On the interplay between Lorentzian Causality and Finsler metrics of Randers type

Transcription

1 On the interplay between Lorentzian Causality and Finsler metrics of Randers type Erasmo Caponio, Miguel Angel Javaloyes and Miguel Sánchez Universidad de Granada WORKSHOP ON FINSLER GEOMETRY AND ITS APPLICATIONS Debrecen, May (2009) E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 1 / 19

2 Interplay between Randers metrics and stationary spacetimes Every splitting (R S, g) of a stationary spacetimes determines a Randers metric R in S. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 2 / 19

3 Interplay between Randers metrics and stationary spacetimes Every splitting (R S, g) of a stationary spacetimes determines a Randers metric R in S Causal properties of the spacetime. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 2 / 19

4 Interplay between Randers metrics and stationary spacetimes Every splitting (R S, g) of a stationary spacetimes determines a Randers metric R in S Causal properties of the spacetime Hopf-Rinow properties of the Randers metric E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 2 / 19

5 Interplay between Randers metrics and stationary spacetimes Every splitting (R S, g) of a stationary spacetimes determines a Randers metric R in S Causal properties of the spacetime Hopf-Rinow properties of the Randers metric There are many splittings associated to the same spacetime E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 2 / 19

6 Interplay between Randers metrics and stationary spacetimes Every splitting (R S, g) of a stationary spacetimes determines a Randers metric R in S Causal properties of the spacetime There are many splittings associated to the same spacetime Hopf-Rinow properties of the Randers metric The Randers metrics associated to different splittings have the same pregeodesics and common properties E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 2 / 19

7 Interplay between Randers metrics and stationary spacetimes Every splitting (R S, g) of a stationary spacetimes determines a Randers metric R in S Causal properties of the spacetime There are many splittings associated to the same spacetime Global hyperbolicity is equivalent to the following condition (A) : B + (p, r) B (p, r) compact p S and r > 0 for the Randers metric R Hopf-Rinow properties of the Randers metric The Randers metrics associated to different splittings have the same pregeodesics and common properties E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 2 / 19

8 Interplay between Randers metrics and stationary spacetimes Every splitting (R S, g) of a stationary spacetimes determines a Randers metric R in S Causal properties of the spacetime There are many splittings associated to the same spacetime Global hyperbolicity is equivalent to the following condition (A) : B + (p, r) B (p, r) compact p S and r > 0 for the Randers metric R Hopf-Rinow properties of the Randers metric The Randers metrics associated to different splittings have the same pregeodesics and common properties Condition (A) implies: (a) convexity of R (b) the existence of f : S R such that R f = R + df is forward and backward complete E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 2 / 19

9 Interplay between Randers metrics and stationary spacetimes E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 3 / 19

10 Interplay between Randers metrics and stationary spacetimes Cauchy horizons of spt are related with the distance function to or from a subset. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 3 / 19

11 Interplay between Randers metrics and stationary spacetimes Cauchy horizons of spt are related with the distance function to or from a subset Differentiability properties of the distance function to a subset are deduced. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 3 / 19

12 Randers metrics E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 4 / 19

13 Randers metrics Randers metrics in a manifold M is a function R : TM R defined as: R(x, v) = h(v, v) + ω x [v] where h is Riemannian and ω a 1-form with ω x h < 1 x M, are basic examples of non-reversible Finsler metrics: R(x, v) R(x, v). E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 4 / 19

14 Randers metrics Randers metrics in a manifold M is a function R : TM R defined as: R(x, v) = h(v, v) + ω x [v] where h is Riemannian and ω a 1-form with ω x h < 1 x M, are basic examples of non-reversible Finsler metrics: R(x, v) R(x, v). Named after the norwegian physicist Gunnar Randers ( ): Randers, G.: On an asymmetrical metric in the fourspace of General Relativity. Phys. Rev. (2) 59, (1941) Gunnar Randers with Albert Einstein E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 4 / 19

