A ‘Regularized’ Schwarzschild Solution

  • Frans R. KlinkhamerEmail author
Part of the Springer Proceedings in Physics book series (SPPHY, volume 170)


An exact solution of the vacuum Einstein field equations over a particular nonsimply-connected manifold is presented. This solution is spherically symmetric and has no curvature singularity. It provides a regularization of the Schwarzschild solution with a curvature singularity at the center.


Curvature Singularity Timelike Curve Antipodal Point Spacetime Model Nonsingular Solution 
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It is a pleasure to thank the participants of the Karl Schwarzschild Meeting on Gravitational Physics (Frankfurt Institute for Advanced Studies, July 2013) for interesting discussions and the organizers for making it all happen.


  1. 1.
    K. Schwarzschild, Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie. Sitzungsberichte der Deutschen Akademie der Wissenschaften zu Berlin, Klasse für Mathematik, Physik, und Technik 189–196 (1916) [scanned version available from]
  2. 2.
    M.D. Kruskal, Maximal extension of Schwarzschild metric. Phys. Rev. 119, 1743–1745 (1960)MathSciNetCrossRefADSzbMATHGoogle Scholar
  3. 3.
    G. Szekeres, On the singularities of a Riemannian manifold. Publ. Math. Debrecen 7, 285–301 (1960)MathSciNetzbMATHGoogle Scholar
  4. 4.
    S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge, 1973)CrossRefzbMATHGoogle Scholar
  5. 5.
    C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation (Freeman, New York, 1973)Google Scholar
  6. 6.
    S. Bernadotte, F.R. Klinkhamer, Bounds on length scales of classical spacetime foam models. Phys. Rev. D 75, 024028 (2007). arXiv:hep-ph/0610216
  7. 7.
    F.R. Klinkhamer, M. Schreck, New two-sided bound on the isotropic Lorentz-violating parameter of modified Maxwell theory. Phys. Rev. D 78, 085026 (2008). arXiv:0809.3217
  8. 8.
    M. Schwarz, Nontrivial Spacetime Topology, Modified Dispersion Relations, and an \(SO(3)\)-Skyrme Model. Ph.D. Thesis, KIT, July 2010. Verlag Dr. Hut, München, Germany (2010)Google Scholar
  9. 9.
    F.R. Klinkhamer, C. Rahmede, Nonsingular spacetime defect. Phys. Rev. D 89, 084064 (2014), arXiv:1303.7219
  10. 10.
    F.R. Klinkhamer, Black-hole solution without curvature singularity. Mod. Phys. Lett. A 28, 1350136 (2013). arXiv:1304.2305
  11. 11.
    F.R. Klinkhamer, Black-hole solution without curvature singularity and closed timelike curves. Acta Phys. Pol. B 45, 5–14 (2014), arXiv:1305.2875
  12. 12.
    F.R. Klinkhamer, A new type of nonsingular black-hole solution in general relativity. Mod. Phys. Lett. A 29, 1430018 (2014), arXiv:1309.7011
  13. 13.
    P. Painlevé, La mécanique classique et la théorie de la relativité. C. R. Acad. Sci. (Paris) 173, 677–680 (1921)zbMATHGoogle Scholar
  14. 14.
    A. Gullstrand, Allgemeine Lösung des statischen Einkörper-problems in der Einsteinschen Gravitationstheorie. Arkiv. Mat. Astron. Fys. 16, 1–15 (1922)Google Scholar
  15. 15.
    K. Martel, E. Poisson, Regular coordinate systems for Schwarzschild and other spherical space-times. Am. J. Phys. 69, 476–480 (2001). arXiv:gr-qc/0001069
  16. 16.
    H. Reissner, Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie. Ann. der Phys. 50, 106–120 (1916)CrossRefADSGoogle Scholar
  17. 17.
    G. Nordström, On the energy of the gravitational field in Einstein’s theory. Proc. Acad. Sci. Amst. 26, 1201–1208 (1918)zbMATHGoogle Scholar
  18. 18.
    F.R.Klinkhamer, Skyrmion spacetime defect. Phys. Rev. D 90, 024007 (2014), arXiv:1402.7048
  19. 19.
    F.R. Klinkhamer, F. Sorba, Comparison of spacetime defects which are homeomorphic but not diffeomorphic. J. Math. Phys. 55, 112503 (2014), arXiv:1404.2901

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikKarlsruher Institut für TechnologieKarlsruheGermany

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