- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Stationary generalizations for the vacuum ring wormhole

Mikhail S. Volkov*

  • *Contact author: mvolkov@univ-tours.fr
PDF

Phys. Rev. D 114, 084031 – Published 13 October, 2026

DOI: https://doi.org/10.1103/mm69-kjkh

Abstract

The ring wormhole is the zero-mass limit of the Kerr metric. Its geometry is locally flat, but the topology is nontrivial, with a throat connecting two asymptotic regions and a distributional curvature singularity on the ring encircling the throat. We construct stationary generalizations of this static wormhole that are different from Kerr and invariant under reflections across the wormhole throat. The problem reduces to solving the vacuum Ernst equations subject to the corresponding symmetry conditions. The slowly rotating perturbative solutions were constructed previously, while we now present a detailed analysis of nonperturbative solutions obtained within a numerical framework. For slow rotation, they exhibit the nonrelativistic relation M∼J2 between the mass and angular momentum, which transforms into the Regge relation J=M2 in the fast-rotation regime, when J→∞ and the ring is stretched without bound by the centrifugal force. However, if the ring size in the static limit is sent to zero at the same time, then M and J remain bounded as the throat linear velocity approaches unity. The wormhole geometry then approaches the extremal Kerr solution, thus “mimicking” it. The wormholes carry a curvature singularity at the ring, but this can be removed by via simple “scalarization” procedure that promotes the vacuum solutions to regular wormholes with a phantom scalar field.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (67)

