OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat vs Zermelo
Generalized Fermat principle and Zermelonavigation: a link between
Lorentzian and Generalized Finslerian Geometries
Miguel Sanchez
Universidad de Granada, IEMath-GR
8th Int. Meeting on Lorentzian Geom. (Malaga, 23/09/2016)
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat vs Zermelo
General aims
Show a correspondence between problems in:
1 Lorentzian Geometry
2 Finslerian and generalized (singular) Finslerian Geometry
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat vs Zermelo
General aims
Applications:
1 For Lorentz: appropriate description of relativistic notions inFinslerian terms
2 For Finsler: new problems and results by using Lorentzianviewpoint
3 Dynamical systems/optimal control:Non singular description of apparently singular problems
Emphasis in the most general viewpoint! extended and singular Finslerand (non-singular) relativistic interpretations
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat vs Zermelo
General aims
Applications:
1 For Lorentz: appropriate description of relativistic notions inFinslerian terms
2 For Finsler: new problems and results by using Lorentzianviewpoint
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. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat vs Zermelo
General aims
Applications:
1 For Lorentz: appropriate description of relativistic notions inFinslerian terms
2 For Finsler: new problems and results by using Lorentzianviewpoint
3 Dynamical systems/optimal control:Non singular description of apparently singular problems
Emphasis in the most general viewpoint! extended and singular Finslerand (non-singular) relativistic interpretations
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat 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. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat 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, Sanchez’11 [CJS11]
(+ others)
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat vs Zermelo
Background: Riemannian-Finsler, Lorentz
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat 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 t 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. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat 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. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat 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. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat vs Zermelo
Zermelo navigation
Riemannian metric 〈·, ·〉: unit spheresSp 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. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat vs Zermelo
Zermelo navigation
Riemannian metric 〈·, ·〉: unit spheresSp 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 〉 < 1M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat 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. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat vs Zermelo
Zermelo navigation
Strong wind 〈Wp,Wp〉 > 1
No Finsler metric but Σ ⊂ TM still makes sense“Wind Riemannian/ Finslerian structure”
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat 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. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat 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 atsome p ∈ M:
K ⊥ Π (and unit): Euclidean indicatrix of Π from the cone
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat vs Zermelo
Remark on projection of cones
Lorentzian is richer (rather than a generalization) of RiemannianChoose a spacelike hyperplane Π and a transversal vector K atsome p ∈ M:
K ⊥ Π (and unit): Euclidean indicatrix of Π from the cone
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat vs Zermelo
Remark on projection of cones
Choose a spacelike hyperplane Π and a transversal direction K atsome p ∈ M:
K timelike but non orthogonal: “Finslerian” indicatrix fromthe cone
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat vs Zermelo
Remark on projection of cones
Choose a spacelike hyperplane Π and a transversal direction K atsome p ∈ M:
K lightlike: Koprina/ “critical wind” indicatrix
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat vs Zermelo
Remark on projection of cones
Choose a spacelike hyperplane Π and a transversal direction K atsome p ∈ M:
K spacelike: “strong wind” indicatrix (+ cone on Π)
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat 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 orlightlike)
Interpretations
f-d timelike (unit) curves ≡ observers
f-d lightlike geod. ≡ light rays
Fermat principle ≡ light arrives fastest/critical
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat 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 orlightlike)
Interpretations
f-d timelike (unit) curves ≡ observers
f-d lightlike geod. ≡ light rays
Fermat principle ≡ light arrives fastest/critical
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat 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 curvesfor 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. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat 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 curvesfor 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. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat vs Zermelo
