Skip to main content
Log in

On the geometry of stationary Galilean spacetimes

  • Research Article
  • Published:
General Relativity and Gravitation Aims and scope Submit manuscript

Abstract

In this work we introduce a new family of non-relativistic spacetimes: standard stationary Galilean spacetimes, which constitute the local geometric model of stationary Galilean spacetimes. We also study the geodesic completeness of stationary Galilean spacetimes as well as the the geometric conditions for these spacetimes that guarantee the existence of a global splitting as a standard stationary Galilean spacetime.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Albujer, A.L., Herrera, J., Rubio, R.M.: New examples of static spacetimes admitting a unique standard decomposition. Gen. Relat. Gravit. 51, 39 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  2. Aledo, J.A., Romero, A., Rubio, R.M.: The existence and uniqueness of standard static splitting. Class. Quantum Gravity 32, 105004 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  3. Bernal, A.N., López, M., Sánchez, M.: Fundamental units of length and time. Found. Phys. 32, 77–108 (2002)

    Article  MathSciNet  Google Scholar 

  4. Bernal, A.N., Sánchez, M.: Leibnizian, Galilean and Newtonian structures of space-time. J. Math. Phys. 44, 1129–1149 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  5. Candela, A.M., Romero, A., Sánchez, M.: Completeness of the trajectories of particles coupled to a general force field. Arch. Ration. Mech. Anal. 208, 255–274 (2013)

    Article  MathSciNet  Google Scholar 

  6. Cartan, E.: Les variètés a conexion affine et la théorie de la relativité généralisée. Ann. Ec. Norm. Sup. 40, 1–25 (1923)

    Article  Google Scholar 

  7. Cartan, E.: Les variètés a conexion affine et la théorie de la relativité généralisée (suite). Ann. Ec. Norm. Sup. 41, 325–412 (1924)

    Google Scholar 

  8. Christian, J.: Exactly soluble sector of quantum gravity. Phys. Rev. D 56, 4844–4877 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  9. de la Fuente, D., Rubio, R.M.: Galilean generalized Robertson-Walker spacetimes: a new family of Galilean geometrical models. J. Math. Phys. 59, 022903 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  10. Dixon, W.G.: On the uniqueness of the Newtonian theory as a geometric theory of gravitation. Commun. Math. Phys. 45, 167–182 (1975)

    Article  ADS  MathSciNet  Google Scholar 

  11. Duggal, K., Sharma, R.: Symmetries of Spacetimes and Riemannian Manifolds. Springer, Berlin (2013)

    MATH  Google Scholar 

  12. Flores, J.L.: The Riemannian and Lorentzian splitting theorems. Atlantis Trans. Geom. 1, 1–20 (2017)

    MathSciNet  MATH  Google Scholar 

  13. Friedrichs, K.: Eine invariante Formulierung des Newtonschen Gravitationsgesetzes und des Grenzüberganges vom Einsteinschen zum Newtonschen Gesetz. Math. Ann. 98, 566–575 (1928)

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  15. Havas, P.: Four-dimensional formulations of Newtonian mechanics and their relation to the special and the general theory of relativity. Rev. Mod. Phys. 36, 938–965 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  16. Künzle, H.P.: Galilei and Lorentz structures on space-time: comparison of the corresponding geometry and physics. Ann. I. H. Poincaré A 17, 337–362 (1972)

    MathSciNet  Google Scholar 

  17. Künzle, H.P.: Covariant Newtonian limts of Lorentz space-times. Gen. Relat. Gravit. 7, 445–457 (1976)

    Article  ADS  Google Scholar 

  18. Malament, D.B.: Topic in the Formulations of General Relativity and Newtonian Gravitation Theory. Chicago Lectures in Physics. University of Chicago Press, Chicago (2012)

    Book  Google Scholar 

  19. O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity, vol. 103. Academic Press, New York (1983)

    MATH  Google Scholar 

  20. Penrose, R.: On gravity’s role in quantum state reduction. Gen. Relat. Gravit. 8, 581–600 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  21. Rodrigues, W.A., Souza, Q., Bozhkov, Y.: The mathematical structure of Newtonian spacetime: classical dynamics and gravitation. Found. Phys. 25, 871–924 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  22. Sachs, R.K., Wu, H.: General Relativity for Mathematicians. Springer, New York (1977)

    Book  Google Scholar 

  23. Sánchez, M., Senovilla, J.M.M.: A note on the uniqueness of global static decompositions. Class. Quantum Gravity 24, 6121 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  24. Spivak, M.: A Comprehensive Introduction to Differential Geometry. Publish or Perish, Houston (1999)

    MATH  Google Scholar 

  25. Trautman, A.: Comparison of Newtonian and Relativistic Theories of Space-Time, Perspectives in Geometry and Relativity. Indiana University Press, Bloomington (1966)

    Google Scholar 

  26. Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Springer, Berlin (1983)

    Book  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referee for his deep reading and valuable suggestions. This article has been partially supported by Spanish MINECO and ERDF Project MTM2016-78807-C2-1-P.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Daniel de la Fuente.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

de la Fuente, D., Pelegrín, J.A.S. & Rubio, R.M. On the geometry of stationary Galilean spacetimes. Gen Relativ Gravit 53, 8 (2021). https://doi.org/10.1007/s10714-020-02772-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10714-020-02772-1

Keywords

Mathematics Subject Classification

Navigation