  1. A. Einstein and N. Rosen, The particle problem in the general theory of relativity, Phys. Rev. 48, 73 (1935).
  2. A. Einstein, B. Podolsky, and N. Rosen, Can quantum mechanical description of physical reality be considered complete?, Phys. Rev. 47, 777 (1935).
  3. J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortschr. Phys. 61, 781 (2013).
  4. I. Javed and E. Wilson-Ewing, Testing wormhole-mediated entanglement with hydrogen, Phys. Rev. Lett. 136, 121501 (2026).
  5. C. W. Misner and J. A. Wheeler, Classical physics as geometry: Gravitation, electromagnetism, unquantized charge, and mass as properties of curved empty space, Ann. Phys. (N.Y.) 2, 525 (1957).
  6. C. W. Misner, Wormhole initial conditions, Phys. Rev. 118, 1110 (1960).
  7. M. Cvetic, G. W. Gibbons, and C. N. Pope, Super-geometrodynamics, J. High Energy Phys. 03 (2015) 029.
  8. B. Abbott et al. (LIGO Scientific and Virgo Collaborations), Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116, 061102 (2016).
  9. M. Morris, K. Thorne, and U. Yurtsever, Wormholes, time machines, and the weak energy condition, Phys. Rev. Lett. 61, 1446 (1988).
  10. M. Visser, Lorentzian Wormholes: From Einstein to Hawking (AIP, Woodbury, NY, 1996).
  11. J. L. Friedman, K. Schleich, and D. M. Witt, Topological censorship, Phys. Rev. Lett. 71, 1486 (1993); 75, 1872(E) (1995).
  12. D. Hochberg and M. Visser, The null energy condition in dynamic wormholes, Phys. Rev. Lett. 81, 746 (1998).
  13. K. Bronnikov, Scalar-tensor theory and scalar charge, Acta Phys. Pol. B4, 251 (1973), https://www.actaphys.uj.edu.pl/fulltext?series=Reg&vol=4&page=251.
  14. H. G. Ellis, Ether flow through a drainhole—a particle model in general relativity, J. Math. Phys. (N.Y.) 14, 104 (1973).
  15. A. Riess et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116, 1009 (1998).
  16. S. Perlmutter et al., Measurements of ω and λ from 42 high-redshift supernovae, Astrophys. J. 517, 565 (1999).
  17. H. Maeda and M. Nozawa, Static and symmetric wormholes respecting energy conditions in Einstein-Gauss-Bonnet gravity, Phys. Rev. D 78, 024005 (2008).
  18. P. Kanti, B. Kleihaus, and J. Kunz, Wormholes in dilatonic Einstein-Gauss-Bonnet theory, Phys. Rev. Lett. 107, 271101 (2011).
  19. M. A. Cuyubamba, R. A. Konoplya, and A. Zhidenko, No stable wormholes in Einstein-dilaton-Gauss-Bonnet theory, Phys. Rev. D 98, 044040 (2018).
  20. K. Bronnikov and S.-W. Kim, Possible wormholes in a brane world, Phys. Rev. D 67, 064027 (2003).
  21. S. V. Sushkov and R. Korolev, Scalar wormholes with nonminimal derivative coupling, Classical Quantum Gravity 29, 085008 (2012).
  22. S. V. Sushkov and M. S. Volkov, Giant wormholes in ghost-free bigravity theory, J. Cosmol. Astropart. Phys. 06 (2015) 017.
  23. J. L. Blázquez-Salcedo, C. Knoll, and E. Radu, Traversable wormholes in Einstein-Dirac-Maxwell theory, Phys. Rev. Lett. 126, 101102 (2021).
  24. R. A. Konoplya and A. Zhidenko, Traversable wormholes in general relativity, Phys. Rev. Lett. 128, 091104 (2022).
  25. V. Dzhunushaliev and V. Folomeev, Rotating wormholes in Einstein-Dirac-Maxwell theory, Gen. Relativ. Gravit. 58, 39 (2026).
  26. S.-W. Kim and H. Lee, Exact solutions of a charged wormhole, Phys. Rev. D 63, 064014 (2001).
  27. J. L. Blázquez-Salcedo, X. Y. Chew, J. Kunz, and D.-H. Yeom, Ellis wormholes in anti-de Sitter space, Eur. Phys. J. C 81, 858 (2021).
  28. H.-C. Kim and W. Lee, Charged wormholes in (anti-)de Sitter spacetime, Phys. Lett. B 873, 140145 (2026).
  29. D. Zipoy, Topology of some spheroidal metrics, J. Math. Phys. (N.Y.) 7, 1137 (1966).
  30. K. A. Bronnikov and J. C. Fabris, Weyl space-times and wormholes in D-dimensional Einstein and dilaton gravity, Classical Quantum Gravity 14, 831 (1997).
  31. G. Clément, D. Gal’tsov, and M. Guenouche, NUT wormholes, Phys. Rev. D 93, 024048 (2016).
  32. G. W. Gibbons and M. S. Volkov, Weyl metrics and wormholes, J. Cosmol. Astropart. Phys. 05 (2017) 039.
  33. G. Clément and D. Gal’tsov, Rotating traversable wormholes in Einstein-Maxwell theory, Phys. Lett. B 838, 137677 (2023).
  34. G. Clément, A wormhole with two black holes, arXiv:2607.19145.
  35. B. Carter, Global structure of the Kerr family of gravitational fields, Phys. Rev. 174, 1559 (1968).
  36. G. W. Gibbons and M. S. Volkov, Zero mass limit of Kerr spacetime is a wormhole, Phys. Rev. D 96, 024053 (2017).
