Skip to main content
SpringerLink
Log in

Conformally Standard Stationary SpaceTimes and Fermat Metrics

  • Conference paper
  • First Online:
Book cover Recent Trends in Lorentzian Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 26))

Abstract

In this chapter, we collect several results for conformally standard stationary spacetimes \((S \times \mathbb{R},g)\) obtained in terms of a Finsler metric of Randers type on the orbit manifold S that we call Fermat metric. This metric is obtained by applying the relativistic Fermat principle and it turns out that it encodes all the causal aspects of the space time.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abbondandolo, A., Schwarz, M.: A smooth pseudo-gradient for the Lagrangian action functional. Adv. Nonlinear Stud. 9(4), 597–623 (2009)

    MathSciNet  MATH  Google Scholar 

  2. Abramowicz, M., Carter, B., Lasota, J.-P.: Optical reference geometry for stationary and static dynamics. Gen. Relativ. Gravit. 20, 1172–1183 (1988)

    Article  MathSciNet  Google Scholar 

  3. Bao, D., Chern, S.-S., Shen, Z.: An Introduction to Riemann-Finsler geometry. Graduate Texts in Mathematics. Springer, New York (2000)

    Book  MATH  Google Scholar 

  4. Bao, D., Robles, C., Shen, Z.: Zermelo navigation on Riemannian manifolds. J. Differ. Geom. 66(3), 377–435 (2004)

    MathSciNet  MATH  Google Scholar 

  5. Bartolo, R., Candela, A.M., Caponio, E.: Normal geodesics connecting two non-necessarily spacelike submanifolds in a stationary spacetime. Adv. Nonlinear Stud. 10(4), 851–866 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian geometry. Monographs and Textbooks in Pure and Applied Mathematics, vol. 202, 2nd edn. Marcel Dekker, New York (1996)

    Google Scholar 

  7. Bernal A.N., Sánchez, M.: On smooth Cauchy hypersurfaces and Geroch’s splitting theorem. Comm. Math. Phys. 243(3), 461–470 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Biliotti L., Javaloyes, M.Á.: t-periodic light rays in conformally stationary spacetimes via Finsler geometry. Houston J. Math. 37(1), 127–146 (2011)

    Google Scholar 

  9. Caponio, E.: Infinitesimal and local convexity of a hypersurface in a semi-Riemannian manifold, these proceedings, Recent Trends in Lorentzian Geometry, Springer Proceedings in Mathematics & Statistics (2013)

    Google Scholar 

  10. Caponio, E.: The index of a geodesic in a Randers space and some remarks about the lack of regularity of the energy functional of a Finsler metric. Acta Math. Acad. Paedagog. Nyházi. (N.S.) 26(2), 265–274 (2010)

    Google Scholar 

  11. Caponio, E., Germinario, A., Sánchez, M.: Geodesics on convex regions of stationary spacetimes and Finslerian Randers spaces. arXiv:1112.3892v1 [math.DG] (2011)

    Google Scholar 

  12. Caponio, E., Javaloyes, M.A.: A remark on the Morse Theorem about infinitely many geodesics between two points. arXiv:1105.3923v1 [math.DG] (2011)

    Google Scholar 

  13. Caponio, E., Javaloyes, M.Á., Masiello, A.: Finsler geodesics in the presence of a convex function and their applications. J. Phys. A 43(13), 135207, 15 (2010)

    Google Scholar 

  14. Caponio, E., Javaloyes, M.Á., 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(3), 857–876 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Caponio, E., Javaloyes, M.Á., Masiello A.: On the energy functional on Finsler manifolds and applications to stationary spacetimes. Math. Ann. 351(2), 365–392 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Caponio, E., Javaloyes, M.A., Sánchez, M.: On the interplay between Lorentzian Causality and Finsler metrics of Randers type. Rev. Mat. Iberoamericana 27(3), 919–952 (2011)

    Article  MATH  Google Scholar 

  17. Dirmeier, A., Plaue, M., Scherfner, M.: Growth conditions, Riemannian completeness and Lorentzian causality. J. Geom. Phys. 62(3), 604–612 (2011)

    MathSciNet  Google Scholar 

  18. Dowker J., Kennedy, G.: Finite temperature and boundary effects in static space-times. J. Phys. A: Math. Gen. 11, 895–920 (1978)

    Article  MathSciNet  Google Scholar 

  19. Flores J.L., Herrera, J.: The c-boundary construction of spacetimes: application to stationary kerr spacetime, these proceedings, Recent Trends in Lorentzian Geometry, Springer Proceedings in Mathematics & Statistics (2013).