15 Stationary spacetimes. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 5 / 19

16 Stationary spacetimes A Lorentzian manifold (M, g) with index 1 (, +,, +). Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 5 / 19

17 Stationary spacetimes A Lorentzian manifold (M, g) with index 1 (, +,, +) timelike if g(v, v) < 0 lightlike if g(v, v) = 0 v TM is causal if g(v, v) 0 spacelike if g(v, v) > 0. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 5 / 19

18 Stationary spacetimes A Lorentzian manifold (M, g) with index 1 (, +,, +) timelike if g(v, v) < 0 lightlike if g(v, v) = 0 v TM is causal if g(v, v) 0 spacelike if g(v, v) > 0. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 5 / 19

19 Stationary spacetimes A Lorentzian manifold (M, g) with index 1 (, +,, +) timelike if g(v, v) < 0 lightlike if g(v, v) = 0 v TM is causal if g(v, v) 0 spacelike if g(v, v) > 0. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 5 / 19

20 Stationary spacetimes A Lorentzian manifold (M, g) with index 1 (, +,, +) timelike if g(v, v) < 0 lightlike if g(v, v) = 0 v TM is causal if g(v, v) 0 spacelike if g(v, v) > 0. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 5 / 19

21 Stationary spacetimes A Lorentzian manifold (M, g) with index 1 (, +,, +) timelike if g(v, v) < 0 lightlike if g(v, v) = 0 v TM is causal if g(v, v) 0 spacelike if g(v, v) > 0 A spacetime is a Lorentzian manifold endowed with a time-orientation. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 5 / 19

22 Stationary spacetimes A Lorentzian manifold (M, g) with index 1 (, +,, +) timelike if g(v, v) < 0 lightlike if g(v, v) = 0 v TM is causal if g(v, v) 0 spacelike if g(v, v) > 0 A spacetime is a Lorentzian manifold endowed with a time-orientation The time-orientation is determined by a timelike vector field T. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 5 / 19

23 Stationary spacetimes A Lorentzian manifold (M, g) with index 1 (, +,, +) timelike if g(v, v) < 0 lightlike if g(v, v) = 0 v TM is causal if g(v, v) 0 spacelike if g(v, v) > 0 A spacetime is a Lorentzian manifold endowed with a time-orientation The time-orientation is determined by a timelike vector field T A causal vector v TM is future-pointing if g(v, T ) < 0 (if g(v, T ) > 0 is past-pointing). Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 5 / 19

24 Stationary spacetimes A Lorentzian manifold (M, g) with index 1 (, +,, +) timelike if g(v, v) < 0 lightlike if g(v, v) = 0 v TM is causal if g(v, v) 0 spacelike if g(v, v) > 0 A spacetime is a Lorentzian manifold endowed with a time-orientation The time-orientation is determined by a timelike vector field T A causal vector v TM is future-pointing if g(v, T ) < 0 (if g(v, T ) > 0 is past-pointing) A stationary spacetime (M, g) is a Lorentzian manifold endowed with a timelike Killing vector field Kerr spacetime. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 5 / 19

25 Lorentzian Causality. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 6 / 19

26 Lorentzian Causality Causality studies if given two points p, q M they are joined by a causal curve. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 6 / 19

27 Lorentzian Causality Causality studies if given two points p, q M they are joined by a causal curve p, q M are chronologically related, and write p q if there exists a future-pointing timelike curve γ from p to q. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 6 / 19

28 Lorentzian Causality Causality studies if given two points p, q M they are joined by a causal curve p, q M are chronologically related, and write p q if there exists a future-pointing timelike curve γ from p to q p, q M are causally related p < q) if there exists a future-pointing causal curve γ from p to q. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 6 / 19

29 Lorentzian Causality Causality studies if given two points p, q M they are joined by a causal curve p, q M are chronologically related, and write p q if there exists a future-pointing timelike curve γ from p to q p, q M are causally related p < q) if there exists a future-pointing causal curve γ from p to q The chronological future of p M is defined as I + (p) = {q M : p q}. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 6 / 19

