¯ Σ ( x 0 , r ) , ¯ B + B − Σ ( x 0 , r ) Σ ( x 0 , r ) ⊂ ˆ Σ ( x 0 , r ) ⊂ ¯ B + B + B + Σ ( x 0 , r ) M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Balls and geodesics for wind Finsler M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Balls and geodesics for wind Finsler w-convexity: c-balls are closed (extend usual convexity) M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Balls and geodesics for wind Finsler Geodesic parametrized by arc length: Σ-admissible curve s.t. γ ( t + ǫ ) ∈ ˆ B + Σ ( γ ( t ) , ǫ ) \ B + Σ ( γ ( t ) , ǫ ) (locally, i.e., for small ǫ > 0) M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Balls and geodesics for wind Finsler Geodesic parametrized by arc length: Σ-admissible curve s.t. γ ( t + ǫ ) ∈ ˆ B + Σ ( γ ( t ) , ǫ ) \ B + Σ ( γ ( t ) , ǫ ) (locally, i.e., for small ǫ > 0) Proposition When ˙ γ ( t ) ∈ A (open): γ geodesic of ( M , Σ) (parametrized by arc length) ⇔ γ (unit) geodesic for either F or F l . M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Going further What about when ˙ γ is Σ-admissible but belongs to ∂ A ? In principle, one could follow but there are technical difficulties (“abnormal” geodesics) � Focus on wind Riemannian (but generalizable to Finslerian) Develop in a “non-singular” way through the spacetime viewpoint M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Definition of SSTK spacetime SSTK s.t: standard with a space-transverse Killing v.f. ( K = ∂ t ) ( R × M , g ), g = − Λ dt 2 + 2 ω dt + g 0 ≡ − (Λ ◦ π ) d t 2 + π ∗ ω ⊗ d t + d t ⊗ π ∗ ω + π ∗ g 0 for Λ (function), ω (1-form), g 0 (Riemannian) on M with Λ > −� ω � 2 0 (Lorentz restriction) Cases: ω = 0 , Λ ≡ 1: Product st : R × M , g = − dt 2 + π ∗ g 0 ≡ − dt 2 + g 0 ω = 0 , Λ > 0 Static st : R × M , g = − Λ dt 2 + g 0 = Λ( − dt 2 + g 0 / Λ) � Conformal to product M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Product/ static case K = ∂ t induces a Riemannian metric g 0 ( ≡ g 0 / Λ) on M M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics SSTK spacetimes SSTK ( R × M , g ) , g = − Λ dt 2 + 2 ω dt + g 0 (with Λ > −� ω � 2 0 ) Cases: Λ ≡ 1, arbitrary ω Normalized (standard) stationary s.t. : R × M , g = − 1 dt 2 + 2 ω dt + g 0 Λ > 0, arbitrary ω Stationary s.t. : R × M , g = − Λ dt 2 + 2 ω dt + g 0 − dt 2 + 2( ω/ Λ) + ( g 0 / Λ) � � � Conformal to normalized Λ M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Stationary case K = ∂ t induces the indicatrix of a Finslerian metric on M M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Stationary case Induces a (pair of) Finslerian metric on M M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Appearance of Finsler For each v ∈ T x M : Future-d. lightlike vector ( F + ( v ) , v ) Past-d. lightlike vector ( − F − ( v ) , v ) where F ± : TM → R , for normalized Λ ≡ 1: � g 0 ( v , v ) + ω ( v ) 2 ± ω ( v ) F ± ( v ) = F ± : Finsler metrics of Randers type, “ Fermat metrics ” M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Appearance of Finsler For each v ∈ T x M : Future-d. lightlike vector ( F + ( v ) , v ) Past-d. lightlike vector ( − F − ( v ) , v ) where F ± : TM → R , for normalized Λ ≡ 1: � g 0 ( v , v ) + ω ( v ) 2 ± ω ( v ) F ± ( v ) = F ± : Finsler metrics of Randers type, “ Fermat metrics ” F − ( v ) = F + ( − v ), F − “reversed metric” of F + ( ≡ F ). M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Expression with wind F : Randers metric with indicatrix Σ = S R + W W : vector field (wind): g 0 ( W , · ) = − ω S R : Riemannian metric indicatrix (sphere bundle) of g R = g 0 / (1 + | W | 2 0 ) M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Expression with wind F : Randers