Revista Matemática Iberoamericana

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

Abstract

We obtain some results in both Lorentz and Finsler geometries, by using a correspondence between the conformal structure (Causality) of standard stationary spacetimes on $M = \mathbb{R} \times S$ and Randers metrics on $S$. In particular: (1) For stationary spacetimes: we give a simple characterization of when $\mathbb{R} \times S$ is causally continuous or globally hyperbolic (including in the latter case, when $S$ is a Cauchy hypersurface), in terms of an associated Randers metric. Consequences for the computability of Cauchy developments are also derived. (2) For Finsler geometry: Causality suggests that the role of completeness in many results of Riemannian Geometry (geodesic connectedness by minimizing geodesics, Bonnet-Myers, Synge theorems) is played by the compactness of symmetrized closed balls in Finslerian Geometry. Moreover, under this condition we show that for any Randers metric $R$ there exists another Randers metric $\tilde R$ with the same pregeodesics and geodesically complete. Even more, results on the differentiability of Cauchy horizons in spacetimes yield consequences for the differentiability of the Randers distance to a subset, and vice versa.

Article information

Source
Rev. Mat. Iberoamericana, Volume 27, Number 3 (2011), 919-952.

Dates
First available in Project Euclid: 9 August 2011

https://projecteuclid.org/euclid.rmi/1312906783

Mathematical Reviews number (MathSciNet)
MR2895339

Zentralblatt MATH identifier
1229.53070

Citation

Caponio, Erasmo; Javaloyes, Miguel Ángel; Sánchez, Miguel. On the interplay between Lorentzian Causality and Finsler metrics of Randers type. Rev. Mat. Iberoamericana 27 (2011), no. 3, 919--952. https://projecteuclid.org/euclid.rmi/1312906783