30 Lorentzian Causality Causality studies if given two points p, q M they are joined by a causal curve p, q M are chronologically related, and write p q if there exists a future-pointing timelike curve γ from p to q p, q M are causally related p < q) if there exists a future-pointing causal curve γ from p to q The chronological future of p M is defined as I + (p) = {q M : p q} The causal future of p M is defined as J + (p) = {q M : p q}. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 6 / 19

31 Lorentzian Causality Causality studies if given two points p, q M they are joined by a causal curve p, q M are chronologically related, and write p q if there exists a future-pointing timelike curve γ from p to q p, q M are causally related p < q) if there exists a future-pointing causal curve γ from p to q The chronological future of p M is defined as I + (p) = {q M : p q} The causal future of p M is defined as J + (p) = {q M : p q} Analogously we define the chronological past I (p) and the causal past J (p).. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 6 / 19

32 The causal ladder Causal properties classify spacetimes depending on the behaviour of causal cones. A spacetime is: Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 7 / 19

33 The causal ladder Causal properties classify spacetimes depending on the behaviour of causal cones. A spacetime is: Chronological if p I + (p) for every p M. Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 7 / 19

34 The causal ladder Causal properties classify spacetimes depending on the behaviour of causal cones. A spacetime is: Chronological if p I + (p) for every p M. Distinguishing if I + (p) = I + (q) or I (p) = I (q) implies p = q Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 7 / 19

35 The causal ladder Causal properties classify spacetimes depending on the behaviour of causal cones. A spacetime is: Chronological if p I + (p) for every p M. Distinguishing if I + (p) = I + (q) or I (p) = I (q) implies p = q Causally continuous if it is distinguishing and the Chronological cones I ± (p) are continuous in p M Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 7 / 19

36 The causal ladder Causal properties classify spacetimes depending on the behaviour of causal cones. A spacetime is: Chronological if p I + (p) for every p M. Distinguishing if I + (p) = I + (q) or I (p) = I (q) implies p = q Causally continuous if it is distinguishing and the Chronological cones I ± (p) are continuous in p M Causally simple if the causal cones J ± (p) are closed for every p M Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 7 / 19

37 The causal ladder Causal properties classify spacetimes depending on the behaviour of causal cones. A spacetime is: Chronological if p I + (p) for every p M. Distinguishing if I + (p) = I + (q) or I (p) = I (q) implies p = q Causally continuous if it is distinguishing and the Chronological cones I ± (p) are continuous in p M Causally simple if the causal cones J ± (p) are closed for every p M Globally hyperbolic if it admits a Cauchy hypersurface (a subset S that meets exactly once every inextendible timelike curve) Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 7 / 19

38 Standard Stationary spacetimes Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 8 / 19

39 Standard Stationary spacetimes Standard Stationary means that M = R S and g((τ, y), (τ, y)) = g 0 (y, y)+2g 0 (δ(x), y)τ β(x)τ 2, where (S, g 0 ) is Riemannian and β(x) > 0. Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 8 / 19

40 Standard Stationary spacetimes Standard Stationary means that M = R S and g((τ, y), (τ, y)) = g 0 (y, y)+2g 0 (δ(x), y)τ β(x)τ 2, where (S, g 0 ) is Riemannian and β(x) > 0. If a stationary spacetime is distinguishing and the Killing field is complete, then it is causally continuous and standard Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 8 / 19

41 Standard Stationary spacetimes Standard Stationary means that M = R S and g((τ, y), (τ, y)) = g 0 (y, y)+2g 0 (δ(x), y)τ β(x)τ 2, where (S, g 0 ) is Riemannian and β(x) > 0. If a stationary spacetime is distinguishing and the Killing field is complete, then it is causally continuous and standard M. A. J. and M. Sánchez, A note on the existence of standard splittings for conformally stationary spacetimes, Classical Quantum Gravity, 25 (2008), pp , 7. Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 8 / 19