metric with indicatrix Σ = S R + W W : vector field (wind): g 0 ( W , · ) = − ω S R : Riemannian metric indicatrix (sphere bundle) of g R = g 0 / (1 + | W | 2 0 ) Necessarily g R ( W , W ) < 1 (mild wind) M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics General SSTK spacetime SSTK ( R × M , g ) , g = − Λ dt 2 + 2 ω dt + g 0 (with Λ > −� ω � 2 0 ) General case: timelike Λ > 0 K := ∂ t Killing and lightlike Λ = 0 spacelike Λ < 0 The projection t : R × M → R time function [for v causal dt ( v ) > 0 defines the future direction] M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Interpretation of K = ∂ t M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Interpretation of K = ∂ t K = ∂ t induces a wind-Riemannian structure M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics SSTK ← → Wind Riemannian Proposition Σ = { v ∈ TM : (1 , v ) is (future-p.) lightlike in T ( R × M ) } is a wind Riemannian structure on M ( Fermat structure of the conformal class of the SSTK) with; Σ computable from: Wind vector W : g 0 ( · , W ) = − ω Riemannian metric g R = g 0 / (Λ + g 0 ( W , W )) Moreover, cone structure on M l computable from the sign. changing metric h (Lorentzian (+ , − , . . . , − ) on M l ) h ( v , v ) = Λ g 0 ( v , v ) + ω ( v ) 2 Conversely, each wind Riemannian structure selects a unique conformal class of SSTK M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Unified viewpoint Merging SSTK and Wind Riemannian for geodesics: Theorem For associated SSTK ↔ Σ , these classes of curves coincide: 1 Projections on M of the future-d. lightlike pregeodesics for SSTK R × M M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Unified viewpoint Merging SSTK and Wind Riemannian for geodesics: Theorem For associated SSTK ↔ Σ , these classes of curves coincide: 1 Projections on M of the future-d. lightlike pregeodesics for SSTK R × M 2 Pregeodesics for wind Riemannian Σ on M M. S´ anchez Generalized Fermat and Zermelo
Overview General Wind Finslerian structures Finslerian and spacetime viewpoints Spacetimes and Wind Riemannian Applications: Fermat and Zermelo Conclusion on geodeics Unified viewpoint Merging SSTK and Wind Riemannian for geodesics: Theorem For associated SSTK ↔ Σ , these classes of curves coincide: 1 Projections on M of the future-d. lightlike pregeodesics for SSTK R × M 2 Pregeodesics for wind Riemannian Σ on M 3 The set of all the pregeodesics for F (locally minimizing F-distance, including critical/Kropina and strong wind regions) F l (on strong wind region M l , locally maximizing) and lightlike for − h (Lorentzian metric on M l ) M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz General Fermat principle Theorem (CJS) Let ( L , g ) be any spacetime and any (smooth embedded) arbitrary arrival curve α . For any piecewise smooth future-directed lightlike curve γ from p 0 to α , such that ˙ γ is not orthogonal to α (at its arrival): γ : [ a , b ] → L is a pregeodesic ⇐ ⇒ it is a critical point of the arrival functional (parameter of α ) Includes classical one (Kovner [Ko], Perlick [Pe]): α timelike Based on a sharp characterization of which vector fields on γ come from a variation by lightlike curves from p 0 to α M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Sharpest Fermat for SSTK Theorem Let ( R × M , g ) SSTK, x 0 , x 1 ∈ M, x 0 � = x 1 , p 0 = ( t 0 , x 0 ) and γ ( s ) = ( ζ ( s ) , x ( s )) lightlike from p 0 to R × { x 1 } . a) γ critical point of the arrival time T = ⇒ pregeodesic. b) γ pregeodesic ⇐ ⇒ (C γ = g ( ∂ t , ˙ γ ) constant and:) M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Sharpest Fermat for SSTK Theorem Let ( R × M , g ) SSTK, x 0 , x 