Theoretical and Mathematical Physics

, Volume 191, Issue 2, pp 661–668 | Cite as

Cosmological models with homogeneous and isotropic spatial sections



The assumption that the universe is homogeneous and isotropic is the basis for the majority of modern cosmological models. We give an example of a metric all of whose spatial sections are spaces of constant curvature but the space–time is nevertheless not homogeneous and isotropic as a whole. We give an equivalent definition of a homogeneous and isotropic universe in terms of embedded manifolds.


homogeneous isotropic universe cosmology 


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  1. 1.
    C. Clarkson, “Establishing homogeneity of the universe in the shadow of dark energy,” Compt. Rendus Phys., 13, 682–718 (2012); arXiv:1204.5505v1 [astro-ph.CO] (2012).ADSCrossRefGoogle Scholar
  2. 2.
    S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley, New York (1972).Google Scholar
  3. 3.
    A. Friedmann, “Über die Krümmung des Raumes,” Z. Phys., 10, 377–386 (1922).ADSCrossRefMATHGoogle Scholar
  4. 4.
    A. Friedmann, “Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes,” Z. Phys., 21, 326–332 (1924).ADSCrossRefMATHGoogle Scholar
  5. 5.
    G. Lemaître, “Un univers homogène de masse constante et de rayon croissant, rendant compte de la Vitesse radiale de nébuleuses extra-galacticues,” Ann. Soc. Sci. Bruxelles A, 47, 49–59 (1927).MATHGoogle Scholar
  6. 6.
    G. Lemaître, “L´Univers en expansion,” Ann. Soc. Sci. Bruxelles A, 53, 51–85 (1933).Google Scholar
  7. 7.
    H. P. Robertson, “On the foundations of relativistic cosmology,” Proc. Nat. Acad. Sci. USA, 15, 822–829 (1929).ADSCrossRefMATHGoogle Scholar
  8. 8.
    H. P. Robertson, “Relativistic cosmology,” Rev. Modern Phys., 5, 62–90 (1933).ADSCrossRefMATHGoogle Scholar
  9. 9.
    H. P. Robertson, “Kinematics and world structure,” Astrophys. J., 82, 284–301 (1935).ADSCrossRefMATHGoogle Scholar
  10. 10.
    R. C. Tolman, “The effect of the annihilation of matter on the wave-length of light from the nebulae,” Proc. Nat. Acad. Sci. USA, 16, 320–337 (1930).ADSCrossRefMATHGoogle Scholar
  11. 11.
    R. C. Tolman, “More complete discussion of the time-dependence of the non-static line element for the universe,” Proc. Nat. Acad. Sci. USA, 16, 409–420 (1930).ADSCrossRefMATHGoogle Scholar
  12. 12.
    D. Hilbert, “Die Grundlagen der Physik,” Math. Ann., 92, 1–32 (1924).MathSciNetCrossRefGoogle Scholar
  13. 13.
    G. Fubini, “Sugli spazii a quattro dimensioni che ammettono un gruppo continuo di movimenti,” Ann. Mat. Pura Appl. Ser. III, 9, 33–90 (1904).CrossRefMATHGoogle Scholar
  14. 14.
    L. P. Eisenhart, Riemannian Geometry, Princeton Univ. Press, Princeton, N. J. (1926).MATHGoogle Scholar
  15. 15.
    R. C. Tolman, “On the estimation of distances in a curved universe with a non-static line element,” Proc. Nat. Acad. Sci. USA, 16, 511–520 (1930).ADSCrossRefMATHGoogle Scholar
  16. 16.
    A. G. Walker, “On Milne’s theory of world-structure,” Proc. London Math. Soc. Ser. 2, 42, 90–127 (1936).ADSMathSciNetMATHGoogle Scholar
  17. 17.
    M. O. Katanaev, “On homogeneous and isotropic universe,” Modern Phys. Lett. A, 30, 1550186 (2015); arXiv:1511.00991v1 [gr-qc] (2015).ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    R. M. Wald, General Relativity, Univ. Chicago Press, Chicago, Ill. (1984).CrossRefMATHGoogle Scholar
  19. 19.
    M. O. Katanaev, “Geometric methods in mathematical physics,” arXiv:1311.0733v3 [math-ph] (2013).MATHGoogle Scholar
  20. 20.
    I. Ya. Aref’eva and I. V. Volovich, “The null energy condition and cosmology,” Theor. Math. Phys., 155, 503–511 (2008).CrossRefMATHGoogle Scholar
  21. 21.
    V. A. Rubakov, “The null energy condition and its violation,” Phys. Usp., 57, 128–142 (2014).ADSCrossRefGoogle Scholar
  22. 22.
    G. A. Alekseev, “Collision of strong gravitational and electromagnetic waves in the expanding universe,” Phys. Rev. D, 93, 061501 (2016).ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    A. K. Gushchin, “L p-estimates for the nontangential maximal function of the solution to a second-order elliptic equation,” Sb. Math., 207, 1384–1409 (2016).MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    A. K. Gushchin, “Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation,” Sb. Math., 206, 1410–1439 (2015).MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    V. V. Zharinov, “Conservation laws, differential identities, and constraints of partial differential equations,” Theor. Math. Phys., 185, 1557–1581 (2015).MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    V. V. Zharinov, “Bäcklund transformations,” Theoret. and Math. Phys., 189, 1681–1692 (2016).ADSCrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia
  2. 2.Kazan (Volga Region) Federal UniversityKazanRussia

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