42 Fermat principle in standard stationary spacetimes E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 9 / 19

43 Fermat principle in standard stationary spacetimes Relativistic Fermat Principle: lightlike pregeodesics are critical points of the arrival time function corresponding to an observer in a suitable class of lightlike curves lightlike curves L 1 =Observer S Pierre de Fermat ( ) E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 9 / 19

44 Fermat principle in standard stationary spacetimes Relativistic Fermat Principle: lightlike pregeodesics are critical points of the arrival time function corresponding to an observer in a suitable class of lightlike curves If you consider as observer s L 1 (s) = (s, x 1 ) in (R S, g), given a lightlike curve γ = (t, x), the arrival time AT(γ) is t(b)=t(a)+ R q b 1 a β g 1 0(ẋ,δ)+ β g 0(ẋ,ẋ)+ 1 β 2 g 0(ẋ,δ) 2 ds. lightlike curves L 1 =Observer S Pierre de Fermat ( ) E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 9 / 19

45 Fermat principle in standard stationary spacetimes Relativistic Fermat Principle: lightlike pregeodesics are critical points of the arrival time function corresponding to an observer in a suitable class of lightlike curves If you consider as observer s L 1 (s) = (s, x 1 ) in (R S, g), given a lightlike curve γ = (t, x), the arrival time AT(γ) is t(b)=t(a)+ R q b 1 a β g 1 0(ẋ,δ)+ β g 0(ẋ,ẋ)+ 1 β 2 g 0(ẋ,δ) 2 ds. lightlike curves L 1 =Observer S Thi is just because g( γ, γ) = 0, that is g 0 (ẋ, ẋ) + 2g 0 (δ(x), ẋ)ṫ β(x)ṫ 2 = 0 Pierre de Fermat ( ) E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 9 / 19

46 Fermat metric and lightlike geodesics. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 10 / 19

47 Fermat metric and lightlike geodesics Let us define the Fermat metric in S as q F (x,v)= 1 β g 1 0(v,δ)+ β g 0(v,v)+ 1 β 2 g 0(v,δ) 2,. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 10 / 19

48 Fermat metric and lightlike geodesics Let us define the Fermat metric in S as q F (x,v)= 1 β g 1 0(v,δ)+ β g 0(v,v)+ 1 β 2 g 0(v,δ) 2, Theorem A curve s γ(s) = (s, x(s)) is a lightlike pregeodesic of (R S, g) iff s x(s) is a Fermat geodesic with unit speed. Consequences:. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 10 / 19

49 Fermat metric and lightlike geodesics Let us define the Fermat metric in S as q F (x,v)= 1 β g 1 0(v,δ)+ β g 0(v,v)+ 1 β 2 g 0(v,δ) 2, Theorem A curve s γ(s) = (s, x(s)) is a lightlike pregeodesic of (R S, g) iff s x(s) is a Fermat geodesic with unit speed. Einstein Ring Consequences: Gravitational lensing can be studied from geodesic connectedness in Fermat metric Gravitational lensing. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 10 / 19

50 Fermat metric and lightlike geodesics Let us define the Fermat metric in S as q F (x,v)= 1 β g 1 0(v,δ)+ β g 0(v,v)+ 1 β 2 g 0(v,δ) 2, Theorem A curve s γ(s) = (s, x(s)) is a lightlike pregeodesic of (R S, g) iff s x(s) is a Fermat geodesic with unit speed. Consequences: Gravitational lensing can be studied from geodesic connectedness in Fermat metric Existence of t-periodic lightlike geodesics is equivalent to existence of Fermat closed geodesics Einstein Ring Gravitational lensing. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 10 / 19

51 Causality through the Fermat metric E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 11 / 19

52 Causality through the Fermat metric Let d the non-symmetric distance in S associated to the Fermat metric E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 11 / 19

53 Causality through the Fermat metric Let d the non-symmetric distance in S associated to the Fermat metric B + (x 0, s) = {p S : d(x 0, p) < s} forward balls E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 11 / 19