1 ∈ M, x 0 � = x 1 , p 0 = ( t 0 , x 0 ) and γ ( s ) = ( ζ ( s ) , x ( s )) lightlike from p 0 to R × { x 1 } . a) γ critical point of the arrival time T = ⇒ pregeodesic. b) γ pregeodesic ⇐ ⇒ (C γ = g ( ∂ t , ˙ γ ) constant and:) (i) C γ < 0 , ˙ x lies in A, x pregeodesic of F parametrized with h ( ˙ x , ˙ x ) = const . , γ is a critical point of T(locally min.) (ii) C γ > 0 , Λ < 0 on all x, x a pregeodesic of F l parametrized with h ( ˙ x , ˙ x ) = const . , γ critical point of T (locally max.) (iii) C γ = 0 , Λ ≤ 0 on all x: whenever Λ < 0 , x lightlike geodesic of h / Λ on M; Λ vanishes on x only at isolated points where ˙ x vanishes. M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Characterization for Zermelo For arbitrary g R , W (generalizing Shen’s et al. [Sh], [BRS]): Solutions x ( s ) of Zermelo’s connecting x 0 , x 1 are (pre)geodesics of Σ and they lie in exactly one of the three previous cases. M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Characterization for Zermelo For arbitrary g R , W (generalizing Shen’s et al. [Sh], [BRS]): Solutions x ( s ) of Zermelo’s connecting x 0 , x 1 are (pre)geodesics of Σ and they lie in exactly one of the three previous cases. If solution exists if: 1 An admissible curve exists from x 0 to x 1 ( ⇐ ⇒ x 0 ≺ x 1 for − h on M , where h ( u , v ) := (1 − g R ( W , W )) g R ( u , v ) + g R ( u , W )( W , v )) M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Characterization for Zermelo For arbitrary g R , W (generalizing Shen’s et al. [Sh], [BRS]): Solutions x ( s ) of Zermelo’s connecting x 0 , x 1 are (pre)geodesics of Σ and they lie in exactly one of the three previous cases. If solution exists if: 1 An admissible curve exists from x 0 to x 1 ( ⇐ ⇒ x 0 ≺ x 1 for − h on M , where h ( u , v ) := (1 − g R ( W , W )) g R ( u , v ) + g R ( u , W )( W , v )) 2 and Σ is w-convex ( ⇐ ⇒ associated SSTK causally simple) M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz General Fermat principle: precise statement Theorem (CJS) ( L , g ) any spacetime , α any arrival curve (smooth, embedded) N p 0 ,α := { γ : [ a , b ] → L | γ piece. smooth f.-d. light. from p 0 to α } Arrival functional: T ( γ ) = α − 1 ( γ ( b )) , ∀ γ ∈ N p 0 ,α γ ∈ N p 0 ,α with ˙ γ ( b ) �⊥ α : pregeodesic ⇐ ⇒ critical point of T M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Lemma 1: critical in terms of admissible v.f. Z Z v.f. on γ admissible : variational v.f. by means of longitudinal curves γ w ∈ N p 0 ,α Lemma 1. γ critical for T ⇔ Z ( b ) = 0, ∀ Z admissible M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Lemma 1: critical in terms of admissible v.f. Z Z v.f. on γ admissible : variational v.f. by means of longitudinal curves γ w ∈ N p 0 ,α Lemma 1. γ critical for T ⇔ Z ( b ) = 0, ∀ Z admissible Proof. Z ( b ) = d dw γ w ( b ) | w =0 = d dw α ( T ( γ w )) | w =0 � d � = dw T ( γ w ) | w =0 α ( T ( γ )) ˙ M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Lemma 2: characterization of admissible Z Lemma 2. Z v.f. on γ , Z ( a ) = 0, with Z ( b ) � ˙ α : ⇒ Z ′ ⊥ ˙ Z admissible ⇐ γ M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Lemma 2: characterization of admissible Z Lemma 2. Z v.f. on γ , Z ( a ) = 0, with Z ( b ) � ˙ α : ⇒ Z ′ ⊥ ˙ Z admissible ⇐ γ ( ⇒ trivial) M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Lemma 2: characterization of admissible Z Lemma 2. Z v.f. on γ , Z ( a ) = 0, with Z ( b ) � ˙ α : ⇒ Z ′ ⊥ ˙ Z admissible ⇐ γ ( ⇒ trivial) • Note: ( ⇐ ) Typical results (i) no lightlike longit. or (ii) geodesic γ ⊥ α non-lightlike M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Lemma 2: characterization of admissible Z Lemma 2. Z