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Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

Generalized Fermat principle and Zermelo navigation: a link between Lorentzian and Generalized Finslerian Geometries

Miguel S´ anchez

Universidad de Granada, IEMath-GR

8th Int. Meeting on Lorentzian Geom. (M´ alaga, 23/09/2016)

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

General aims

Show a correspondence between problems in:

1 Lorentzian Geometry 2 Finslerian and generalized (singular) Finslerian Geometry

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

General aims

Applications:

1 For Lorentz: appropriate description of relativistic notions in

Finslerian terms

2 For Finsler: new problems and results by using Lorentzian

viewpoint

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

General aims

Applications:

1 For Lorentz: appropriate description of relativistic notions in

Finslerian terms

2 For Finsler: new problems and results by using Lorentzian

viewpoint

3 Dynamical systems/optimal control:

Non singular description of apparently singular problems Emphasis in the most general viewpoint extended and singular Finsler

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

General aims

Applications:

1 For Lorentz: appropriate description of relativistic notions in

Finslerian terms

2 For Finsler: new problems and results by using Lorentzian

viewpoint

3 Dynamical systems/optimal control:

Non singular description of apparently singular problems Emphasis in the most general viewpoint extended and singular Finsler and (non-singular) relativistic interpretations

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

General aims

Focus on a pair of variational goals: Generalization of relativistic Fermat principle Solution to generalized Zermelo problem (navigation in arbitrary wind)

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

Joint work with E Caponio and MA Javaloyes

Main reference: Caponio, Javaloyes, S. arxiv 1407.5494 [CJS] Previous work Caponio, Javaloyes, Masiello’11 [CJM] Caponio, Javaloyes, S´ anchez’11 [CJS11] (+ others)

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

Background: Riemannian-Finsler, Lorentz

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

Finslerian elements

Finsler metric F : TM → R: generalization of Riemann’s replace pointwise Euclidean scalar products by norms Fp

1 Smooth: F smooth outside zero section or, equally: 1 Smooth indicatrix (set of unit spheres)

Σ := F −1(1)(⊂ TM)

2 Transversality Σ ⋔ TpM for all p ∈ M 2 Strong convexity of pointwise indicatrices (unit spheres)

Σp = F −1

p (1) ovaloids (II > 0 in particular strictly convex)

bound the unit (open) ball Bp = F −1

p ([0, 1)) 3 No reversibility assumed: Fp(λvp) = λFp(v) just for λ ≥ 0

(and even 0p ∈ Bp no barycenter) non-symmetric distance

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

Finslerian geodesics

Energy functional: E(γ) = (1/2)

• F 2(γ′(s))ds

Geodesics:

Critical points of E (for length pregeodesics) Locally minimize: energy, non-symmetric distance

An example for interpretations: mobile

Σp: maximum velocity depending on p (Riemann. case) and direction (properly Finsler) Length of (unit) curves ≡ arrival time at maximum speed [non-reversible] (Pre)geodesics ≡ locally fastest paths [non-reversible] Some cases:

hill (Matsumoto), mild wind (Zermelo, Shen et al.)

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

Zermelo Navigation

• Plane/Zeppelin in the air with a (stationary) wind
• Submarine in the sea dragged by a (stationary) current

Zermelo problem: find fastest path between two points

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

Zermelo navigation

Riemannian metric ·, ·: unit spheres Sp maximum speed zeppelin/air Vector field W : velocity of the wind respect to Earth Finsler model indicatrix Σp = Sp + Wp (Randers) metric Z Z-geodesics solve Zermelo’s...