54 Causality through the Fermat metric Let d the non-symmetric distance in S associated to the Fermat metric B + (x 0, s) = {p S : d(x 0, p) < s} forward balls B (x 0, s) = {p S : d(x 0, p) < s} backward balls E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 11 / 19

55 Causality through the Fermat metric Let d the non-symmetric distance in S associated to the Fermat metric B + (x 0, s) = {p S : d(x 0, p) < s} forward balls B (x 0, s) = {p S : d(x 0, p) < s} backward balls Define the symmetrized distance d s (p, q) = 1 (d(p, q) + d(q, p)) 2 and B s (x, r) = {p S : d s (x, p) < r} E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 11 / 19

56 Causality through the Fermat metric Let d the non-symmetric distance in S associated to the Fermat metric B + (x 0, s) = {p S : d(x 0, p) < s} forward balls B (x 0, s) = {p S : d(x 0, p) < s} backward balls Define the symmetrized distance d s (p, q) = 1 (d(p, q) + d(q, p)) 2 and B s (x, r) = {p S : d s (x, p) < r} Let (R S, g) be a standard stationary spacetime. Then I ± (t 0, x 0 ) = s>0 {t 0 ± s} B ± (x 0, s), E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 11 / 19

57 Causality through the Fermat metric Let d the non-symmetric distance in S associated to the Fermat metric B + (x 0, s) = {p S : d(x 0, p) < s} forward balls B (x 0, s) = {p S : d(x 0, p) < s} backward balls Define the symmetrized distance d s (p, q) = 1 (d(p, q) + d(q, p)) 2 and B s (x, r) = {p S : d s (x, p) < r} Let (R S, g) be a standard stationary spacetime. Then I ± (t 0, x 0 ) = s>0 {t 0 ± s} B ± (x 0, s), R (s, p) (0, p) ( t, p) B + (p, s) I + (0, p) I (0, p) S B (p, t) E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 11 / 19

58 Causality through the Fermat metric Theorem Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 12 / 19

59 Causality through the Fermat metric Theorem Let (R S, g) be a standard stationary spacetime. Then (R S, g) is causally continuous and Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 12 / 19

60 Causality through the Fermat metric Theorem Let (R S, g) be a standard stationary spacetime. Then (R S, g) is causally continuous and (a) the following assertions become equivalent: (i) (R S, g) is causally simple, (ii) J + (p) is closed for all p, (iii) J (p) is closed for all p and (iv) the associated Finsler manifold (S, F ) is convex, Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 12 / 19

61 Causality through the Fermat metric Theorem Let (R S, g) be a standard stationary spacetime. Then (R S, g) is causally continuous and (a) the following assertions become equivalent: (i) (R S, g) is causally simple, (ii) J + (p) is closed for all p, (iii) J (p) is closed for all p and (iv) the associated Finsler manifold (S, F ) is convex, (b) it is globally hyperbolic if and only if B + (x, r) B (x, r) is compact for every x S and r > 0. Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 12 / 19

62 Causality through the Fermat metric Theorem Let (R S, g) be a standard stationary spacetime. Then (R S, g) is causally continuous and (a) the following assertions become equivalent: (i) (R S, g) is causally simple, (ii) J + (p) is closed for all p, (iii) J (p) is closed for all p and (iv) the associated Finsler manifold (S, F ) is convex, (b) it is globally hyperbolic if and only if B + (x, r) B (x, r) is compact for every x S and r > 0. (c) a slice {t 0 } S, t 0 R, is a Cauchy hypersurface if and only if the Fermat metric F on S is forward and backward complete. Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 12 / 19

63 Randers metrics with the same geodesics. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 13 / 19

64 Randers metrics with the same geodesics R Let R and R Randers metrics and g and g standard stationary metrics. Define the relations R R R R = df for some f, where f is always a smooth real function on S. S f S S f = {(f (x), x) : x S} φ f : S S f x (f (x), x). Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 13 / 19