    Google Scholar 

  20. Flores, J.L., Herrera, J., Sánchez, M.: Gromov, Cauchy and causal boundaries for Riemannian, Finslerian and Lorentzian manifolds. arXiv:1011.1154v2 [math.DG], in Mem. Amer. Math. Soc. (to appear)

    Google Scholar 

  21. Flores, J.L., Herrera, J., Sánchez, M.: On the final definition of the causal boundary and its relation with the conformal boundary. Adv. Theor. Math. Phys. 15(4), 991–1058 (2011)

    MathSciNet  MATH  Google Scholar 

  22. Fortunato, D., Giannoni, F., Masiello, A.: A Fermat principle for stationary space-times and applications to light rays. J. Geom. Phys. 15(2), 159–188 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  23. Giambò, R., Giannoni, F., Piccione, P.: Genericity of nondegeneracy for light rays in stationary spacetimes. Comm. Math. Phys. 287(3), 903–923 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Giambò, R., Javaloyes, M.A.: Addendum to: Genericity of nondegeneracy for light rays in stationary spacetimes. Comm. Math. Phys. 295(1), 289–291 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gibbons, G., Perry, M.: Black holes and thermal green functions. Proc. Roy. Soc. London, Ser. A 358, 467–494 (1978)

    MathSciNet  Google Scholar 

  26. Gibbons, G.W., Herdeiro, C.A.R., Warnick, C.M., Werner, M.C.: Stationary metrics and optical Zermelo-Randers-Finsler geometry. Phys. Rev. D 79(4), 044022, 21 (2009)

    Google Scholar 

  27. Ingarden, R.S.: On the geometrically absolute optical representation in the electron microscope. Trav. Soc. Sci. Lett. Wrocław. Ser. B. (45), 60 (1957)

    Google Scholar 

  28. Javaloyes, M.A., Lichtenfelz, L., Piccione, P.: Almost isometries of non-reversible metrics with applications to stationary spacetimes. arXiv:1205.4539v1 [math.DG] (2012)

    Google Scholar 

  29. Javaloyes, M.A., Sánchez, M.: A note on the existence of standard splittings for conformally stationary spacetimes. Class. Quantum Grav. 25(16), 168001, 7 (2008)

    Google Scholar 

  30. Javaloyes, M.A., Sánchez, M.: On the definition and examples of Finsler metrics. arXiv:1111.5066v2 [math.DG] (2011), in Ann. Scuola Norm. Sup. Pisa Cl. Sci. (to appear)

    Google Scholar 

  31. Kobayashi, S., Nomizu, K.: Foundations of differential geometry. Wiley Classics Library, vol. I. Wiley, New York (1996) Reprint of the 1963 original, A Wiley-Interscience Publication

    Google Scholar 

  32. Kovner, I.: Fermat principle in gravitational fields. Astrophys. J. 351, 114–120 (1990)

    Article  Google Scholar 

  33. Landau, L., Lifshitz, E.: The classical theory of fields, 2nd edn. Pergamon Press, Addison-Wesley, Oxford, Reading, MA (1962)

    MATH  Google Scholar 

  34. Levi-Civita, T.: La teoria di Einstein e il principio di Fermat. Nuovo Cimento 16, 105–114 (1918)

    Article  MATH  Google Scholar 

  35. Levi-Civita, T.: The Absolute Differential Calculus. Blackie & Son Limited, London (1927)

    MATH  Google Scholar 

  36. Lichnerowicz, A.: Théories relativistes de la gravitation et de l’électromagnétisme. Relativité générale et théories unitaires. Masson et Cie, Paris (1955)

    MATH  Google Scholar 

  37. Lichnerowicz, A., Thiry, Y.: Problèmes de calcul des variations liés à la dynamique classique et à la théorie unitaire du champ. C. R. Acad. Sci. Paris 224, 529–531 (1947)

    MathSciNet  MATH  Google Scholar 

  38. Masiello, A.: An alternative variational principle for geodesics of a Randers metric. Adv. Nonlinear Stud. 9(4), 783–801 (2009)

    MathSciNet  MATH  Google Scholar 

  39. Masiello, A., Piccione, P.: Shortening null geodesics in Lorentzian manifolds. Applications to closed light rays. Differ. Geom. Appl. 8(1), 47–70 (1998)

    MathSciNet  MATH  Google Scholar 

  40. Matsumoto, M.: On C-reducible Finsler spaces. Tensor (N.S.) 24, 29–37 (1972) Commemoration volumes for Prof. Dr. Akitsugu Kawaguchi’s seventieth birthday, Vol. I

