Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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).
S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley, New York (1972).
A. Friedmann, “Über die Krümmung des Raumes,” Z. Phys., 10, 377–386 (1922).
A. Friedmann, “Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes,” Z. Phys., 21, 326–332 (1924).
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).
G. Lemaître, “L´Univers en expansion,” Ann. Soc. Sci. Bruxelles A, 53, 51–85 (1933).
H. P. Robertson, “On the foundations of relativistic cosmology,” Proc. Nat. Acad. Sci. USA, 15, 822–829 (1929).
H. P. Robertson, “Relativistic cosmology,” Rev. Modern Phys., 5, 62–90 (1933).
H. P. Robertson, “Kinematics and world structure,” Astrophys. J., 82, 284–301 (1935).
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).
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).
D. Hilbert, “Die Grundlagen der Physik,” Math. Ann., 92, 1–32 (1924).
G. Fubini, “Sugli spazii a quattro dimensioni che ammettono un gruppo continuo di movimenti,” Ann. Mat. Pura Appl. Ser. III, 9, 33–90 (1904).
L. P. Eisenhart, Riemannian Geometry, Princeton Univ. Press, Princeton, N. J. (1926).
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).
A. G. Walker, “On Milne’s theory of world-structure,” Proc. London Math. Soc. Ser. 2, 42, 90–127 (1936).
M. O. Katanaev, “On homogeneous and isotropic universe,” Modern Phys. Lett. A, 30, 1550186 (2015); arXiv:1511.00991v1 [gr-qc] (2015).
R. M. Wald, General Relativity, Univ. Chicago Press, Chicago, Ill. (1984).
M. O. Katanaev, “Geometric methods in mathematical physics,” arXiv:1311.0733v3 [math-ph] (2013).
I. Ya. Aref’eva and I. V. Volovich, “The null energy condition and cosmology,” Theor. Math. Phys., 155, 503–511 (2008).
V. A. Rubakov, “The null energy condition and its violation,” Phys. Usp., 57, 128–142 (2014).
G. A. Alekseev, “Collision of strong gravitational and electromagnetic waves in the expanding universe,” Phys. Rev. D, 93, 061501 (2016).
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).
A. K. Gushchin, “Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation,” Sb. Math., 206, 1410–1439 (2015).
V. V. Zharinov, “Conservation laws, differential identities, and constraints of partial differential equations,” Theor. Math. Phys., 185, 1557–1581 (2015).
V. V. Zharinov, “Bäcklund transformations,” Theoret. and Math. Phys., 189, 1681–1692 (2016).
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 191, No. 2, pp. 219–227, May, 2017.
Rights and permissions
About this article
Cite this article
Katanaev, M.O. Cosmological models with homogeneous and isotropic spatial sections. Theor Math Phys 191, 661–668 (2017). https://doi.org/10.1134/S0040577917050063
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577917050063