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

Zermelo navigation

Riemannian metric ·, ·: unit spheres Sp maximum speed zeppelin/air Vector field W : velocity of the wind respect to Earth Finsler model indicatrix Σp = Sp + Wp (Randers) metric Z Z-geodesics solve Zermelo’s... under mild wind, W , W < 1

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

Zermelo navigation

Note: if the wind is not mild... f. ex. critical Wp, Wp = 1 “Singular” Finsler metric: 0p ∈ Σp (“Kropina metric”) Forbidden directions unreachable regions

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

Zermelo navigation

Strong wind Wp, Wp > 1 No Finsler metric but Σ ⊂ TM still makes sense “Wind Riemannian/ Finslerian structure”

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

Lorentzian manifolds and spacetimes

Lorentz metric g, (−, +, · · · +) Cone structure (conformal class): vp ∈ TpM \ {0} timelike g(vp, vp) < 0, lightlike g(vp, vp) = 0; (≤ 0, causal) spacelike g(vp, vp) > 0 Spacetime: g + time orientation

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

Remark on projection of cones

Lorentzian is richer (rather than a generalization) of Riemannian

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

Remark on projection of cones

Lorentzian is richer (rather than a generalization) of Riemannian Choose a spacelike hyperplane Π and a transversal vector K at some p ∈ M: K ⊥ Π (and unit): Euclidean indicatrix of Π from the cone

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

Remark on projection of cones

Choose a spacelike hyperplane Π and a transversal direction K at some p ∈ M: K timelike but non orthogonal: “Finslerian” indicatrix from the cone

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

Remark on projection of cones

Choose a spacelike hyperplane Π and a transversal direction K at some p ∈ M: K lightlike: Koprina/ “critical wind” indicatrix

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

Remark on projection of cones

Choose a spacelike hyperplane Π and a transversal direction K at some p ∈ M: K spacelike: “strong wind” indicatrix (+ cone on Π)

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

Lorentzian geodesics

Geodesics: Critical points of E(γ) = (1/2)

• g(γ′(s), γ′(s))ds

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 Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

Lorentzian geodesics

Geodesics: Critical points of E(γ) = (1/2)

• g(γ′(s), γ′(s))ds

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 Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz 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 Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz 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 Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz 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 Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz 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 Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz Fermat vs Zermelo

Link classical Fermat/Zermelo

Now, for mild wind, let x, y ∈ M: “Vertical” lines R × {x}, R × {y} are timelike

• bservers

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 αy(s) = (s, y) ∈ R × {y}

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz 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 Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz 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 Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz 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 Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz 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 Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz 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 Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Introduction Background: Riemann-Finsler, Lorentz 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 Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian Conclusion on geodeics

Notion of wind Finslerian structure

Definition For a vector space V : —Wind Minkowskian structure: Compact strongly convex smooth hypersurface ΣV embedded in V —Unit ball B Bounded open domain B enclosed by ΣV —Conic domain A : region determined half lines from 0 to B. 0 ∈ B ⇒ A = V 0 ∈ ΣV ⇒ A = half space 0 ∈ ¯ B ⇒ properly conic A

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian Conclusion on geodeics

Notion of wind Finslerian structure

Definition For a manifold M: — Wind Finslerian str.: smooth hypersurface Σ ֒ → TM: Σp = Σ ∩ TpM is wind Minkowski in TpM (+transversality) — Ball at p: Bp ⊂ TpM ( Ap) — Conic domain A := ∪pAp — Region of strong wind: Ml := {p ∈ M : 0 / ∈ ¯ Bp} — Properly conic domain: Al := Σ ∩ π−1(Ml)

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian Conclusion on geodeics

Notion of wind Finslerian structure

Proposition Any Σ is the displacement of the indicatrix of Finsler metric F0 along some vector field W : F0

• v

Z(v) − W

• = 1,

(v ∈ Σ ⇐ ⇒ Z(v) is a solution) — Uniqueness if 0p is required to be the barycentre of each Fp — Wind Riemannian: displacement of F0 = √gR (ellipsoids)

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian 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) Fl : Al → [0, +∞) Fl is a Lorentz-Finsler metric in the region Ml of strong wind with F < Fl. Moreover, a cone structure appears

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian Conclusion on geodeics

Notion of wind Finslerian structure

Cone structure on Ml: Limit region F = Fl: Cone ∪ A: “Σ-admissible vectors” it characterizes accessibility from x0 to x1 (x0 ≺ x1) For wind Riemannian, associated to a Lorentzian metric Curvatures for F and Fl are computable [JV]

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian Conclusion on geodeics

Balls and geodesics for wind Finsler

No “distance” dF for Σ redefinitions of balls and geodesics for any wind Finsler

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian Conclusion on geodeics

Balls and geodesics for wind Finsler

No “distance” dF for Σ redefinitions of balls and geodesics for any wind Finsler Σ admissible γ from x0 to x: γ′ in a closure of A(⊃ Al). (Forward/backwards) wind balls [mild wind: usual open balls] B+

Σ (x0, r) = {x ∈ M | ∃ γ Σ-a. x0 to x : ℓF(γ) < r < ℓFl(γ)},

B−

Σ (x0, r) = {x ∈ M | ∃ γ Σ-a. x to x0 : ℓF(γ) < r < ℓFl(γ)}.