65 Randers metrics with the same geodesics R Let R and R Randers metrics and g and g standard stationary metrics. Define the relations R R R R = df for some f, where f is always a smooth real function on S. S f S S f = {(f (x), x) : x S} Then R R if and only if the associated stationary metrics are different splittings of the same spacetime φ f : S S f x (f (x), x). Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 13 / 19

66 Generalized Hopf-Rinow theorem. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 14 / 19

67 Generalized Hopf-Rinow theorem Theorem (Accurate Hopf-Rinow for Randers metrics) Let (S, R) a Randers manifold and given a function f : S R define R f (x, v) = R(x, v) df x (v). The following conditions are equivalent: Heinz Hopf ( ) E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 14 / 19

68 Generalized Hopf-Rinow theorem Theorem (Accurate Hopf-Rinow for Randers metrics) Let (S, R) a Randers manifold and given a function f : S R define R f (x, v) = R(x, v) df x (v). The following conditions are equivalent: (A) the intersection B + (x, r) B (x, r) of (S, R) is compact for every r > 0 and x S Heinz Hopf ( ) E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 14 / 19

69 Generalized Hopf-Rinow theorem Theorem (Accurate Hopf-Rinow for Randers metrics) Let (S, R) a Randers manifold and given a function f : S R define R f (x, v) = R(x, v) df x (v). The following conditions are equivalent: (A) the intersection B + (x, r) B (x, r) of (S, R) is compact for every r > 0 and x S (B) the symmetrized closed balls B s (x, r) of (S, R) are compact for every r > 0 and x S Heinz Hopf ( ) E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 14 / 19

70 Generalized Hopf-Rinow theorem Theorem (Accurate Hopf-Rinow for Randers metrics) Let (S, R) a Randers manifold and given a function f : S R define R f (x, v) = R(x, v) df x (v). The following conditions are equivalent: (A) the intersection B + (x, r) B (x, r) of (S, R) is compact for every r > 0 and x S (B) the symmetrized closed balls B s (x, r) of (S, R) are compact for every r > 0 and x S (C) there exists f such that R f is geodesically complete Heinz Hopf ( ). Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 14 / 19

71 Generalized Hopf-Rinow theorem Theorem (Accurate Hopf-Rinow for Randers metrics) Let (S, R) a Randers manifold and given a function f : S R define R f (x, v) = R(x, v) df x (v). The following conditions are equivalent: (A) the intersection B + (x, r) B (x, r) of (S, R) is compact for every r > 0 and x S (B) the symmetrized closed balls B s (x, r) of (S, R) are compact for every r > 0 and x S (C) there exists f such that R f is geodesically complete (D) there exists f and p S such that the forward and the backward exponentials of R f are defined in T p S Heinz Hopf ( ). Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 14 / 19

72 Generalized Hopf-Rinow theorem Theorem (Accurate Hopf-Rinow for Randers metrics) Let (S, R) a Randers manifold and given a function f : S R define R f (x, v) = R(x, v) df x (v). The following conditions are equivalent: (A) the intersection B + (x, r) B (x, r) of (S, R) is compact for every r > 0 and x S (B) the symmetrized closed balls B s (x, r) of (S, R) are compact for every r > 0 and x S (C) there exists f such that R f is geodesically complete (D) there exists f and p S such that the forward and the backward exponentials of R f are defined in T p S (E) there exists f such that the quasi-metric d f associated to R f is forward and backward complete Heinz Hopf ( ). Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 14 / 19

73 Generalized Hopf-Rinow theorem Theorem (Accurate Hopf-Rinow for Randers metrics) Let (S, R) a Randers manifold and given a function f : S R define R f (x, v) = R(x, v) df x (v). The following conditions are equivalent: (A) the intersection B + (x, r) B (x, r) of (S, R) is compact for every r > 0 and x S (B) the symmetrized closed balls B s (x, r) of (S, R) are compact for every r > 0 and x S (C) there exists f such that R f is geodesically complete (D) there exists f and p S such that the forward and the backward exponentials of R f are defined in T p S (E) there exists f such that the quasi-metric d f associated to R f is forward and backward complete In such a case, (S, R) is convex. Heinz Hopf ( ). Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 14 / 19

