Killing vector fields and a homogeneous isotropic universe

Some basic theorems on Killing vector fields are reviewed. In particular, the topic of a constant curvature space is examined. A detailed proof is given for a theorem describing the most general form of the metric of a homogeneous isotropic space-time. Although this theorem can be considered commonly known, its complete proof is difficult to find in the literature. An example metric is presented which, while all its spatial cross sections correspond to constant curvature spaces, still is not homogeneous and isotropic as a whole. An equivalent definition of a homogeneous isotropic space-time in terms of embedded manifolds is also given.

Keywords: Killing vector field, homogeneous universe, isotropic universe, Friedmann metric
PACS: 04.20.−q
DOI: 10.3367/UFNe.2016.05.037808
URL: https://ufn.ru/en/articles/2016/7/d/
Web of Science

000386357600004
Scopus

2-s2.0-84991721633
ADS NASA

2016PhyU...59..689K
Citation: Katanaev M O "Killing vector fields and a homogeneous isotropic universe" Phys. Usp. 59 689–700 (2016)

BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Received: 4th, December 2015, accepted: 16th, May 2016

Оригинал: Катанаев М О «Векторные поля Киллинга и однородная и изотропная вселенная» УФН 186 763–775 (2016); DOI: 10.3367/UFNr.2016.05.037808

References (23) ↓ Cited by (20) Similar articles (20)

Friedman A Z. Phys. 10 377 (1922); Per. na russk. yaz., Fridman A A Zhurn. Russk. Fiz.-khim. Obshch-va Ch. Fiz. 56 (1) 59 (1924); Fridman A A Usp. Fiz. Nauk 80 439 (1963); Fridman A A Usp. Fiz. Nauk 93 280 (1967)
Friedmann A Z. Phys. 21 326 (1924); Per. na russk. yaz., Fridman A A Usp. Fiz. Nauk 80 447 (1963)
Lemaître G Ann. Soc. Sci. Bruxelles A 47 49 (1927); Per. na angl. yaz., Lemaître G Mon. Not. R. Astron. Soc. 91 483 (1931)
Lemaître G Ann. Soc. Sci. Bruxelles A 53 51 (1933); Per. na angl. yaz., Lemaître G Gen. Rel. Grav. 29 641 (1997)
Robertson H P Proc. Natl. Acad. Sci. USA 15 822 (1929)
Robertson H P Rev. Mod. Phys. 5 62 (1933)
Robertson H P Astrophys. J. 82 284 (1935)
Tolman R C Proc. Natl. Acad. Sci. USA 16 320 (1930)
Tolman R C Proc. Natl. Acad. Sci. USA 16 409 (1930)
Tolman R C Proc. Natl. Acad. Sci. USA 16 511 (1930)
Walker A G Proc. London Math. Soc. 2 42 90 (1936)
Hilbert D Math. Ann. 15 1 (1924)
Fubini G Ann. Mat. 3 9 33 (1904)
Eisenhart L P Riemannian Geometry (Princeton: Princeton Univ. Press, 1926); Per. na russk. yaz., Eizenkhart L P Rimanova Geometriya (M.: IL, 1948)
Landau L D, Lifshits E M Teoriya Polya (M.: Nauka, 1967); Per. na angl. yaz., Landau L D, Lifshitz E M The Classical Theory Of Fields 2nd ed. (New York: Pergamon, 1962)
Weinberg S Gravitation And Cosmology. Principles And Applications Of The General Theory Of Relativity (New York: Wiley, 1972)
Hawking S W, Ellis G F R The Large Scale Structure Of Space-Time (Cambridge: Univ. Press, 1973)
Misner C W, Thorne K S, Wheeler J A Gravitation (San Francisco: W.H. Freeman, 1973)
Wald R M General Relativity (Chicago: Univ. of Chicago Press, 1984)
Kobayashi S, Nomizu K Foundations Of Differential Geometry Vol. 1, 2 (New York: Interscience Publ., 1963, 1969); Per. na russk. yaz., Kobayasi Sh, Nomidzu K Osnovy Differentsial’noi Geometrii Vol. 1, 2 (M.: Nauka, 1981)
Dubrovin B A, Novikov S P, Fomenko A T Sovremennaya Geometriya. Metody i Prilozheniya 4-e izd. (M.: Nauka, 1998); Per. na angl. yaz., Dubrovin B A, Fomenko A T, Novikov S P Modern Geometry — Methods And Applications 2nd ed. (New York: Springer-Verlag, 1992)
Wolf J A Spaces Of Constant Curvature (Berkley, Calif.: Univ. of California Press, 1972)
Katanaev M O arXiv:1311.0733