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian Conclusion on geodeics

Balls and geodesics for wind Finsler

No “distance” dF for Σ redefinitions of balls and geodesics for any wind Finsler Σ admissible γ from x0 to x: γ′ in a closure of A(⊃ Al). (Forward/backwards) wind balls [mild wind: usual open balls] B+

Σ (x0, r) = {x ∈ M | ∃ γ Σ-a. x0 to x : ℓF(γ) < r < ℓFl(γ)},

B−

Σ (x0, r) = {x ∈ M | ∃ γ Σ-a. x to x0 : ℓF(γ) < r < ℓFl(γ)}.

Wind c-balls: ˆ B+

Σ (x0, r) = {x ∈ M | ∃ γ Σ-a. x0 to x : ℓF(γ) ≤ r ≤ ℓFl(γ)},

ˆ B−

Σ (x0, r) = {x ∈ M | ∃ γ Σ-a. x to x0 : ℓF(γ) ≤ r ≤ ℓFl(γ)}.

Closed balls: (usual closures) ¯ B+

Σ (x0, r), ¯

B−

Σ (x0, r)

B+

Σ (x0, r) ⊂ ˆ

B+

Σ (x0, r) ⊂ ¯

B+

Σ (x0, r)

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian Conclusion on geodeics

Balls and geodesics for wind Finsler

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian 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 Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian 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 Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian 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 Fl.

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian 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 Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian Conclusion on geodeics

Definition of SSTK spacetime

SSTK s.t: standard with a space-transverse Killing v.f. (K = ∂t) (R × M, g), g = −Λdt2 + 2ωdt + g0 ≡ −(Λ ◦ π)dt2 + π∗ω ⊗ dt + dt ⊗ π∗ω + π∗g0 for Λ (function), ω (1-form), g0 (Riemannian) on M with Λ > −ω2

0 (Lorentz restriction)

Cases: ω = 0, Λ ≡ 1: Product st : R × M, g = −dt2 + π∗g0 ≡ −dt2 + g0 ω = 0, Λ > 0 Static st : R × M, g = −Λdt2 + g0 = Λ(−dt2 + g0/Λ) Conformal to product

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian Conclusion on geodeics

Product/ static case

K = ∂t induces a Riemannian metric g0(≡ g0/Λ) on M

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian Conclusion on geodeics

SSTK spacetimes

SSTK (R × M, g), g = −Λdt2 + 2ωdt + g0 (with Λ > −ω2

0)

Cases: Λ ≡ 1, arbitrary ω Normalized (standard) stationary s.t. : R × M, g = −1dt2 + 2ωdt + g0 Λ > 0, arbitrary ω Stationary s.t. : R × M, g = −Λdt2 + 2ωdt + g0 Conformal to normalized Λ

• −dt2 + 2(ω/Λ) + (g0/Λ)
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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian Conclusion on geodeics

Stationary case

K = ∂t induces the indicatrix of a Finslerian metric on M

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian Conclusion on geodeics

Stationary case

Induces a (pair of) Finslerian metric on M

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian Conclusion on geodeics

Appearance of Finsler

For each v ∈ TxM: Future-d. lightlike vector (F +(v), v) Past-d. lightlike vector (−F −(v), v) where F ± : TM → R, for normalized Λ ≡ 1: F ±(v) =

• g0(v, v) + ω(v)2 ± ω(v)

F ±: Finsler metrics of Randers type, “Fermat metrics”

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian Conclusion on geodeics

Appearance of Finsler

For each v ∈ TxM: Future-d. lightlike vector (F +(v), v) Past-d. lightlike vector (−F −(v), v) where F ± : TM → R, for normalized Λ ≡ 1: F ±(v) =

• g0(v, v) + ω(v)2 ± ω(v)

F ±: Finsler metrics of Randers type, “Fermat metrics” F −(v) = F +(−v), F − “reversed metric” of F + (≡ F).

