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Cosmologies with a magnetic field, dust, and Λ

M. Žofka* and K. Rosquist†

  • *Contact author: martin.zofka@matfyz.cuni.cz
  • †Contact author: kr@fysik.su.se
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Phys. Rev. D 114, 024018 – Published 7 July, 2026

DOI: https://doi.org/10.1103/3ghv-bvxp

Abstract

We investigate exact solutions of the Einstein-Maxwell equations with the cosmological constant where the source of the gravitational field consists of a magnetic field and dust. In particular, we restrict our study to the case of Bianchi type III models. All these solutions either start with a singularity and then expand, or they are initially collapsing and end at a singularity. We discuss the physical meaning of the parameters appearing in the metrics and examine the possible subcases and the relative importance of the dust and the magnetic field throughout the evolution of the solution.

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References (37)

  1. R. Durrer and A. Neronov, Cosmological magnetic fields: Their generation, evolution and observation, Astron. Astrophys. Rev. 21, 62 (2013).
  2. D. Paoletti, J. Chluba, F. Finelli, and J. A. Rubiño-Martín, Constraints on primordial magnetic fields from their impact on the ionization history with Planck 2018, Mon. Not. R. Astron. Soc. 517, 3916 (2022).
  3. I. Vovk, A. Korochkin, A. Neronov, and D. Semikoz, Constraint on intergalactic magnetic field from Fermi/LAT observations of the “pair echo” of GRB 221009A, Astron. Astrophys. 683, A25 (2024).
  4. V. A. Acciari et al., A lower bound on intergalactic magnetic fields from time variability of 1ES 0229+200 from magic and Fermi/LAT observations, Astron. Astrophys. 670, A145 (2023).
  5. Planck Collaboration, Planck 2015 results—XIX. Constraints on primordial magnetic fields, Astron. Astrophys. 594, A19 (2016).
  6. D. Grasso and H. R. Rubinstein, Magnetic fields in the early universe, Phys. Rep. 348, 163 (2001).
  7. T. Papanikolaou and K. N. Gourgouliatos, Constraining supermassive primordial black holes with magnetically induced gravitational waves, Phys. Rev. D 108, 063532 (2023).
  8. R. Kantowski and R. K. Sachs, Some spatially homogeneous anisotropic relativistic cosmological models, J. Math. Phys. (N.Y.) 7, 443 (1966).
  9. G. F. R. Ellis, Dynamics of pressure-free matter in general relativity, J. Math. Phys. (N.Y.) 8, 1171 (1967).
  10. Ø. Grøn and E. Eriksen, A dust-filled Kantowski-Sachs universe with Λ>0, Phys. Lett. 121A, 217 (1987).
  11. J. R. Gair, Kantowski–Sachs universes with counter-rotating dust, Classical Quantum Gravity 19, 6345 (2002).
  12. R. Kantowski, Some relativistic cosmological models, Ph.D. thesis, University of Texas at Austin, Austin, TX, 1966.
  13. A. S. Kompaneyets and A. Chernov, Solution of the gravitation equations for a homogeneous anisotropic model, J. Exp. Theor. Phys. 47, 1303 (1965), https://jetp.ras.ru/cgi-bin/e/index/e/20/5/p1303?a=list.
  14. C. B. Collins, Global structure of the Kantowski–Sachs cosmological models, J. Math. Phys. (N.Y.) 18, 2116 (1977).
  15. E. Weber, Kantowski–Sachs cosmological models approaching isotropy, J. Math. Phys. (N.Y.) 25, 3279 (1984).
  16. J. M. Stewart and G. F. R. Ellis, Solutions of Einstein’s equations for a fluid which exhibit local rotational symmetry, J. Math. Phys. (N.Y.) 9, 1072 (1968).
  17. J. B. Griffiths and J. Podolský, Exact Space-Times in Einstein’s General Relativity, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2009).