74 Convexity of Finsler metrics E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 15 / 19

75 Convexity of Finsler metrics In fact, condition (A) generalizes forward and backward completeness for any Finsler metric and it is enough to prove Palais-Smale condition of the energy functional Paul Finsler ( ) E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 15 / 19

76 Convexity of Finsler metrics In fact, condition (A) generalizes forward and backward completeness for any Finsler metric and it is enough to prove Palais-Smale condition of the energy functional (A) Convexity holds for any Finsler metric Paul Finsler ( ) E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 15 / 19

77 Convexity of Finsler metrics In fact, condition (A) generalizes forward and backward completeness for any Finsler metric and it is enough to prove Palais-Smale condition of the energy functional (A) Convexity holds for any Finsler metric Morse theory can be developed assuming condition (A) (Remember the talk by Erasmo Caponio) Paul Finsler ( ) E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 15 / 19

78 Convexity of Finsler metrics In fact, condition (A) generalizes forward and backward completeness for any Finsler metric and it is enough to prove Palais-Smale condition of the energy functional (A) Convexity holds for any Finsler metric Morse theory can be developed assuming condition (A) (Remember the talk by Erasmo Caponio) Condition (A) implies that the symmetrized distance is complete Paul Finsler ( ) E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 15 / 19

79 Convexity of Finsler metrics In fact, condition (A) generalizes forward and backward completeness for any Finsler metric and it is enough to prove Palais-Smale condition of the energy functional (A) Convexity holds for any Finsler metric Morse theory can be developed assuming condition (A) (Remember the talk by Erasmo Caponio) Condition (A) implies that the symmetrized distance is complete The converse is not true Paul Finsler ( ) E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 15 / 19

80 Convexity of Finsler metrics In fact, condition (A) generalizes forward and backward completeness for any Finsler metric and it is enough to prove Palais-Smale condition of the energy functional (A) Convexity holds for any Finsler metric Morse theory can be developed assuming condition (A) (Remember the talk by Erasmo Caponio) Condition (A) implies that the symmetrized distance is complete The converse is not true Does symmetrized distance completeness imply convexity? Paul Finsler ( ) E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 15 / 19

81 Cut loci of Randers metrics E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 16 / 19

82 Cut loci of Randers metrics (S, R) Randers and C S closed E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 16 / 19

83 Cut loci of Randers metrics (S, R) Randers and C S closed ρ C : S R + the distance function from C to p (the infinum of the length of curves joining C to p E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 16 / 19

84 Cut loci of Randers metrics (S, R) Randers and C S closed ρ C : S R + the distance function from C to p (the infinum of the length of curves joining C to p A minimizing segment is a unit speed geodesic such that ρ C (γ(s)) = s E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 16 / 19

85 Cut loci of Randers metrics (S, R) Randers and C S closed ρ C : S R + the distance function from C to p (the infinum of the length of curves joining C to p A minimizing segment is a unit speed geodesic such that ρ C (γ(s)) = s minimizing geodesic E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 16 / 19

86 Cut loci of Randers metrics (S, R) Randers and C S closed ρ C : S R + the distance function from C to p (the infinum of the length of curves joining C to p A minimizing segment is a unit speed geodesic such that ρ C (γ(s)) = s Cut C is the cut locus, the points x S \ C where the minimizing segment do not minimize anymore minimizing geodesic Cut locus is the dot line E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 16 / 19

87 Cut loci of Randers metrics (S, R) Randers and C S closed ρ C : S R + the distance function from C to p (the infinum of the length of curves joining C to p A minimizing segment is a unit speed geodesic such that ρ C (γ(s)) = s Cut C is the cut locus, the points x S \ C where the minimizing segment do not minimize anymore This function is studied when C is a C 2,1 loc boundary in: Y. Li and L. Nirenberg, The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations, Comm. Pure Appl. Math.,(2005). minimizing geodesic Cut locus is the dot line E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 16 / 19

