Skip to main content
SpringerLink
Log in

Chronological Spacetimes without Lightlike Lines are Stably Causal

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

The statement of the title is proved. It implies that under physically reasonable conditions, spacetimes which are free from singularities are necessarily stably causal and hence admit a time function. Read as a singularity theorem it states that if there is some form of causality violation on spacetime then either it is the worst possible, namely violation of chronology, or there is a singularity. The analogous result: “Non-totally vicious spacetimes without lightlike rays are globally hyperbolic” is also proved, and its physical consequences are explored.

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

Access this article

Log in via an institution

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. Akolia G.M., Joshi P., Vyas U.: On almost causality. J. Math. Phys. 22, 1243–1247 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  2. Beem J.K.: Conformal changes and geodesic completeness. Commun. Math. Phys. 49, 179–186 (1976)

    Article  MATH  ADS  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  4. Bernal A.N., Sánchez M.: Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. Commun. Math. Phys. 257, 43–50 (2005)

    Article  MATH  ADS  Google Scholar 

  5. Borde A.: Geodesic focusing, energy conditions and singularities. Class. Quant. Grav. 4, 343–356 (1987)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. Chicone C., Ehrlich P.: Line integration of Ricci curvature and conjugate points in Lorentzian and Riemannian manifolds. Manus. Math. 31, 297–316 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  7. Geroch R.: Domain of dependence. J. Math. Phys. 11, 437–449 (1970)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. Geroch, R., Horowitz, G.T.: Global structure of spacetimes. In: Vol. General relativity: An Einstein centenary survey, Cambridge: Cambridge University Press, 1979, pp. 212–292

  9. Hawking S.W.: The existence of cosmic time functions. Proc. Roy. Soc. London, series A 308, 433–435 (1968)

    Article  ADS  Google Scholar 

  10. Hawking S.W.: Chronology protection conjecture. Phys. Rev. D 46, 603–611 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  11. Hawking S.W., Ellis G.F.R.: The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge (1973)

    MATH  Google Scholar 

  12. Hawking S.W., Penrose R.: The singularities of gravitational collapse and cosmology. Proc. Roy. Soc. Lond. A 314, 529–548 (1970)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. Krasnikov S.: No time machines in classical general relativity. Class. Quant. Grav. 19, 4109–4129 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Kriele M.: The structure of chronology violating sets with compact closure. Class. Quant. Grav. 6, 1607–1611 (1989)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. Minguzzi E.: Limit curve theorems in Lorentzian geometry. J. Math. Phys. 49, 092501 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  16. Minguzzi E.: The causal ladder and the strength of K-causality. I. Class. Quant. Grav. 25, 015009 (2008)

    Article  MathSciNet  Google Scholar 

  17. Minguzzi E.: The causal ladder and the strength of K-causality. II. Class. Quant. Grav. 25, 015010 (2008)

    Article  MathSciNet  Google Scholar 

  18. Minguzzi E.: Non-imprisonment conditions on spacetime. J. Math. Phys. 49, 062503 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  19. Minguzzi E.: Weak distinction and the optimal definition of causal continuity. Class. Quant. Grav. 25, 075015 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  20. Minguzzi, E., Sánchez, M.: The causal hierarchy of spacetimes. In: H. Baum, D. Alekseevsky (eds.), Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys., Zurich: Eur. Math. Soc. Publ. House, 2008, pp. 299–358

  21. Newman R.P.A.C.: Black holes without singularities. Gen. Rel. Grav. 21, 981–995 (1989)

    Article  MATH  ADS  Google Scholar 

  22. O’Neill B.: Semi-Riemannian Geometry. Academic Press, San Diego (1983)

    MATH  Google Scholar 

  23. Penrose, R.: Singularities and time-asymmetry. In: General relativity: An Einstein centenary survey, Cambridge: Cambridge University Press, 1979, pp. 581–638

  24. Seifert H.: The causal boundary of space-times. Gen. Rel. Grav. 1, 247–259 (1971)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. Seifert H.J.: Smoothing and extending cosmic time functions. Gen. Rel. Grav. 8, 815–831 (1977)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. Senovilla J.M.M.: Singularity theorems and their consequences. Gen. Rel. Grav. 30, 701–848 (1998)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  27. Sorkin R.D., Woolgar E.: A causal order for spacetimes with C 0 Lorentzian metrics: proof of compactness of the space of causal curves. Class. Quant. Grav. 13, 1971–1993 (1996)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  28. Thorne, K.: Closed Timelike Curves. In: General Relativity and Gravitation, Bristol: Inst. of Phys. Publ., 1993, pp. 295–315

  29. Tipler F.J.: Singularities and causality violation. Ann. Phys. 108, 1–36 (1977)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  30. Tipler F.J.: General relativity and conjugate ordinary differential equations. J. Diff. Eq. 30, 165–174 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  31. Tipler F.J.: On the nature of singularities in general relativity. Phys. Rev. D 15, 942–945 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  32. Visser M.: Lorentzian Wormholes. Springer-Verlag, New York (1996)

    Google Scholar 

  33. Woodhouse N.M.J.: The differentiable and causal structures of space-time. J. Math. Phys. 14, 495–501 (1973)

    Article  MATH  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica Applicata, Università degli Studi di Firenze, Via S. Marta 3, I-50139, Firenze, Italy

    E. Minguzzi

Authors
  1. E. Minguzzi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Minguzzi.

Additional information

Communicated by G.W. Gibbons

Rights and permissions

Reprints and permissions

About this article

Cite this article

Minguzzi, E. Chronological Spacetimes without Lightlike Lines are Stably Causal. Commun. Math. Phys. 288, 801–819 (2009). https://doi.org/10.1007/s00220-009-0784-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-009-0784-6

Keywords

Search

Navigation