  37. B. H. Voorhees, Static axially symmetric gravitational fields, Phys. Rev. D 2, 2119 (1970).
  38. G. W. Gibbons and M. S. Volkov, Ring wormholes via duality rotations, Phys. Lett. B 760, 324 (2016).
  39. V. P. Frolov, P. Krtous, and A. Zelnikov, Ring wormholes and time machines, Phys. Rev. D 108, 024034 (2023).
  40. I. A. Sarmiento-Alvarado, L. Bixano, and T. Matos, Axially symmetric wormholes, Classical Quantum Gravity 43, 175004 (2026).
  41. H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers, and E. Herlt, Exact Solutions of Einstein’s Field Equations, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2003).
  42. V. Belinski and E. Verdaguer, Gravitational Solitons, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2005).
  43. A. Eris and M. Gurses, Stationary axially-symmetric solutions of Einstein-Maxwell massless scalar field equations, J. Math. Phys. (N.Y.) 18, 1303 (1977).
  44. P. E. Kashargin and S. V. Sushkov, Slowly rotating wormholes: The first order approximation, Gravitation Cosmol. 14, 80 (2008).
  45. P. E. Kashargin and S. V. Sushkov, Slowly rotating scalar field wormholes: The second order approximation, Phys. Rev. D 78, 064071 (2008).
  46. B. Kleihaus and J. Kunz, Rotating Ellis wormholes in four dimensions, Phys. Rev. D 90, 121503 (2014).
  47. X. Y. Chew, B. Kleihaus, and J. Kunz, Geometry of spinning Ellis wormholes, Phys. Rev. D 94, 104031 (2016).
  48. M. S. Volkov, Stationary generalizations for the Bronnikov-Ellis wormhole and for the vacuum ring wormhole, Phys. Rev. D 104, 124064 (2021).
  49. H.-a. Shinkai and S. A. Hayward, Fate of the first traversible wormhole: Black hole collapse or inflationary expansion, Phys. Rev. D 66, 044005 (2002).
  50. B. Azad, J. L. Blazquez-Salcedo, F. S. Khoo, and J. Kunz, Are slowly rotating Ellis-Bronnikov wormholes stable?, Phys. Lett. B 848, 138349 (2024).
  51. B. Azad, J. L. Blázquez-Salcedo, F. S. Khoo, and J. Kunz, Radial perturbations of Ellis-Bronnikov wormholes in slow rotation up to second order, Phys. Rev. D 109, 124051 (2024).
  52. F. J. Ernst, New formulation of the axially symmetric gravitational potential, Phys. Rev. 167, 1175 (1968).
  53. A. H. Taub, Empty space-times admitting a three parameter group of motions, Ann. Math. 53, 472 (1951).
  54. L. Smarr, Mass formula for Kerr black holes, Phys. Rev. Lett. 30, 71 (1973); 30, 521(E) (1973).
  55. F. Hecht, New development in freefem++, J. Numer. Math. 20, 251 (2012).
  56. A. J. Amsel, G. T. Horowitz, D. Marolf, and M. M. Roberts, Uniqueness of extremal Kerr and Kerr-Newman black holes, Phys. Rev. D 81, 024033 (2010).
  57. T. Damour and S. N. Solodukhin, Wormholes as black hole foils, Phys. Rev. D 76, 024016 (2007).
  58. L. Smarr, Surface geometry of charged rotating black holes, Phys. Rev. D 7, 289 (1973).
  59. G. Clement, Selfgravitating cosmic rings, Phys. Lett. B 449, 12 (1999).
  60. C. Hoenselaers, W. Kinnersley, and B. C. Xanthopoulos, Symmetries of the stationary Einstein-Maxwell equations. 6. Transformations which generate asymptotically flat space-times, J. Math. Phys. (N.Y.) 20, 2530 (1979).
  61. J. M. Bardeen and G. T. Horowitz, The extreme Kerr throat geometry: A vacuum analog of AdS2× S2, Phys. Rev. D 60, 104030 (1999).
  62. R. P. Geroch, Multipole moments. II. Curved space, J. Math. Phys. (N.Y.) 11, 2580 (1970).
  63. R. Hansen, Multipole moments of stationary spacetimes, J. Math. Phys. (N.Y.) 15, 46 (1974).
  64. L. Landau and E. Lifshitz, The Classical Theory of Fields, Course of Theoretical Physics (Butterworth-Heinemann, Oxford, England, 1975).
  65. G. Fodor, C. Hoenselaers, and Z. Perjes, Multipole moments of axisymmetric systems in relativity, J. Math. Phys. (N.Y.) 30, 2252 (1989).
  66. G. Fodor, E. d. S. C. Filho, and B. Hartmann, Calculation of multipole moments of axistationary electrovacuum spacetimes, Phys. Rev. D 104, 064012 (2021).
  67. T. P. Sotiriou and T. A. Apostolatos, Corrected multipole moments of axisymmetric electrovacuum spacetimes, Classical Quantum Gravity 21, 5727 (2004).

Outline

Information

Phys. Rev. D 114, 084031– Published 13 October, 2026
  • Received 18 June 2026
  • Accepted 10 September 2026
Crossmark - Check for updates button

© 2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation