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 Spanish Relativity Meeting ERE2009 Bilbao, September 7-11 (2009)

2 Interplay between Randers metrics and stationary spacetimes (R S, l) is a standard stationary spacetime l((τ,y),(τ,y))=g 0 (y,y)+ L 1 =Observer 2g 0 (δ,y)τ β(x)τ 2, lightlike geodesic (t, x) where β(x) > 0. S

3 Interplay between Randers metrics and stationary spacetimes (R S, l) is a standard stationary spacetime L 1 =Observer l((τ,y),(τ,y))=g 0 (y,y)+ lightlike curves (t, x) where β(x) > 0. 2g 0 (δ,y)τ β(x)τ 2, S

4 Interplay between Randers metrics and stationary spacetimes (R S, l) is a standard stationary spacetime L 1 =Observer l((τ,y),(τ,y))=g 0 (y,y)+ lightlike curves (t, x) where β(x) > 0. 2g 0 (δ,y)τ β(x)τ 2, Fermat Principle S

5 Interplay between Randers metrics and stationary spacetimes (R S, l) is a standard stationary spacetime l((τ,y),(τ,y))=g 0 (y,y)+ lightlike geodesic (t, x) L 1 =Observer where β(x) > 0. 2g 0 (δ,y)τ β(x)τ 2, Fermat geodesic x S S is naturally endowed with a Randers metric F called the Fermat metric F (x,v)= 1 β g 0(v,δ)+ q 1 β g 0(v,v)+ 1 β 2 g 0(v,δ) 2

6 Interplay between Randers metrics and stationary spacetimes (R S, l) is a standard stationary spacetime l((τ,y),(τ,y))=g 0 (y,y)+ L 1 =Observer 2g 0 (δ,y)τ β(x)τ 2, lightlike geodesic (t, x) where β(x) > 0. Fermat geodesic x S This is because arrival time of lightlike curves is AT (γ)= R β g 0(ẋ,δ)+ q 1 β g 0(ẋ,ẋ)+ 1 β 2 g 0(ẋ,δ) 2 ds

7 Interplay between Randers metrics and stationary spacetimes L 1 =Observer Causal properties of (R S, l) lightlike geodesic (t, x) Fermat geodesic x S proper- Hopf-Rinow ties of (S, F )

8 Interplay between Randers metrics and stationary spacetimes H + (A) Cauchy horizons of a subset A contained in a slice {t 0 } S A

9 Interplay between Randers metrics and stationary spacetimes H + (A) Cauchy horizons of a subset A contained in a slice {t 0 } S A

10 Interplay between Randers metrics and stationary spacetimes H + (A) Cauchy horizons of a subset A contained in a slice {t 0 } S A are the graph of the distance function to the complementary A c in (S, F )

11 Interplay between Randers metrics and stationary spacetimes H + (A) Differential properties of the Cauchy horizons in (R S, l) A Differential properties of the distance function to a subset in (S, F )

12 Causal condition to have a standard splitting Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious

13 Causal condition to have a standard splitting A spacetime is Stationary if it admits a timelike Killing field. Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious

14 Causal condition to have a standard splitting A spacetime is Stationary if it admits a timelike Killing field. How restrictive is to consider standard stationary spacetimes rather than stationary? Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious

15 Causal condition to have a standard splitting A spacetime is Stationary if it admits a timelike Killing field. How restrictive is to consider standard stationary spacetimes rather than stationary? 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

16 Causal condition to have a standard splitting A spacetime is Stationary if it admits a timelike Killing field. How restrictive is to consider standard stationary spacetimes rather than stationary? 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. Theorem (M. A. J.- M. Sánchez) If a stationary spacetime L is distinguishing and the timelike 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

17 Causal condition to have a standard splitting A spacetime is Stationary if it admits a timelike Killing field. How restrictive is to consider standard stationary spacetimes rather than stationary? 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. Theorem (M. A. J.- M. Sánchez) If a stationary spacetime L is distinguishing and the timelike 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

18 Causality through the Fermat metric Theorem Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious

19 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

20 Causality through the Fermat metric Theorem Let (R S, g) be a standard stationary spacetime. Then (R S, g) is causally continuous and (a) (R S, g) is causally simple iff the associated Finsler manifold (S, F ) is convex, Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious

21 Causality through the Fermat metric Theorem Let (R S, g) be a standard stationary spacetime. Then (R S, g) is causally continuous and (a) (R S, g) is causally simple iff 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

22 Causality through the Fermat metric Theorem Let (R S, g) be a standard stationary spacetime. Then (R S, g) is causally continuous and (a) (R S, g) is causally simple iff 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

23 Causality through the Fermat metric Theorem Let (R S, g) be a standard stationary spacetime. Then (R S, g) is causally continuous and (a) (R S, g) is causally simple iff 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. B + (p, r) = {q : d(p, q) r} and B (p, r) = {q : d(q, p) r} d(p, q) d(q, p)!!!! Globally hyperbolic Causally simple Causally continuous Stably causal Strongly causal Distinguishing Causal Chronological Non-totally vicious

24 Cauchy developments and distance function to a subset

25 Cauchy developments and distance function to a subset Theorem Let (R S, g) be a standard stationary spacetime such that {t 0 } S is Cauchy, and A t0 = {t 0 } A. Then

26 Cauchy developments and distance function to a subset Theorem Let (R S, g) be a standard stationary spacetime such that {t 0 } S is Cauchy, and A t0 = {t 0 } A. Then H + (A t0 ) = {(t, y) : inf x / A d(x, y) = t t 0 }