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian Conclusion on geodeics

Expression with wind

F: Randers metric with indicatrix Σ = SR + W W : vector field (wind): g0(W , ·) = −ω SR: Riemannian metric indicatrix (sphere bundle) of gR = g0/(1 + |W |2

0)

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian Conclusion on geodeics

Expression with wind

F: Randers metric with indicatrix Σ = SR + W W : vector field (wind): g0(W , ·) = −ω SR: Riemannian metric indicatrix (sphere bundle) of gR = g0/(1 + |W |2

0)

Necessarily gR(W , W ) < 1 (mild wind)

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian Conclusion on geodeics

General SSTK spacetime

SSTK (R × M, g), g = −Λdt2 + 2ωdt + g0 (with Λ > −ω2

0)

General case: K := ∂t Killing and    timelike Λ > 0 lightlike Λ = 0 spacelike Λ < 0 The projection t : R × M → R time function [for v causal dt(v) > 0 defines the future direction]

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian Conclusion on geodeics

Interpretation of K = ∂t

• M. S´

anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian Conclusion on geodeics

Interpretation of K = ∂t

K = ∂t induces a wind-Riemannian structure

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian 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 : g0(·, W ) = −ω Riemannian metric gR = g0/(Λ + g0(W , W ))

Moreover, cone structure on Ml computable from the sign. changing metric h (Lorentzian (+, −, . . . , −) on Ml)

h(v, v) = Λg0(v, v) + ω(v)2

Conversely, each wind Riemannian structure selects a unique conformal class of SSTK

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian 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

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian 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

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo General Wind Finslerian structures Spacetimes and Wind Riemannian 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) Fl (on strong wind region Ml, locally maximizing) and lightlike for −h (Lorentzian metric on Ml)

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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 p0 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 p0 to α

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... and some of the applications to Lorentz

Sharpest Fermat for SSTK

Theorem Let (R × M, g) SSTK, x0, x1 ∈ M, x0 = x1, p0 = (t0, x0) and γ(s) = (ζ(s), x(s)) lightlike from p0 to R × {x1}. a) γ critical point of the arrival time T = ⇒ pregeodesic. b) γ pregeodesic ⇐ ⇒ (Cγ = g(∂t, ˙ γ) constant and:)

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... and some of the applications to Lorentz

Sharpest Fermat for SSTK

Theorem Let (R × M, g) SSTK, x0, x1 ∈ M, x0 = x1, p0 = (t0, x0) and γ(s) = (ζ(s), x(s)) lightlike from p0 to R × {x1}. a) γ critical point of the arrival time T = ⇒ pregeodesic. b) γ pregeodesic ⇐ ⇒ (Cγ = g(∂t, ˙ γ) constant and:) (i) Cγ < 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) Cγ > 0, Λ < 0 on all x, x a pregeodesic of Fl parametrized with h( ˙ x, ˙ x) = const., γ critical point of T (locally max.) (iii) Cγ = 0, Λ ≤ 0 on all x: whenever Λ < 0, x lightlike geodesic

• f h/Λ on M; Λ vanishes on x only at isolated points where ˙

x vanishes.

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... and some of the applications to Lorentz

Characterization for Zermelo

For arbitrary gR, W (generalizing Shen’s et al. [Sh], [BRS]): Solutions x(s) of Zermelo’s connecting x0, x1 are (pre)geodesics of Σ and they lie in exactly one of the three previous cases.