v.f. on γ , Z ( a ) = 0, with Z ( b ) � ˙ α : ⇒ Z ′ ⊥ ˙ Z admissible ⇐ γ Sketch ( ⇐ ): Neighborhood of γ covered by a finite number of coordinates which looks like a t -dependent SSTK and: (a) γ nowhere orthogonal to ∂ t (b) α � ∂ t at γ ( b ). M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Lemma 2: characterization of admissible Z Lemma 2. Z v.f. on γ , Z ( a ) = 0, with Z ( b ) � ˙ α : = Z ′ ⊥ ˙ Z admissible ⇐ γ Neighborhood of γ covered by a finite number of coordinates which looks like a t -dependent SSTK and: (a) γ nowhere orthogonal to ∂ t (b) α � ∂ t at γ ( b ). Put Z = ( Y , W ) in each local splitting R × S W : fixed endpoint variation for x ( s ) = Π S ( γ ( s )) M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Lemma 2: characterization of admissible Z Lemma 2. Z v.f. on γ , Z ( a ) = 0, with Z ( b ) � ˙ α : = Z ′ ⊥ ˙ Z admissible ⇐ γ Neighborhood of γ covered by a finite number of coordinates which looks like a t -dependent SSTK and: (a) γ nowhere orthogonal to ∂ t (b) α � ∂ t at γ ( b ). Put Z = ( Y , W ) in each local splitting R × S W : fixed endpoint variation for x ( s ) = Π S ( γ ( s )) Lift this variation imposing longitudinal curves in N p 0 ,α � diff. eqn. for t coordinate M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Lemma 2: characterization of admissible Z Lemma 2. Z v.f. on γ , Z ( a ) = 0, with Z ( b ) � ˙ α : = Z ′ ⊥ ˙ Z admissible ⇐ γ Neighborhood of γ covered by a finite number of coordinates which looks like a t -dependent SSTK and: (a) γ nowhere orthogonal to ∂ t (b) α � ∂ t at γ ( b ). Put Z = ( Y , W ) in each local splitting R × S W : fixed endpoint variation for x ( s ) = Π S ( γ ( s )) Lift this variation imposing longitudinal curves in N p 0 ,α � diff. eqn. for t coordinate Check: (i) consistency eqn. from Z ′ ⊥ ˙ γ , Z ( b ) � ˙ α and (b) (ii) non-degeneracy eqn. (uniqueness) because of (a) � constructed admissible v.f. agrees Z M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Lemma 3: construction of admissible Z Lemma 3. Explicit construction of such admissible Z : — Choose U along γ at no point orthogonal — For each W along γ with W ( a ) = W ( b ) = 0, put: Z W ( s ) = W ( s ) + f W ( s ) U ( s ) , where � s � s g ( W ′ , ˙ g ( U ′ , ˙ γ ) γ ) f W ( s ) = − e − ρ ( s ) γ ) e ρ d µ with ρ ( s ) = γ ) d µ g ( U , ˙ g ( U , ˙ a a M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Lemma 3: construction of admissible Z Lemma 3. Explicit construction of such admissible Z : — Choose U along γ at no point orthogonal — For each W along γ with W ( a ) = W ( b ) = 0, put: Z W ( s ) = W ( s ) + f W ( s ) U ( s ) , where � s � s g ( W ′ , ˙ g ( U ′ , ˙ γ ) γ ) f W ( s ) = − e − ρ ( s ) γ ) e ρ d µ with ρ ( s ) = γ ) d µ g ( U , ˙ g ( U , ˙ a a Sketch of proof. Z W is admissible: check g ( Z ′ W , ˙ γ ) = 0 (eqn for f W ) M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Lemma 3: construction of admissible Z Lemma 3. Explicit construction of admissible Z : — Choose U along γ at no point orthogonal — For each W along γ with W ( a ) = W ( b ) = 0, put: Z W ( s ) = W ( s ) + f W ( s ) U ( s ) Sketch. Any admissible Z is some Z W : M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Lemma 3: construction of admissible Z Lemma 3. Explicit construction of admissible Z : — Choose U along γ at no point orthogonal — For each W along γ with W ( a ) = W ( b ) = 0, put: Z W ( s ) = W ( s ) + f W ( s ) U ( s ) Sketch. Any admissible Z is some Z W : 1 Define W ( s ) = Z ( s ) − ( c ( s − a ) / ( b − a )) U ( s ) with c s.t. Z ( b ) = cU ( b ). M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Lemma 3: construction of admissible Z Lemma 3. Explicit construction of admissible Z : — Choose U along γ at no point orthogonal — For each W along γ with W ( a ) = W ( b ) = 0, put: Z W ( s ) = W ( s ) + f W ( s ) U ( s ) Sketch. Any admissible Z is some Z W : 1 Define W ( s ) = Z ( s ) − ( c ( s − a ) / ( b − a )) U with c s.t. Z ( b ) = cU ( b ). 