27 Cauchy developments and distance function to a subset Theorem Let (R S, g) be a standard stationary spacetime such that {t 0 } S is Cauchy, and A t0 = {t 0 } A. Then H + (A t0 ) = {(t, y) : inf x / A d(x, y) = t t 0 } H (A t0 ) = {(t, y) : inf x / A d(y, x) = t t 0 }

28 Cauchy developments and distance function to a subset Theorem Let (R S, g) be a standard stationary spacetime such that {t 0 } S is Cauchy, and A t0 = {t 0 } A. Then H + (A t0 ) = {(t, y) : inf x / A d(x, y) = t t 0 } H (A t0 ) = {(t, y) : inf x / A d(y, x) = t t 0 } inf x / A d(x,y)=d(a c,y) inf x / A d(y,x)=d(y,a c )

29 Cauchy developments and distance function to a subset Theorem Let (R S, g) be a standard stationary spacetime such that {t 0 } S is Cauchy, and A t0 = {t 0 } A. Then H + (A t0 ) = {(t, y) : inf x / A d(x, y) = t t 0 } H (A t0 ) = {(t, y) : inf x / A d(y, x) = t t 0 } Cauchy horizons can be seen as the graph of the distance function to a subset!!!! inf x / A d(x,y)=d(a c,y) inf x / A d(y,x)=d(y,a c )

30 Measure of the crease set

31 Measure of the crease set any point in H + (A) admits a generator : a lightlike geodesic segment contained in H + (A) which is past-inextedible or has a past endpoint in the boundary of A.

32 Measure of the crease set any point in H + (A) admits a generator : a lightlike geodesic segment contained in H + (A) which is past-inextedible or has a past endpoint in the boundary of A. Let H + mul (A) be the set of points p H+ (A) \ A admitting more than one generator.

33 Measure of the crease set any point in H + (A) admits a generator : a lightlike geodesic segment contained in H + (A) which is past-inextedible or has a past endpoint in the boundary of A. Let H + mul (A) be the set of points p H+ (A) \ A admitting more than one generator. As a consequence of a Theorem for Finsler metrics by Li and Nirenberg (Comm. Pure Appl. Math. 2005): YanYan Li and Louis Nirenberg

34 Measure of the crease set any point in H + (A) admits a generator : a lightlike geodesic segment contained in H + (A) which is past-inextedible or has a past endpoint in the boundary of A. Let H + mul (A) be the set of points p H+ (A) \ A admitting more than one generator. As a consequence of a Theorem for Finsler metrics by Li and Nirenberg (Comm. Pure Appl. Math. 2005): Theorem (R S, g) (n + 1)-standard stationary, with S Cauchy an Ω S, open connected with C 2,1 loc -boundary Ω. If A t0 = {t 0 } A and B is bounded then h n 1 ((R B) H + mul (A)) < + YanYan Li and Louis Nirenberg

35 Measure of the crease set any point in H + (A) admits a generator : a lightlike geodesic segment contained in H + (A) which is past-inextedible or has a past endpoint in the boundary of A. Let H + mul (A) be the set of points p H+ (A) \ A admitting more than one generator. As a consequence of a Theorem for Finsler metrics by Li and Nirenberg (Comm. Pure Appl. Math. 2005): Achronal curve γ Cauchy surface S Theorem (R S, g) (n + 1)-standard stationary, with S Cauchy an Ω S, open connected with C 2,1 loc -boundary Ω. If A t0 = {t 0 } A and B is bounded then h n 1 ((R B) H + mul (A)) < + YanYan Li and Louis Nirenberg

36 Measure of the crease set any point in H + (A) admits a generator : a lightlike geodesic segment contained in H + (A) which is past-inextedible or has a past endpoint in the boundary of A. Let H + mul (A) be the set of points p H+ (A) \ A admitting more than one generator. As a consequence of a Theorem for Finsler metrics by Li and Nirenberg (Comm. Pure Appl. Math. 2005): Theorem (R S, g) (n + 1)-standard stationary, with S Cauchy an Ω S, open connected with C 2,1 loc -boundary Ω. If A t0 = {t 0 } A and B is bounded then h n 1 ((R B) H + mul (A)) < + Achronal curve γ J (γ) Cauchy surface S A = J (γ) S YanYan Li and Louis Nirenberg

37 Open problems

38 Open problems (1) Is there any relation between the flag curvature of the Fermat metric and the Weyl tensor of the spacetime?:

39 Open problems (1) Is there any relation between the flag curvature of the Fermat metric and the Weyl tensor of the spacetime?: (2) In the paper G. W. Gibbons, C. A. R. Herdeiro, C. M. Warnick, M. C. Werner, Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry., Phys.Rev.D79: ,2009 the authors show that Fermat metrics with constant flag curvature correspond with locally conformally flat stationary spacetimes, but the converse is not true.

40 Open problems (1) Is there any relation between the flag curvature of the Fermat metric and the Weyl tensor of the spacetime?: (2) In the paper G. W. Gibbons, C. A. R. Herdeiro, C. M. Warnick, M. C. Werner, Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry., Phys.Rev.D79: ,2009 the authors show that Fermat metrics with constant flag curvature correspond with locally conformally flat stationary spacetimes, but the converse is not true. (3) Which is the condition in the Fermat metric that characterizes conformally flatness for the stationary spacetime?

41 Bibliography

42 Bibliography More information in: E. Caponio, M. A. J. and M. Sánchez, The interplay between Lorentzian causality and Finsler metrics of Randers type., arxiv: , preprint E. Caponio, M. A. J. and A. Masiello, On the energy functional on Finsler manifolds and applications to stationary spacetimes, arxiv: , preprint 2007.