Skip to main content
Log in

On conformal Lorentzian length spaces

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

Abstract

Recently, Lorentzian length spaces have been introduced inspired by length spaces. One of the main objects of study in these spaces is a time separation function \(\tau \), which is closely linked to their causal structure. In analogy to the metric d in length spaces, \(\tau \) can express basic notions and many results in the setting of Lorentzian length spaces. In this paper, the concept of conformal Lorentzian length spaces is introduced and a novel version of limit curve theorem is proven. Finally, the global hyperbolic and causally simple Lorentzian length spaces are characterized.

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.

Fig. 1

Similar content being viewed by others

References

  1. Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian Geometry. Marcel Dekker, New York (1996)

    MATH  Google Scholar 

  2. Beran, T., Ohanyan, A., Rott, F., Solis, D.A.: The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature. Lett. Math. Phys. 113, 48 (2023). https://doi.org/10.1007/s11005-023-01668-w

    Article  MathSciNet  MATH  Google Scholar 

  3. Beran, T., Napper, L., Rott, F.: Alexandrov’s patchwork and the Bonnet-Myers Theorem for Lorentzian length spaces. (2023) arXiv:2302.11615

  4. Bombelli, L., Lee, J., Meyer, D., Sorkin, R.D.: Spacetime as a causal set. Phys. Rev. Lett. 59(5), 521–524 (1987). https://doi.org/10.1103/PhysRevLett.59.521

    Article  MathSciNet  Google Scholar 

  5. Borchers, H.J., Sen, R.N.: Theory of ordered spaces. Comm. Math. Phys. 132(3), 593–611 (1990). https://doi.org/10.1007/BF02156539

    Article  MathSciNet  MATH  Google Scholar 

  6. Burtscher, A., García-Heveling, L.: Time functions on Lorentzian length spaces (2021). arXiv preprint arXiv:2108.02693

  7. Busemann, H.: Timelike spaces. Dissertationes Math. (Rozprawy Mat.) 53, 52 (1967)

    MathSciNet  MATH  Google Scholar 

  8. Friedrich, H.: Conformal Einstein evolution. In: The conformal structure of space-time: Geometry analysis, numerics, pp. 1–50. Springer, Berlin Heidelberg (2002)

    Google Scholar 

  9. Galloway, G.: Existence of CMC Cauchy surfaces and spacetime splitting. Pure Appl. Math. Q. 15(2), 667–682 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hau, L.A., Cabrera Pacheco, A.J., Solis, D.. A.: On the causal hierarchy of Lorentzian length spaces. Classical Quantum Gravity 37(21), 215013, 22 (2020). https://doi.org/10.1088/1361-6382/abb25f

    Article  MathSciNet  MATH  Google Scholar 

  11. Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time. Cambridge University Press, Cambridge (1973)

    Book  MATH  Google Scholar 

  12. Kronheimer, E.H., Penrose, R.: On the structure of causal spaces. Proc. Cambridge Philos. Soc. 63, 481–501 (1967). https://doi.org/10.1017/S030500410004144X

    Article  MathSciNet  MATH  Google Scholar 

  13. Kunzinger, M., Samann, C.: Lorentzian length spaces. Ann. Global Anal. Geom. 54(3), 399–447 (2018). https://doi.org/10.1007/s10455-018-9633-1

    Article  MathSciNet  MATH  Google Scholar 

  14. Kunzinger, M., Steinbauer, R.: Null distance and convergence of Lorentzian length spaces. Ann. Henri Poincar 23, 4319–4342 (2022). https://doi.org/10.1007/s00023-022-01198-6

    Article  MathSciNet  MATH  Google Scholar 

  15. Minguzzi, E.: Limit curve theorems in Lorentzian geometry. J. Math. Phys. 49(9), 09250118 (2008). https://doi.org/10.1063/1.2973048

    Article  MathSciNet  MATH  Google Scholar 

  16. Minguzzi, E.: Characterization of some causality conditions through the continuity of the Lorentzian distance. J. Geom. Phys. 59(7), 827–833 (2009). https://doi.org/10.1016/j.geomphys.2009.03.007

    Article  MathSciNet  MATH  Google Scholar 

  17. Minguzzi, E.: Lorentzian causality theory. Living Rev. Relativ. 22(1), 3 (2019). https://doi.org/10.1007/s41114-019-0019-x

    Article  MATH  Google Scholar 

  18. Minguzzi, E.: Causality theory for closed cone structures with applications. Rev. Math. Phys. 31(5), 1930001, 139 (2019). https://doi.org/10.1142/S0129055X19300012

    Article  MathSciNet  MATH  Google Scholar 

  19. Sormani, C., Vega, C.: Null distance on a spacetime. Classical Quantum Gravity 33(8), 085001, 29 (2016). https://doi.org/10.1088/0264-9381/33/7/085001

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the referee for the careful and fruitful comments which led to the improvement of the original article.

Author information

Authors and Affiliations

Authors

Contributions

The authors contributed equally to the final version of manuscript.

Corresponding author

Correspondence to Neda Ebrahimi.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

The authors approve all the ethics of Springer. The authors read and approved the final manuscript.

Consent for publication

The authors give their consent for publication of the manuscript.

Data availability

‘Not applicable’.

Code availability

‘Not applicable’.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ebrahimi, N., Vatandoost, M. & Pourkhandani, R. On conformal Lorentzian length spaces. Anal.Math.Phys. 13, 93 (2023). https://doi.org/10.1007/s13324-023-00855-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13324-023-00855-1

Keywords

Mathematics Subject Classification

Navigation