2 Z and Z W admissible ⇒ Z − Z W admissible... M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Lemma 3: construction of admissible Z Lemma 3. Explicit construction of admissible Z : — Choose U along γ at no point orthogonal — For each W along γ with W ( a ) = W ( b ) = 0, put: Z W ( s ) = W ( s ) + f W ( s ) U ( s ) Sketch. Any admissible Z is some Z W : 1 Define W ( s ) = Z ( s ) − ( c ( s − a ) / ( b − a )) U with c s.t. Z ( b ) = cU ( b ). 2 Z and Z W admissible ⇒ Z − Z W admissible ... 3 ... but Z − Z W = ( f W ( s ) − c ( s − a ) / ( b − a )) U =: p ( s ) U M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Lemma 3: construction of admissible Z Lemma 3. Explicit construction of admissible Z : — Choose U along γ at no point orthogonal — For each W along γ with W ( a ) = W ( b ) = 0, put: Z W ( s ) = W ( s ) + f W ( s ) U ( s ) Sketch. Any admissible Z is some Z W : 1 Define W ( s ) = Z ( s ) − ( c ( s − a ) / ( b − a )) U with c : Z ( b ) = cU ( b ). 2 Z and Z W admissible ⇒ Z − Z W admissible ... 3 ... but Z − Z W = ( f W ( s ) − c ( s − a ) / ( b − a )) U =: pU 4 As 0 = g (( pU ) ′ , ˙ γ ) + pg ( U ′ , ˙ γ ) = ˙ pg ( U , ˙ γ ) and p ( a ) = 0 ⇒ p ≡ 0 M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz General Fermat principle: proof Sketch proof of theorem Lemma 1: γ critical for T ⇔ Z ( b ) = 0 for all admissible Z M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz General Fermat principle: proof Lemma 1: γ critical for T ⇔ Z ( b ) = 0 for all admissible Z Lemma 3 (chosen U ): Z = Z W = W + f W U ⇔ f W ( b ) = 0 , as W ( b ) = 0 (= W ( a )) M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz General Fermat principle: proof Lemma 1: γ critical for T ⇔ Z ( b ) = 0 for all admissible Z Lemma 3 (chosen U ): Z = Z W = W + f W U ⇔ f W ( b ) = 0 (as W ( b ) = 0 = W ( a )) Using the explicit formula for f W : � b g ( W ′ , ˙ γ ) ⇔ γ ) e ρ d µ = 0 . g ( U , ˙ a M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz General Fermat principle: proof Lemma 1: γ critical for T ⇔ Z ( b ) = 0 for all admissible Z Lemma 3 (chosen U ): Z = Z W = W + f W U ⇔ f W ( b ) = 0 (as W ( b ) = 0 = W ( a )) Using the explicit formula for f W : � b g ( W ′ , ˙ γ ) ⇔ γ ) e ρ d µ = 0 . a g ( U , ˙ Integrating by parts (with smooth W vanishing at breaks) � b γ ) ′ ) d µ = 0, for some function ϕ ⇔ a g ( W , ( ϕ ˙ M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz General Fermat principle: proof Lemma 1: γ critical for T ⇔ Z ( b ) = 0 for all admissible Z Lemma 3 (chosen U ): Z = Z W = W + f W U ⇔ f W ( b ) = 0 (as W ( b ) = 0 = W ( a )) Using the explicit formula for f W : � b g ( W ′ , ˙ γ ) ⇔ γ ) e ρ d µ = 0 . g ( U , ˙ a Integrating by parts (with smooth W vanishing at breaks) � b γ ) ′ ) d µ = 0, for some function ϕ ⇔ a g ( W , ( ϕ ˙ Using standard variational arguments: γ ) ′ = 0 (well-known characterization of pregeodesics) ⇔ ( ϕ ˙ M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Characterization of the causal ladder SSTK are always stably continuous ( t time function) Causal continuity characterizable in terms of the associated wind Finslerian structure Causal simplicity and global hyperbolicity especially interesting M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Causal simplicity of SSTK Causal simplicity (for SSTK spacetimes, J ± ( p ) closed) ⇐ ⇒ w-convexity (c-balls are