88 Cauchy horizons E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 17 / 19

89 Cauchy horizons Construct a standard stationary spacetime with R (the reverse metric of R) as Fermat metric E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 17 / 19

90 Cauchy horizons Construct a standard stationary spacetime with R (the reverse metric of R) as Fermat metric H = {( ρ C (x), x) : x S \ C} is a future horizon, that is, an achronal, closed, future null geodesically ruled topological hypersurface. E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 17 / 19

91 Cauchy horizons Construct a standard stationary spacetime with R (the reverse metric of R) as Fermat metric H = {( ρ C (x), x) : x S \ C} is a future horizon, that is, an achronal, closed, future null geodesically ruled topological hypersurface. There are several results for the differentiability of future horizons: J. K. Beem and A. Królak, Cauchy horizon end points and differentiability, J. Math. Phys., 39 (1998), pp P. T. Chruściel, J. H. G. Fu, G. J. Galloway, and R. Howard, On fine differentiability properties of horizons and applications to Riemannian geometry, J. Geom. Phys., 41 (2002), pp E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 17 / 19

92 Cauchy horizons Construct a standard stationary spacetime with R (the reverse metric of R) as Fermat metric H = {( ρ C (x), x) : x S \ C} is a future horizon, that is, an achronal, closed, future null geodesically ruled topological hypersurface. There are several results for the differentiability of future horizons: J. K. Beem and A. Królak, Cauchy horizon end points and differentiability, J. Math. Phys., 39 (1998), pp P. T. Chruściel, J. H. G. Fu, G. J. Galloway, and R. Howard, On fine differentiability properties of horizons and applications to Riemannian geometry, J. Geom. Phys., 41 (2002), pp E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 17 / 19

93 Cut loci of Randers metrics via Cauchy horizons Putting all together we obtain: E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 18 / 19

94 Cut loci of Randers metrics via Cauchy horizons Putting all together we obtain: Theorem ρ C is differentiable at p S \ C iff it is crossed by exactly one minimizing segment. E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 18 / 19

95 Cut loci of Randers metrics via Cauchy horizons Putting all together we obtain: Theorem ρ C is differentiable at p S \ C iff it is crossed by exactly one minimizing segment. Corollary The n-dimensional Haussdorf measure of Cut C is zero. E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 18 / 19

96 Open problems. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 19 / 19

97 Open problems (1) Does it hold Generalized Hopf-Rinow theorem for any Finsler metric?. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 19 / 19

98 Open problems (1) Does it hold Generalized Hopf-Rinow theorem for any Finsler metric? (2) and the results for the distance ρ C from a closed subset?. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 19 / 19

99 Open problems (1) Does it hold Generalized Hopf-Rinow theorem for any Finsler metric? (2) and the results for the distance ρ C from a closed subset? More information in: E. Caponio, M. A. Javaloyes and M. Sánchez, The interplay between Lorentzian causality and Finsler metrics of Randers type., arxiv: , preprint E. Caponio, M. A. Javaloyes and A. Masiello, On the energy functional on Finsler manifolds and applications to stationary spacetimes, arxiv: , preprint Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 19 / 19

100 Open problems (1) Does it hold Generalized Hopf-Rinow theorem for any Finsler metric? (2) and the results for the distance ρ C from a closed subset? More information in: E. Caponio, M. A. Javaloyes and M. Sánchez, The interplay between Lorentzian causality and Finsler metrics of Randers type., arxiv: , preprint E. Caponio, M. A. Javaloyes and A. Masiello, On the energy functional on Finsler manifolds and applications to stationary spacetimes, arxiv: , preprint Thank you for your attention!!!!!! E. Caponio, M. A. Javaloyes, M. Sánchez (*) Interplay between Lorentzian and Randers metrics 19 / 19