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... and some of the applications to Lorentz

Characterization for Zermelo

For arbitrary gR, W (generalizing Shen’s et al. [Sh], [BRS]): Solutions x(s) of Zermelo’s connecting x0, x1 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 x0 to x1

(⇐ ⇒ x0 ≺ x1 for −h on M , where h(u, v) := (1 − gR(W , W ))gR(u, v) + gR(u, W )(W , v))

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... and some of the applications to Lorentz

Characterization for Zermelo

For arbitrary gR, W (generalizing Shen’s et al. [Sh], [BRS]): Solutions x(s) of Zermelo’s connecting x0, x1 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 x0 to x1

(⇐ ⇒ x0 ≺ x1 for −h on M , where h(u, v) := (1 − gR(W , W ))gR(u, v) + gR(u, W )(W , v))

2 and Σ is w-convex

(⇐ ⇒ associated SSTK causally simple)

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... and some of the applications to Lorentz

General Fermat principle: precise statement

Theorem (CJS) (L, g) any spacetime, α any arrival curve (smooth, embedded) Np0,α := {γ : [a, b] → L|γ piece. smooth f.-d. light. from p0 to α} Arrival functional: T(γ) = α−1(γ(b)), ∀γ ∈ Np0,α γ ∈ Np0,α with ˙ γ(b) ⊥ α: pregeodesic ⇐ ⇒ critical point of T

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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 γw ∈ Np0,α Lemma 1. γ critical for T ⇔ Z(b) = 0, ∀Z admissible

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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 γw ∈ Np0,α Lemma 1. γ critical for T ⇔ Z(b) = 0, ∀Z admissible Proof. Z(b) = d dw γw(b) |w=0= d dw α(T(γw)) |w=0 = d dw T(γw) |w=0

• ˙

α(T(γ))

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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 admissible ⇐ ⇒ Z ′ ⊥ ˙ γ

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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 admissible ⇐ ⇒ Z ′ ⊥ ˙ γ (⇒ trivial)

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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 admissible ⇐ ⇒ Z ′ ⊥ ˙ γ (⇒ trivial)

• Note: (⇐) Typical results

(i) no lightlike longit. or (ii) geodesic γ ⊥ α non-lightlike

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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 admissible ⇐ ⇒ Z ′ ⊥ ˙ 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).

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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 admissible ⇐ = Z ′ ⊥ ˙ 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) = ΠS(γ(s))

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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 admissible ⇐ = Z ′ ⊥ ˙ 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) = ΠS(γ(s)) Lift this variation imposing longitudinal curves in Np0,α

• diff. eqn. for t coordinate
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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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 admissible ⇐ = Z ′ ⊥ ˙ 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) = ΠS(γ(s)) Lift this variation imposing longitudinal curves in Np0,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

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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: ZW (s) = W (s) + fW (s)U(s), where fW (s) = −e−ρ(s) s

a

g(W ′, ˙ γ) g(U, ˙ γ) eρdµ with ρ(s) = s

a

g(U′, ˙ γ) g(U, ˙ γ) dµ

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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: ZW (s) = W (s) + fW (s)U(s), where fW (s) = −e−ρ(s) s

a

g(W ′, ˙ γ) g(U, ˙ γ) eρdµ with ρ(s) = s

a

g(U′, ˙ γ) g(U, ˙ γ) dµ Sketch of proof. ZW is admissible: check g(Z ′

W , ˙

γ) = 0 (eqn for fW )

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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: ZW (s) = W (s) + fW (s)U(s)

• Sketch. Any admissible Z is some ZW :
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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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: ZW (s) = W (s) + fW (s)U(s)

• Sketch. Any admissible Z is some ZW :

1 Define W (s) = Z(s) − (c(s − a)/(b − a))U(s)

with c s.t. Z(b) = cU(b).

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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: ZW (s) = W (s) + fW (s)U(s)

• Sketch. Any admissible Z is some ZW :

1 Define W (s) = Z(s) − (c(s − a)/(b − a))U

with c s.t. Z(b) = cU(b).

2 Z and ZW admissible ⇒ Z − ZW admissible...

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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: ZW (s) = W (s) + fW (s)U(s)

• Sketch. Any admissible Z is some ZW :

1 Define W (s) = Z(s) − (c(s − a)/(b − a))U

with c s.t. Z(b) = cU(b).