Link Fermat/Zermelo
Start with Zermelo on M and represent graphs of curves adding acoordinate “time” as a dimension more
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat vs Zermelo
Link Fermat/Zermelo
Maximum velocities: add a “unit of time” to all theindicatrices cone structure compatible with a (conformalclass of) Lorentz g (independent of t)
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat 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×Mfrom (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 observerαy (s) = (s, y) ∈ R× {y}
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat 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 geodesicThe relation with spacetimes holds Fermat principle solves Zermelo problem
Overall goal basics on wind Finslerian, spacetimes andFinsler/Lorentz correspondence (including Randers/stationaryspacetimes)
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat 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 geodesicThe relation with spacetimes holds Fermat principle solves Zermelo problem
Overall goal basics on wind Finslerian, spacetimes andFinsler/Lorentz correspondence (including Randers/stationaryspacetimes)
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat 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 geodesicThe relation with spacetimes holds
Fermat principle solves Zermelo problem
Overall goal basics on wind Finslerian, spacetimes andFinsler/Lorentz correspondence (including Randers/stationaryspacetimes)
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat 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 geodesicThe relation with spacetimes holds Fermat principle solves Zermelo problem
Overall goal basics on wind Finslerian, spacetimes andFinsler/Lorentz correspondence (including Randers/stationaryspacetimes)
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat 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 geodesicThe relation with spacetimes holds Fermat principle solves Zermelo problem
Overall goal basics on wind Finslerian, spacetimes andFinsler/Lorentz correspondence (including Randers/stationaryspacetimes)
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
IntroductionBackground: Riemann-Finsler, LorentzFermat vs Zermelo
Plan
General wind Finslerian structures+ Spacetime viewpoint
Applications: generalized Fermat and Zermelo (and more)
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
General Wind Finslerian structuresSpacetimes and Wind RiemannianConclusion on geodeics
Notion of wind Finslerian structure
Definition
For a vector space V :—Wind Minkowskian structure: Compact strongly convexsmooth 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 6∈ B ⇒ properly conic A
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
General Wind Finslerian structuresSpacetimes and Wind RiemannianConclusion 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. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
General Wind Finslerian structuresSpacetimes and Wind RiemannianConclusion 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)
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
General Wind Finslerian structuresSpacetimes and Wind RiemannianConclusion 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 regionMl of strong wind with F < Fl . Moreover, a cone structureappears
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
General Wind Finslerian structuresSpacetimes and Wind RiemannianConclusion 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 metricCurvatures for F and Fl are computable [JV]
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
General Wind Finslerian structuresSpacetimes and Wind RiemannianConclusion 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 : `F (γ) < r < `Fl
(γ)},B−Σ (x0, r) = {x ∈ M | ∃ γ Σ-a. x to x0 : `F (γ) < r < `Fl
(γ)}.
Wind c-balls:
B+Σ (x0, r) = {x ∈ M | ∃ γ Σ-a. x0 to x : `F (γ) ≤ r ≤ `Fl
(γ)},B−Σ (x0, r) = {x ∈ M | ∃ γ Σ-a. x to x0 : `F (γ) ≤ r ≤ `Fl
(γ)}.
Closed balls: (usual closures) B+Σ (x0, r), B−Σ (x0, r)
B+Σ (x0, r) ⊂ B+
Σ (x0, r) ⊂ B+Σ (x0, r)
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
General Wind Finslerian structuresSpacetimes and Wind RiemannianConclusion 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 : `F (γ) < r < `Fl
(γ)},B−Σ (x0, r) = {x ∈ M | ∃ γ Σ-a. x to x0 : `F (γ) < r < `Fl
(γ)}.
Wind c-balls:
B+Σ (x0, r) = {x ∈ M | ∃ γ Σ-a. x0 to x : `F (γ) ≤ r ≤ `Fl
(γ)},B−Σ (x0, r) = {x ∈ M | ∃ γ Σ-a. x to x0 : `F (γ) ≤ r ≤ `Fl
(γ)}.
Closed balls: (usual closures) B+Σ (x0, r), B−Σ (x0, r)
B+Σ (x0, r) ⊂ B+
Σ (x0, r) ⊂ B+Σ (x0, r)
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
General Wind Finslerian structuresSpacetimes and Wind RiemannianConclusion 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 : `F (γ) < r < `Fl
(γ)},B−Σ (x0, r) = {x ∈ M | ∃ γ Σ-a. x to x0 : `F (γ) < r < `Fl
(γ)}.
Wind c-balls:
B+Σ (x0, r) = {x ∈ M | ∃ γ Σ-a. x0 to x : `F (γ) ≤ r ≤ `Fl
(γ)},B−Σ (x0, r) = {x ∈ M | ∃ γ Σ-a. x to x0 : `F (γ) ≤ r ≤ `Fl
(γ)}.