closed) M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Causal simplicity of SSTK Causal simplicity (for SSTK spacetimes, J ± ( p ) closed) ⇐ ⇒ w-convexity (c-balls are closed) Variational methods type Fortunato et al. [FGM], become applicable providing results on existence and multiplicity M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Causal simplicity of SSTK Causal simplicity (for SSTK spacetimes, J ± ( p ) closed) ⇐ ⇒ w-convexity (c-balls are closed) Variational methods type Fortunato et al. [FGM], become applicable providing results on existence and multiplicity Applications even for stationary s.t.: Extension of previous results Applications to gravitational lensing [CGS] M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Causal simplicity of SSTK Causal simplicity (for SSTK spacetimes, J ± ( p ) closed) ⇐ ⇒ w-convexity (c-balls are closed) Variational methods type Fortunato et al. [FGM], become applicable providing results on existence and multiplicity Applications even for stationary s.t.: Extension of previous results Applications to gravitational lensing [CGS] ...now extensible to SSTK M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz Chracterization of global hyperbolicity Global hyperbolicity ( J + ( p ) ∩ J − ( q ) compact) equivalent to any of 1 All intersections ¯ Σ ( x 0 , r 1 ) ∩ ¯ B + B − Σ ( x 1 , r 2 ) compact 2 All intersections ˆ Σ ( x 0 , r 1 ) ∩ ˆ B + B − Σ ( x 1 , r 2 ) 3 In the case of K timelike (stationary/Randers): Compactness of ¯ B + s ( p , r ) Spacelike slices S t = { ( t , x ) : x ∈ R × M } Cauchy hypers. (crossed exactly once by any inextendible causal curve) equivalent to any of: 1 All closed ¯ Σ ( x , r ), ¯ B + B − Σ ( x , r ) compact 2 All c-balls ˆ B + Σ ( x , r ), ˆ B − Σ ( x , r ) compact 3 Σ (forward and backward) geodesically complete M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz An unexpected application for Riemann, Finsler & Lorentz Application to Riemann/Finsler/wind Finsler Geom. [FHS] Relativistic notion of causal boundary � New notion of boundary extending classical Cauchy, Gromov and Busemann for Riemannian and Finslerian Geometries, now extensible to wind Finsler M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz An unexpected application for Riemann, Finsler & Lorentz Application to Riemann/Finsler/wind Finsler Geom. [FHS] Relativistic notion of causal boundary � New notion of boundary extending classical Cauchy, Gromov and Busemann for Riemannian and Finslerian Geometries, now extensible to wind Finsler Application to Lorentz Geom. [FHS]: description of the c-boundary of static/ stationary/ SSTK s.t. in terms of Riemannian [FHa]/ Finslerian/ wind Finslerian elements M. S´ anchez Generalized Fermat and Zermelo
Overview Fermat and Zermelo Finslerian and spacetime viewpoints Generalized Fermat: Sketch of proof Applications: Fermat and Zermelo ... and some of the applications to Lorentz ...and some other applications to Finsler [CJS11] 1 To weaken completeness by compactness of balls ¯ B + s ( p , r ) (Heine-Borel) in classical Finsler theorems such as Myers 2 Characterization of the differentiability of the distance from a subset d ( C , · ) with applications to Hamilton Jacobi equation (extended by Tanaka & Sabau [TS]) 3 Properties of completeness in classes of projectively related metrics (extended by Matveev ’12) 4 Properties of the Hausdorff dimension for the cut locus, extending a previous result of Lee & Nirenberg ’06 [LN] 5 Appropriate description of Randers manifolds of constant flag curvature [CJS14] and Javaloyes & Vit´ orio [JV] M. S´ anchez Generalized Fermat and Zermelo
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