2 Z and ZW admissible ⇒ Z − ZW admissible ... 3 ... but Z − ZW = (fW (s) − c(s − a)/(b − a)) U =: p(s)U

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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: ZW (s) = W (s) + fW (s)U(s)

• Sketch. Any admissible Z is some ZW :

1 Define W (s) = Z(s) − (c(s − a)/(b − a))U

with c: Z(b) = cU(b).

2 Z and ZW admissible ⇒ Z − ZW admissible ... 3 ... but Z − ZW = (fW (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

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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 = ZW = W + fW U ⇔ fW (b) = 0, as W (b) = 0 (= W (a))

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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 = ZW = W + fW U ⇔ fW (b) = 0 (as W (b) = 0 = W (a)) Using the explicit formula for fW : ⇔ b

a g(W ′, ˙ γ) g(U, ˙ γ) eρdµ = 0.

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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 = ZW = W + fW U ⇔ fW (b) = 0 (as W (b) = 0 = W (a)) Using the explicit formula for fW : ⇔ b

a g(W ′, ˙ γ) g(U, ˙ γ) eρdµ = 0.

Integrating by parts (with smooth W vanishing at breaks) ⇔ b

a g(W , (ϕ˙

γ)′)dµ = 0, for some function ϕ

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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 = ZW = W + fW U ⇔ fW (b) = 0 (as W (b) = 0 = W (a)) Using the explicit formula for fW : ⇔ b

a g(W ′, ˙ γ) g(U, ˙ γ) eρdµ = 0.

Integrating by parts (with smooth W vanishing at breaks) ⇔ b

a g(W , (ϕ˙

γ)′)dµ = 0, for some function ϕ Using standard variational arguments: ⇔ (ϕ˙ γ)′ = 0 (well-known characterization of pregeodesics)

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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)

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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]

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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 ¯

B+

Σ (x0, r1) ∩ ¯

B−

Σ (x1, r2) compact

2 All intersections ˆ

B+

Σ (x0, r1) ∩ ˆ

B−

Σ (x1, r2)

3 In the case of K timelike (stationary/Randers):

Compactness of ¯ B+

s (p, r)

Spacelike slices St = {(t, x) : x ∈ R × M} Cauchy hypers. (crossed exactly once by any inextendible causal curve) equivalent to any of:

1 All closed ¯

B+

Σ (x, r), ¯

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

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Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... 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´

• rio [JV]
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Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... and some of the applications to Lorentz

References

[CJS] Caponio, Javaloyes, S´ anchez: arxiv 1407.5494 Lorentz-Finsler [CJS11] Caponio, Javaloyes, S´ anchez: Rev. Mat. Iberoam (2011) [CJM] Caponio, Javaloyes, Masiello: Math. Ann. (2011) + [FHS] Flores, Herrera, S´ anchez: Memoirs AMS (2013) Fermat’s principle, visibility and lensing [Ko] Kovner: Astroph. J. (1990) [Pe] Perlick: Class. Quant. Grav (1990) [FGM] Fortunato, Giannoni, Masiello, J. Geom. Phys. (1995) [CGS] Caponio, Germinario, S´ anchez, J. Geom. Anal. (2016)

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Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... and some of the applications to Lorentz

References

Zermelo’s navigation [Sh] Shen: Canadian J. Math. (2003) [BRS] Bao, Robles, Shen: J. Diff. Geom. (2004) [YS] Yoshikawa, Sabau: Geom. Dedicata (2014) [JV] Javaloyes, Vit´

• rio, arXiv:1412.0465.

Related Finslerian problems [FHa] Flores, Harris: Class. Quant. Grav. (2007) [JV] Javaloyes, Vit´

• rio, in progress

[LN] Li, Nirenberg: Comm. Pure Appl. Math. (2005) [Ma] Matveev: Springer Proc. Math. & Stat. 26 (2013) [TS] Tanaka, Sabau: arXiv:1207.0918

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anchez Generalized Fermat and Zermelo

Overview Finslerian and spacetime viewpoints Applications: Fermat and Zermelo Fermat and Zermelo Generalized Fermat: Sketch of proof ... and some of the applications to Lorentz

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• M. S´

anchez Generalized Fermat and Zermelo