References

• Bangert, V.: On the existence of closed geodesics on two-spheres. Internat. J. Math. 4 (1993), no. 1, 1-10.
• Bao, D., Chern, S.-S. and Shen, Z.: An introduction to Riemann-Finsler geometry. Graduate Texts in Mathematics 200. Springer-Verlag, New York, 2000.
• Bao, D., Robles, C. and Shen, Z.: Zermelo navigation on Riemannian manifolds. J. Differential Geom. 66 (2004), no. 3, 377-435.
• Bartolo, R., Candela, A. and Caponio, E.: Normal geodesics connecting two non-necessarily spacelike submanifolds in a stationary spacetime. Adv. Nonlinear Stud. 10 (2010), no. 4, 851-866.
• Bartolo, R., Caponio, E., Germinario, A. and Sánchez, M.: Convex domains of Finsler and Riemannian manifold. Calc. Var. Partial Differential Equations 40 (2010), no. 3-4, 335-356.
• Beem, J.K., Ehrlich, P.E. and Easley, K.L.: Global Lorentzian geometry. Monographs and Textbooks in Pure and Applied Mathematics 202. Marcel Dekker, New York, 1996.
• Beem, J.K. and Królak, A.: Cauchy horizon end points and differentiability. J. Math. Phys. 39 (1998), no. 11, 6001-6010.
• Bernal, A.N. and Sánchez, M.: On smooth Cauchy hypersurfaces and Geroch's splitting theorem. Comm. Math. Phys. 243 (2003), no. 3, 461-470.
• Bernal, A.N. and Sánchez, M.: Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. Comm. Math. Phys. 257 (2005), no. 1, 43-50.
• Bernal, A.N. and Sánchez, M.: Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions. Lett. Math. Phys. 77 (2006), no. 2, 183-197.
• Bernal, A.N. and Sánchez, M.: Globally hyperbolic spacetimes can be defined as “causal” instead of “strongly causal”. Classical Quanum Gravity 24 (2007), no. 3, 745-749.
• Biliotti, L. and Javaloyes, M.A.: $t$-periodic light rays in conformally stationary spacetimes via Finsler geometry. Houston J. Math. 37 (2011), no. 1, 127-146.
• Candela, A.M., Flores, J.L. and Sánchez, M.: Global hyperbolicity and Palais-Smale condition for action functionals in stationary spacetimes. Adv. Math. 218 (2008), no. 2, 515-536.
• Caponio, E., Javaloyes, M.A. and Masiello, A.: On the energy functional on Finsler manifolds and applications to stationary spacetimes. Math. Ann., in press. DOI: 10.1007/s00208-010-0602-7.
• Caponio, E., Javaloyes, M.A. and Masiello, A.: Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric. Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), no. 3, 857-876.
• Chruściel, P.T., Fu, J.H.G., Galloway, G.J. and Howard, R.: On fine differentiability properties of horizons and applications to Riemannian geometry. J. Geom. Phys. 41 (2002), no. 1-2, 1-12.
• Chruściel, P.T. and Galloway, G.J.: Horizons non-differentiable on a dense set. Comm. Math. Phys. 193 (1998), no. 2, 449-470.
• Dazord, P.: Propiétés globales des géodésiques des Espaces de Finsler. Theses, Université de Lyon, 1969.
• Flores, J.L., Herrera, J. and Sánchez, M.: Gromov, Cauchy and causal boundaries for Riemannian, Finslerian and Lorentzian manifolds. arXiv:1011.1154v1 [math.DG], 2010.
• Flores, J.L. and Sánchez, M.: Geodesics in stationary spacetimes. Application to Kerr spacetime. Int. J. Theor. Phys. Group Theory Nonlinear Opt. 8 (2002), no. 3, 319-336. (also republished in Theoretical Physics 2002 Part 2, 141-158. Horizons in World Physics 243, Nova Science).
• Flores, J.L. and Sánchez, M.: The causal boundary of wave-type spacetimes. J. High Energy Phys. (2008), no. 3, 036, 43 pp.
• Franks, J.: Geodesics on $S^22$ and periodic points of annulus homeomorphisms. Invent. Math. 108 (1992), no. 2, 403-418.
• Geroch, R.: Domain of dependence. J. Math. Phys. 11 (1970), 437-449.
• Gibbons, G.W., Herdeiro, C.A.R., Warnick, C.M. and Werner, M.C.: Stationary metrics and optical Zermelo-Randers-Finsler geometry. Phys. Rev. D 79 (2009), no. 4, 044022, 21 pp.
• Gromov, M.: Metric structures for Riemannian and non-Riemannian spaces. Progress in Mathematics 152. Birkhäuser, Boston, MA, 1999.
• Hawking, S.W. and Ellis, G.F.R.: The large scale structure of space-time. Cambridge Monographs on Mathematical Physics 1. Cambridge University Press, London-New York, 1973.
• Javaloyes, M.A. and Sánchez, M.: A note on the existence of standard splittings for conformally stationary spacetimes. Classical Quantum Gravity 25 (2008), no. 16, 168001, 7 pp.
• Katok, A.B.: Ergodic perturbations of degenerate integrable Hamiltonian systems. Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 539-576.
• Kovner, I.: Fermat principles for arbitrary space-times. Astrophysical Journal 351 (1990), 114-120.
• Li, Y. and Nirenberg, L.: The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations. Comm. Pure Appl. Math. 58 (2005), no. 1, 85-146.
• Masiello, A.: Variational methods in Lorentzian geometry. Pitman Research Notes in Mathematics Series 309. Longman Scientific & Technical, Harlow, New York, 1994.
• Mawhin, J. and Willem, M.: Critical point theory and Hamiltonian systems. Applied Math. Sciences 74. Springer-Verlag, New York, 1989.
• Minguzzi, E. and Sánchez, M.: The causal hierarchy of spacetimes. In Recent developments in pseudo-Riemannian geometry, 299-358. ESI Lect. Math. Phys. Eur. Math. Soc., Zürich, 2008.
• Nomizu, K. and Ozeki, H.: The existence of complete Riemannian metrics. Proc. Amer. Math. Soc. 12 (1961), 889-891.
• O'Neill, B.: Semi-Riemannian geometry. Pure and Applied Mathematics 103. Academic Press, New York, 1983.
• O'Neill, B.: The geometry of Kerr black holes. A K Peters, Wellesley, MA, 1995.
• Penrose, R.: Techniques of differential topology in relativity. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics 7. SIAM, Philadelphia, Pa., 1972.
• Perlick, V.: On Fermat's principle in general relativity. I. The general case. Classical Quantum Gravity 7 (1990), no. 8, 1319-1331.
• Rademacher, H.-B.: Nonreversible Finsler metrics of positive flag curvature. In A sampler of Riemann-Finsler geometry, 261-302. Math. Sci. Res. Inst. Publ. 50. Cambridge Univ. Press, Cambridge, 2004.
• Rademacher, H.-B.: A sphere theorem for non-reversible Finsler metrics. Math. Ann. 328 (2004), no. 3, 373-387.
• Sachs, R.K. and Wu, H.H.: General relativity for mathematicians. Graduate Texts in Math. 48. Springer-Verlag, New York-Heidelberg, 1977.
• Sánchez, M.: Some remarks on causality theory and variational methods in Lorentzian manifolds. Conf. Semin. Mat. Univ. Bari 265 (1997), 1-12.
• Sánchez, M.: Causal hierarchy of spacetimes, temporal functions and smoothness of Geroch's splitting. A revision. Mat. Contemp. 29 (2005), 127-155.
• Sánchez, M.: On the geometry of static spacetimes. Nonlinear Analysis 63 (2005), 455-463.
• Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C. and Held, C.E.: Exact solutions of Einstein's field equations. Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge, 2003.
• Ziller, W.: Geometry of the Katok examples. Ergodic Theory Dynam. Systems 3 (1983), no. 1, 135-157.