Closed balls: (usual closures) B+Σ (x0, r), B−Σ (x0, r)
B+Σ (x0, r) ⊂ B+
Σ (x0, r) ⊂ B+Σ (x0, r)
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
General Wind Finslerian structuresSpacetimes and Wind RiemannianConclusion on geodeics
Balls and geodesics for wind Finsler
w-convexity: c-balls are closed (extend usual convexity)
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
General Wind Finslerian structuresSpacetimes and Wind RiemannianConclusion on geodeics
Balls and geodesics for wind Finsler
w-convexity: c-balls are closed (extend usual convexity)M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
General Wind Finslerian structuresSpacetimes and Wind RiemannianConclusion on geodeics
Balls and geodesics for wind Finsler
Geodesic parametrized by arc length: Σ-admissible curve s.t.γ(t + ε) ∈ B+
Σ (γ(t), ε) \ B+Σ (γ(t), ε) (locally, i.e., for small ε > 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. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
General Wind Finslerian structuresSpacetimes and Wind RiemannianConclusion on geodeics
Balls and geodesics for wind Finsler
Geodesic parametrized by arc length: Σ-admissible curve s.t.γ(t + ε) ∈ B+
Σ (γ(t), ε) \ B+Σ (γ(t), ε) (locally, i.e., for small ε > 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. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
General Wind Finslerian structuresSpacetimes and Wind RiemannianConclusion on geodeics
Going further
What about when γ is Σ-admissible but belongs to ∂A?
In principle, one could follow but there are technicaldifficulties (“abnormal” geodesics) Focus on wind Riemannian (but generalizable to Finslerian)
Develop in a “non-singular” way through the spacetimeviewpoint
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
General Wind Finslerian structuresSpacetimes and Wind RiemannianConclusion 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 withΛ > −‖ω‖2
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
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
General Wind Finslerian structuresSpacetimes and Wind RiemannianConclusion on geodeics
Product/ static case
K = ∂t induces a Riemannian metric g0(≡ g0/Λ) on M
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
General Wind Finslerian structuresSpacetimes and Wind RiemannianConclusion on geodeics
SSTK spacetimes
SSTK (R×M, g), g = −Λdt2 + 2ωdt + g0 (with Λ > −‖ω‖20)
Cases:
Λ ≡ 1, arbitrary ωNormalized (standard) stationary s.t. :R×M, g = −1dt2 + 2ωdt + g0
Λ > 0, arbitrary ωStationary s.t. : R×M, g = −Λdt2 + 2ωdt + g0
Conformal to normalized Λ(−dt2 + 2(ω/Λ) + (g0/Λ)
)
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
General Wind Finslerian structuresSpacetimes and Wind RiemannianConclusion on geodeics
Stationary case
K = ∂t induces the indicatrix of a Finslerian metric on MM. Sanchez Generalized Fermat and Zermelo
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Stationary case
Induces a (pair of) Finslerian metric on MM. Sanchez Generalized Fermat and Zermelo
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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 ).
M. Sanchez Generalized Fermat and Zermelo
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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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Expression with wind
F : Randers metric with indicatrix
Σ = SR + W
W : vector field (wind):g0(W , ·) = −ωSR : Riemannian metric indicatrix (sphere bundle) ofgR = g0/(1 + |W |20)
Necessarily gR(W ,W ) < 1 (mild wind)
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Expression with wind
F : Randers metric with indicatrix
Σ = SR + W
W : vector field (wind):g0(W , ·) = −ωSR : Riemannian metric indicatrix (sphere bundle) ofgR = g0/(1 + |W |20)
Necessarily gR(W ,W ) < 1 (mild wind)
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General SSTK spacetime
SSTK (R×M, g), g = −Λdt2 + 2ωdt + g0 (with Λ > −‖ω‖20)
General case:
K := ∂t Killing and
timelike Λ > 0lightlike Λ = 0spacelike Λ < 0
The projection t : R×M → R time function[for v causal dt(v) > 0 defines the future direction]
M. Sanchez Generalized Fermat and Zermelo
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Interpretation of K = ∂t
M. Sanchez Generalized Fermat and Zermelo
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Interpretation of K = ∂t
K = ∂t induces a wind-Riemannian structure
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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 theconformal class of the SSTK) with;
Σ computable from:
Wind vector W : g0(·,W ) = −ωRiemannian 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 uniqueconformal class of SSTK
M. Sanchez Generalized Fermat and Zermelo
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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 forSSTK 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/Kropinaand strong wind regions)Fl (on strong wind region Ml , locally maximizing) andlightlike for −h (Lorentzian metric on Ml)
M. Sanchez Generalized Fermat and Zermelo
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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 forSSTK 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/Kropinaand strong wind regions)Fl (on strong wind region Ml , locally maximizing) andlightlike for −h (Lorentzian metric on Ml)
M. Sanchez Generalized Fermat and Zermelo
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General Wind Finslerian structuresSpacetimes and Wind RiemannianConclusion 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 forSSTK 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/Kropinaand strong wind regions)Fl (on strong wind region Ml , locally maximizing) andlightlike for −h (Lorentzian metric on Ml)
M. Sanchez Generalized Fermat and Zermelo
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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 fromp0 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 γcome from a variation by lightlike curves from p0 to α
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Sharpest Fermat for SSTK
Theorem
Let (R×M, g) SSTK, x0, x1 ∈ M, x0 6= x1, p0 = (t0, x0) andγ(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 withh(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 parametrizedwith h(x , x) = const., γ critical point of T (locally max.)