  18. H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, Exact Solutions of Einstein’s Field Equations, 2nd ed., Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2003).
  19. A. A. Coley and B. O. J. Tupper, Two-fluid Friedmann-Robertson-Walker cosmologies and their numerical predictions, Can. J. Phys. 64, 204 (1986).
  20. V. Faraoni, S. Jose, and S. Dussault, Multi-fluid cosmology in Einstein gravity: Analytical solutions, Gen. Relativ. Gravit. 53, 109 (2021).
  21. K. S. Thorne, Primordial element formation, primordial magnetic fields, and the isotropy of the Universe, J. Math. Phys. (N.Y.) 148, 51 (1967).
  22. J. P. Vajk and P. G. Eltgroth, Spatially homogeneous anisotropic cosmological models containing relativistic fluid and magnetic field, J. Math. Phys. (N.Y.) 11, 2212 (1970).
  23. A. G. Doroshkevich, Model of a universe with a uniform magnetic field, Astrophysics 1, 138 (1965).
  24. I. S. Shikin, A homogeneous anisotropic cosmological model with a magnetic field, Dokl. Akad. Nauk SSSR 171, 73 (1966), https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=dan&paperid=32670&option_lang=eng.
  25. G. A. Alekseev, Influence of electromagnetic fields on the evolution of initially homogeneous and isotropic universe, Proc Steklov Inst Math / Trudy Matematicheskogo instituta imeni VA Steklova 281, 129 (2013).
  26. N. Tomimura and I. Waga, Inhomogeneous cosmological models: New exact solutions with dust, isotropic radiation, and electromagnetic field, Astrophys. J. 317, 52 (1987).
  27. T. Singh and R. B. S. Yadav, An anisotropic magnetohydrodynamic universe in general relativity, Indian J. Pure Appl. Math. 11, 917 (1980), https://insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20005a21_917.pdf.
  28. R. Casadio, A. Kamenshchik, P. Petriakova, and P. Mavrogiannis, Bianchi cosmologies, magnetic fields, and singularities, Phys. Rev. D 108, 084059 (2023).
  29. S. Ram, Spatially homogeneous cosmological models of Bianchi type III, J. Math. Phys. (N.Y.) 30, 757 (1989).
  30. S. Pal, D. Maity, and T. Q. Do, Magnetogenesis from anisotropic universe, Phys. Rev. D 109, 083507 (2024).
  31. D. Lorenz, Exact Bianchi-Kantowski-Sachs solutions of Einstein’s field equations, J. Phys. A 16, 575 (1983).
  32. M. A. H. MacCallum, A. Moussiaux, P. Tombal, and J. Demaret, On the general solution for a ‘diagonal’ vacuum Bianchi type III model with a cosmological constant, J. Phys. A 15, 1757 (1982).
  33. I. S. Gradshteyn, I. M. Ryzhik, Y. V. Geronimus, and M. Y. Tseytlin, Table of Integrals, Series, and Products, 7th ed., edited by A. Jeffrey and D. Zwillinger (Academic Press, New York, 2007).

  34. There is an error in (3.156.4) in the 8th edition of Gradshteyn and Ryzhik [35]: it defines χ≔arcsinηba2+b2a2+η2, which has the wrong asymptotics for η→∞. We confirmed the result from the 7th edition [33] by integrating numerically.

  35. I. S. Gradshteyn, I. M. Ryzhik, Y. V. Geronimus, and M. Y. Tseytlin, Table of Integrals, Series, and Products, 8th ed., edited by D. Zwillinger and V. H. Moll (Academic Press, New York, 2015).
  36. W. B. Bonnor, Static magnetic fields in general relativity, Proc. Phys. Soc. London Sect. A 67, 225 (1954).
  37. M. Melvin, Pure magnetic and electric geons, Phys. Lett. 8, 65 (1964).

Outline

Information

Phys. Rev. D 114, 024018– Published 7 July, 2026
  • Received 22 May 2025
  • Accepted 9 June 2026
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