On the topology of spherically symmetric space-times

• Autorzy: J. Szenthe
• Języki publikacji: EN

Abstrakty

EN

Spherically symmetric space-times have attained considerable attention ever since the early beginnings of the theory of general relativity. In fact, they have appeared already in the papers of K. Schwarzschild [12] and W. De Sitter [5] which were published in 1916 and 1917 respectively soon after Einstein's epoch-making work [7] in 1915. The present survey is concerned mainly with recent results pertainig to the toplogy of spherically symmetric space-times. Definition. By space-time a connected time-oriented 4-dimensional Lorentz manifold is meant. If (M,<,>) is a space-time, and Φ: SO(3)×M→M an isometric action such that the maximal dimension of its orbits is equal to 2, then the action Φ is said to be spherical and the space-time is said to be spherically symmetric [8]; [11]. Likewise, isometric actions Ψ: O(3)×M→M are also considered ([10], p. 365; [4]) which will be called quasi-spherical if the maximal dimension of its orbits is 2 and then the space-time is said to be quasi-spherically symmetric here. Each quasi-spherical action yields a spherical one by restricting it to the action of SO(3); the converse of this statement will be considered elsewhere. The main results concerning spherically symmetric space-times are generally either of local character or pertaining to topologically restricted simple situations [14], and earlier results of global character are scarce [1], [4], [6], [13]. A report on recent results concerning the global geometry of spherically symmetric space-times [16] is presented below.

Słowa kluczowe

EN

Lorentz mainfolds; general relativity; isometric actions; 53C50; 57S25; 83C40

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2004
• Tom: 2
• Numer: 5
• Strony: 725-731

Daty

wydano

2004-10-01

online

2004-10-01

Twórcy

autor

J. Szenthe

• Eötvös University

Bibliografia

• [1] P.G. Bergmann, M. Cahen and A.B. Komar: “Spherically symmetric gravitational fields”, J. Math. Phys., (1965), pp 1–5.
• [2] G.D. Birkhoff: Relativity and Modern Physics, Cambridge, 1923.
• [3] G.E. Bredon: Introduction to Compact Transformation Groups, New York, 1972.
• [4] C.J.S. Clarke: “Spherical symmetry does not imply direct product”, Class. Quantum Grav., Vol. 4, (1987), pp. 37–40. http://dx.doi.org/10.1088/0264-9381/4/3/001
• [5] W. De Sitter: “On Einstein's theory of gravitation and its astronomical consequences”, Mon. Not. Roy. Astr. Soc., Vol. 76, (1917), p. 699.
• [6] J. Ehlers: Relativity, Astrophysics and Cosmology, Dordrecht, 1973.
• [7] A. Einstein: “Zur allgemeinen Relativitätstheorie”, Sitzungsb. Preuss. Akad. Wiss.; Phys.-Math. Kl., (1915), pp. 778–779.
• [8] S.W. Hawking and G.F.R. Ellis: The Large Scale Structure of Space-time, London, 1973.
• [9] S. Kobayashi and K. Nomizu: Foundations of Differential Geometry I, II, New York, 1963, 1969.
• [10] B. O'Neill: Semi-Riemannian Geometry with Applications to Relativity, New York, 1983.
• [11] R.K. Sachs and H. Wu: General Relativity for Mathematicians, New York, 1977.
• [12] K. Schwarzschild: “Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie”, Sitzungsb. Preuss. Akad. Wiss.; Phys.-Math. Kl., (1916), pp. 189–196.
• [13] R. Siegl: “Some underlying manifolds of the Schwarzschild solution”, Class. Quantum Grav., Vol. 9, (1992), pp. 239–240. http://dx.doi.org/10.1088/0264-9381/9/1/021
• [14] J.L. Synge: Relativity: The General Theory, Amsterdam, 1960.
• [15] J. Szenthe: “A construction of transverse submanifolds”, Univ. Iagell. Acta Math., Vol. 41, (2003), To appear.
• [16] J. Szenthe: “On the global geometry of spherically symmetric space-times”, Math. Proc. Camb. Phil. Soc., Vol. 137, (2004), pp. 297–306. http://dx.doi.org/10.1017/S030500410400790X

Identyfikatory

DOI

10.2478/BF02475973