(iii) Cγ = 0, Λ ≤ 0 on all x : whenever Λ < 0, x lightlike geodesicof h/Λ on M; Λ vanishes on x only at isolated points where xvanishes.
M. Sanchez Generalized Fermat and Zermelo
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Sharpest Fermat for SSTK
Theorem
Let (R×M, g) SSTK, x0, x1 ∈ M, x0 6= x1, p0 = (t0, x0) andγ(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 withh(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 parametrizedwith h(x , x) = const., γ critical point of T (locally max.)
(iii) Cγ = 0, Λ ≤ 0 on all x : whenever Λ < 0, x lightlike geodesicof h/Λ on M; Λ vanishes on x only at isolated points where xvanishes.
M. Sanchez Generalized Fermat and Zermelo
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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 threeprevious cases.
If solution exists if:
1 An admissible curve exists from x0 to x1
(⇐⇒ x0 ≺ x1 for −h on M , whereh(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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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 threeprevious cases.
If solution exists if:
1 An admissible curve exists from x0 to x1
(⇐⇒ x0 ≺ x1 for −h on M , whereh(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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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 threeprevious cases.
If solution exists if:
1 An admissible curve exists from x0 to x1
(⇐⇒ x0 ≺ x1 for −h on M , whereh(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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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) 6⊥ α:pregeodesic ⇐⇒ critical point of T
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Lemma 1: critical in terms of admissible v.f. Z
Z v.f. on γ admissible: variational v.f. by means of longitudinalcurves γw ∈ Np0,α
Lemma 1. γ critical for T ⇔ Z (b) = 0, ∀Z admissible
Proof.
Z (b) =d
dwγw (b) |w=0=
d
dwα(T (γw )) |w=0
=
(d
dwT (γw ) |w=0
)α(T (γ))
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Lemma 1: critical in terms of admissible v.f. Z
Z v.f. on γ admissible: variational v.f. by means of longitudinalcurves γw ∈ Np0,α
Lemma 1. γ critical for T ⇔ Z (b) = 0, ∀Z admissible
Proof.
Z (b) =d
dwγw (b) |w=0=
d
dwα(T (γw )) |w=0
=
(d
dwT (γw ) |w=0
)α(T (γ))
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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
M. Sanchez Generalized Fermat and Zermelo
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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
M. Sanchez Generalized Fermat and Zermelo
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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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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 coordinateswhich looks like a t-dependent SSTK and:(a) γ nowhere orthogonal to ∂t(b) α ‖ ∂t at γ(b).
M. Sanchez Generalized Fermat and Zermelo
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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 coordinateswhich 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× SW : 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
M. Sanchez Generalized Fermat and Zermelo
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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 coordinateswhich 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× SW : 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
M. Sanchez Generalized Fermat and Zermelo
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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 coordinateswhich 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× SW : 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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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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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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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).
M. Sanchez Generalized Fermat and Zermelo
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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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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))Uwith c s.t. Z (b) = cU(b).
2 Z and ZW admissible ⇒ Z − ZW admissible...
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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))Uwith 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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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))Uwith 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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Sketch proof of theorem
Lemma 1: γ critical for T⇔ Z (b) = 0 for all admissible Z
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General Fermat principle: proof
Lemma 1: γ critical for T⇔ Z (b) = 0 for all admissible Z
Lemma 3 (chosen U): Z = ZW = W + fWU⇔ fW (b) = 0, as W (b) = 0 (= W (a))
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General Fermat principle: proof
Lemma 1: γ critical for T⇔ Z (b) = 0 for all admissible Z
Lemma 3 (chosen U): Z = ZW = W + fWU⇔ fW (b) = 0 (as W (b) = 0 = W (a))
Using the explicit formula for fW :
⇔∫ ba
g(W ′,γ)g(U,γ) e
ρdµ = 0.