    Google Scholar 

  41. Matsumoto, M.: On Finsler spaces with Randers’ metric and special forms of important tensors. J. Math. Kyoto Univ. 14, 477–498 (1974)

    MathSciNet  MATH  Google Scholar 

  42. Mawhin, J., Willem, M.: Critical point theory and Hamiltonian systems. Applied Mathematical Sciences, vol. 74. Springer, New York (1989)

    Google Scholar 

  43. Mercuri, F.: The critical points theory for the closed geodesics problem. Math. Z. 156(3), 231–245 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  44. Minguzzi, E., Sánchez, M.: The causal hierarchy of spacetimes. In: Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys., pp. 299–358. Eur. Math. Soc., Zürich (2008)

    Google Scholar 

  45. Miron, R.: The geometry of Ingarden spaces. Rep. Math. Phys. 54(2), 131–147 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  46. O’Neill, B.: Semi-Riemannian geometry. Pure and Applied Mathematics, vol. 103. Academic [Harcourt Brace Jovanovich Publishers], New York (1983) With applications to relativity

    Google Scholar 

  47. Perlick, V.: On Fermat’s principle in general relativity. I. The general case. Class. Quantum Grav. 7, 1319–1331 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  48. Perlick, V.: On Fermat’s principle in general relativity. II. The conformally stationary case. Class. Quantum Grav. 7(10), 1849–1867 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  49. Perlick, V.: Fermat principle in Finsler spacetimes. Gen. Relat. Gravit. 38(2), 365–380 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  50. Pham, Q.: Inductions électromagnétiques en rélativité général et principe de Fermat. Arch. Ration. Mech. Anal. 1, 54–80 (1957)

    Article  MathSciNet  Google Scholar 

  51. Randers, G.: On an asymmetrical metric in the fourspace of general relativity. hys. Rev. 59(2), 195–199 (1941)

    Google Scholar 

  52. Sánchez, M.: Some remarks on causality theory and variational methods in Lorenzian manifolds. Conf. Semin. Mat. Univ. Bari (265), ii+12 (1997)

    Google Scholar 

  53. Shibata, C., Shimada, H., Azuma, M., Yasuda, H.: On Finsler spaces with Randers’ metric. Tensor (N.S.) 31(2), 219–226 (1977)

    Google Scholar 

  54. Synge, J.: An alternative treatment of Fermat’s principle for a stationary gravitational field. Philos. Mag. J. Sci. 50, 913–916 (1925)

    Article  MATH  Google Scholar 

  55. Warner, F.: The conjugate locus of a Riemannian manifold. Amer. J. Math. 87, 575–604 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  56. Weyl, H.: Zur gravitationstheorie. Ann. Phys. (Berlin) 54, 117–145 (1917)

    Google Scholar 

  57. Yasuda, H., Shimada, H.: On Randers spaces of scalar curvature. Rep. Math. Phys. 11(3), 347–360 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  58. Zaustinsky, E.: Spaces with non-symmetric distance. Mem. Amer. Math. Soc. 34, 1–91 (1959)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The author appreciates very useful suggestions given by the anonymous referees and would like to acknowledge the careful reading of the first version and the advices of Erasmo Caponio, Jose Luis Flores, Jonatan Herrera, and Miguel Sánchez.

The author is partially supported by Regional Junta de Andalucía Grant P09-FQM-4496, by MICINN project MTM2009-10418, and by Fundación Séneca project 04540/GERM/06.

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, Espinardo, 30100, Murcia, Spain

    Miguel Angel Javaloyes

Authors
  1. Miguel Angel Javaloyes
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miguel Angel Javaloyes .

Editor information

Editors and Affiliations

  1. , Dept of Geometry and Topology, University of Granada, Campus Fuentenueva s/n, Granada, E18071, Granada, Spain

    Miguel Sánchez

  2. Departamento de Mátematica Aplicada, Universidad de Granada, Campus de Fuentenueva, Granada, 18071, Spain

    MIguel Ortega

  3. , Dept. of Geometry and Topology, University of Granada, Campus Fuentenueva s/n, Granada, E-18071, Granada, Spain

    Alfonso Romero

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media New York

About this paper

Cite this paper

Javaloyes, M.A. (2012). Conformally Standard Stationary SpaceTimes and Fermat Metrics. In: Sánchez, M., Ortega, M., Romero, A. (eds) Recent Trends in Lorentzian Geometry. Springer Proceedings in Mathematics & Statistics, vol 26. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4897-6_9

Download citation

Publish with us

Policies and ethics

Search

Navigation