Gravitation and Cosmology

, Volume 25, Issue 2, pp 116–121 | Cite as

Space-time Singularities vs. Topologies in the Zeeman—Göbel Class

  • Kyriakos PapadopoulosEmail author
  • B. K. Papadopoulos


We first observe that the Path topology of Hawking, King and MacCarthy is an analogue, in curved space-times, of a topology that was suggested by Zeeman as an alternative topology to his so-called Fine topology in Minkowski space-time. We then review a result of a recent paper on spaces of paths and the Path topology, and see that there are at least five more topologies in the class \(\mathfrak{Z}-\mathfrak{G}\) of Zeeman-Göbel topologies which admit a countable basis, incorporate the causal and conformal structures, but the Limit Curve Theorem (LCT) fails to hold. The “problem” that the LCT does not hold can be resolved by “adding back” the light cones in the basic-open sets of these topologies, and create new basic open sets for new topologies. But, the main question is: do we really need the LCT. to hold, and why? Why is the manifold topology, under which the group of homeomorphisms of a space-time is vast and of no physical significance (Zeeman), more preferable than an appropriate topology in the class \(\mathfrak{Z}-\mathfrak{G}\) under which a homeomorphism is an isometry (Göbel)? Since topological conditions that come as a result of a causality requirement are key in the existence of singularities in general relativity, the global topological conditions that one will supply the space-time manifold might play an important role in describing the transition from a quantum nonlocal theory to a classical local theory.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



K.P. would like to thank Robert Low for his comments on the interval topology, Fabio Scardigli for discussions on the topological structure of spacetime, and Nikolaos Kalogeropoulos for inspiring discussions on quantum gravity, based on his article [23].


  1. 1.
    E.H. Kronheimer and R. Penrose, “On the structure of causal spaces,” Proc. Camb. Phil. Soc. 63, 481 (1967).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    R. Penrose, “Techniques of differential topology in relativity,” in: CBMS-NSF Regional Conference Series in Applied Mathematics, 1972.Google Scholar
  3. 3.
    Chris Good and Kyriakos Papadopoulos, “A topological characterization of ordinals: van Dalen and Wattel revisited,” Topology Appl. 159, 1565–1572 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kyriakos Papadopoulos, “On the orderability problem and the interval topology,” in: Topics in Mathematical Analysis and Applications, in “Optimization and Its Applications” Springer Series, Eds. T. Rassias and L. Toth (Springer-Verlag, 2014).Google Scholar
  5. 5.
    Gerhard Gierz, Karl Heinrich Hoffmann, Klaus Keimel, Jimmie D. Lawson, Michael W. Mislove, and Dana S. Scott, A Compendium of Continuous Lattices (Springer-Verlag, 1980).CrossRefzbMATHGoogle Scholar
  6. 6.
    Robert J. Low, “Spaces of paths and the path topology,” J. Math. Phys. 57, 092503 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    E. C. Zeeman, “The topology of Minkowski space,” Topology 6, 161–170 (1967).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Göbel, “Zeeman topologies on space-times of general relativity theory,” Comm. Math. Phys. 46, 289–307 (1976).Google Scholar
  9. 9.
    E. C. Zeeman, “Causality implies the Lorentz group,” J. Math. Phys. 5, 490–493 (1964).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ignatios Antoniadis, Spiros Cotsakis, and Kyriakos Papadopoulos, “The causal order on the ambient boundary,” Mod. Phys. Lett. A, 31, Issue 20 (2016).Google Scholar
  11. 11.
    G. M. Reed, “The intersection topology w.r.t. the real line and the countable ordinals,” Trans. Am. Math. Society 297, 509–520 (1986).MathSciNetzbMATHGoogle Scholar
  12. 12.
    Kyriakos Papadopoulos, “On the possibility of singularities on the ambient boundary,” Int. J. Geom. Methods in Mod. Phys. 14, No. 10 (2017).Google Scholar
  13. 13.
    S. W. Hawking, A. R. King, and P. J. McCarthy, “A new topology for curved space-time which incorporates the causal, differential, and conformal structures,” J. Math. Phys. 17, 174–181 (1976).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kyriakos B. Papadopoulos, Santanu Acharjee, and Basil K. Papadopoulos, “The order on the light cone and its induced topology,” Int. J. Geom. Methods in Mod. Phys. 15, 1850069 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kyriakos B. Papadopoulos and Basil K. Padopoulos, “On two topologies that were suggested by Zeeman,” arXiv: 1706.07488.Google Scholar
  16. 16.
    Kyriakos Papadopoulos, “On properties of nests: some answers and questions,” Questions and Answers in General Topology 33, No. 2 (2015).Google Scholar
  17. 17.
    Kyriakos Papadopoulos, “Nests and their role in the orderability problem,” in: Mathematical Analysis, Approximation Theory and Their Applications, Eds. Th. M. Rassians and V. Gupta (Springer, 2016), pp. 517–533.CrossRefGoogle Scholar
  18. 18.
    David B. Malament, “The class of continuous timelike curves determines the topology of space-time,” J. Math. Phys. 19, 1399 (1977).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    R. Geroch, “Domain of dependence,” J. Math. Phys. 11, 437–449 (1970).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Seth Major, “On recovering continuum topology from a causal set,” J. Math. Phys. 48, 032501, (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Onkar Parrikar and Sumati Surya, “Causal topology in future and past distinguishing space-times,” Class. Quantum Grav. 28, No 15 (2011).Google Scholar
  22. 22.
    R. D. Sorkin, “Does locality fail at intermediate length scales?” in Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter, Ed. D. Oriti (Cambridge Univ. Press, Cambridge, UK. 2009), pp. 26–43.CrossRefGoogle Scholar
  23. 23.
    Nikolaos Kalogeropoulos, “Systolic aspects of black hole entropy,” arXiv: 1711.09963.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Department of MathematicsKuwait UniversitySafatKuwait
  2. 2.Department of Civil EngineeringDemocritus University of ThraceKomotiniGreece

Personalised recommendations