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General Fermat principle: proof
Lemma 1: γ critical for T⇔ Z (b) = 0 for all admissible Z
Lemma 3 (chosen U): Z = ZW = W + fWU⇔ fW (b) = 0 (as W (b) = 0 = W (a))
Using the explicit formula for fW :
⇔∫ ba
g(W ′,γ)g(U,γ) e
ρdµ = 0.
Integrating by parts (with smooth W vanishing at breaks)
⇔∫ ba g(W , (ϕγ)′)dµ = 0, for some function ϕ
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General Fermat principle: proof
Lemma 1: γ critical for T⇔ Z (b) = 0 for all admissible Z
Lemma 3 (chosen U): Z = ZW = W + fWU⇔ fW (b) = 0 (as W (b) = 0 = W (a))
Using the explicit formula for fW :
⇔∫ ba
g(W ′,γ)g(U,γ) e
ρdµ = 0.
Integrating by parts (with smooth W vanishing at breaks)
⇔∫ ba g(W , (ϕγ)′)dµ = 0, for some function ϕ
Using standard variational arguments:⇔ (ϕγ)′ = 0 (well-known characterization of pregeodesics)
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Characterization of the causal ladder
SSTK are always stably continuous (t time function)
Causal continuity characterizable in terms of the associatedwind Finslerian structure
Causal simplicity and global hyperbolicity especially interesting
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OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
Fermat and ZermeloGeneralized 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], becomeapplicable providing results on existence and multiplicityApplications even for stationary s.t.:
Extension of previous results
Applications to gravitational lensing [CGS]
...now extensible to SSTK
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
Fermat and ZermeloGeneralized 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], becomeapplicable 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. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
Fermat and ZermeloGeneralized 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], becomeapplicable providing results on existence and multiplicityApplications even for stationary s.t.:
Extension of previous results
Applications to gravitational lensing [CGS]
...now extensible to SSTK
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
Fermat and ZermeloGeneralized 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], becomeapplicable providing results on existence and multiplicityApplications even for stationary s.t.:
Extension of previous results
Applications to gravitational lensing [CGS]
...now extensible to SSTK
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
Fermat and ZermeloGeneralized 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 B+Σ (x0, r1) ∩ 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 B+Σ (x , r), B−
Σ (x , r) compact3 Σ (forward and backward) geodesically complete
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
Fermat and ZermeloGeneralized 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 FinslerianGeometries, now extensible to wind Finsler
Application to Lorentz Geom. [FHS]:description of the c-boundary of static/ stationary/ SSTK s.t. interms of Riemannian [FHa]/ Finslerian/ wind Finslerian elements
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
Fermat and ZermeloGeneralized 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 FinslerianGeometries, now extensible to wind Finsler
Application to Lorentz Geom. [FHS]:description of the c-boundary of static/ stationary/ SSTK s.t. interms of Riemannian [FHa]/ Finslerian/ wind Finslerian elements
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
Fermat and ZermeloGeneralized 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 asubset d(C , ·) with applications to Hamilton Jacobi equation(extended by Tanaka & Sabau [TS])
3 Properties of completeness in classes of projectively relatedmetrics (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 flagcurvature [CJS14] and Javaloyes & Vitorio [JV]
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
Fermat and ZermeloGeneralized Fermat: Sketch of proof... and some of the applications to Lorentz
References
[CJS] Caponio, Javaloyes, Sanchez: arxiv 1407.5494
Lorentz-Finsler
[CJS11] Caponio, Javaloyes, Sanchez: Rev. Mat. Iberoam (2011)[CJM] Caponio, Javaloyes, Masiello: Math. Ann. (2011)+ [FHS] Flores, Herrera, Sanchez: 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, Sanchez, J. Geom. Anal. (2016)
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
Fermat and ZermeloGeneralized 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, Vitorio, arXiv:1412.0465.
Related Finslerian problems
[FHa] Flores, Harris: Class. Quant. Grav. (2007)[JV] Javaloyes, Vitorio, in progress[LN] Li, Nirenberg: Comm. Pure Appl. Math. (2005)[Ma] Matveev: Springer Proc. Math. & Stat. 26 (2013)[TS] Tanaka, Sabau: arXiv:1207.0918
M. Sanchez Generalized Fermat and Zermelo
OverviewFinslerian and spacetime viewpoints
Applications: Fermat and Zermelo
Fermat and ZermeloGeneralized Fermat: Sketch of proof... and some of the applications to Lorentz
Thanks!
THANK YOU FOR
YOUR ATTENTION
M. Sanchez Generalized Fermat and Zermelo