On the topology of spherically symmetric space-times

J. Szenthe

Open Mathematics (2004)

  • Volume: 2, Issue: 5, page 725-731
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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J. Szenthe. "On the topology of spherically symmetric space-times." Open Mathematics 2.5 (2004): 725-731. <http://eudml.org/doc/268747>.

@article{J2004,
abstract = {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.},
author = {J. Szenthe},
journal = {Open Mathematics},
keywords = {53C50; 57S25; 83C40},
language = {eng},
number = {5},
pages = {725-731},
title = {On the topology of spherically symmetric space-times},
url = {http://eudml.org/doc/268747},
volume = {2},
year = {2004},
}

TY - JOUR
AU - J. Szenthe
TI - On the topology of spherically symmetric space-times
JO - Open Mathematics
PY - 2004
VL - 2
IS - 5
SP - 725
EP - 731
AB - 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.
LA - eng
KW - 53C50; 57S25; 83C40
UR - http://eudml.org/doc/268747
ER -

References

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  1. [1] P.G. Bergmann, M. Cahen and A.B. Komar: “Spherically symmetric gravitational fields”, J. Math. Phys., (1965), pp 1–5. Zbl0125.21005
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  5. [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. [6] J. Ehlers: Relativity, Astrophysics and Cosmology, Dordrecht, 1973. 
  7. [7] A. Einstein: “Zur allgemeinen Relativitätstheorie”, Sitzungsb. Preuss. Akad. Wiss.; Phys.-Math. Kl., (1915), pp. 778–779. 
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  10. [10] B. O'Neill: Semi-Riemannian Geometry with Applications to Relativity, New York, 1983. 
  11. [11] R.K. Sachs and H. Wu: General Relativity for Mathematicians, New York, 1977. Zbl0373.53001
  12. [12] K. Schwarzschild: “Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie”, Sitzungsb. Preuss. Akad. Wiss.; Phys.-Math. Kl., (1916), pp. 189–196. Zbl46.1296.02
  13. [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. [14] J.L. Synge: Relativity: The General Theory, Amsterdam, 1960. Zbl0090.18504
  15. [15] J. Szenthe: “A construction of transverse submanifolds”, Univ. Iagell. Acta Math., Vol. 41, (2003), To appear. Zbl1065.53056
  16. [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 Zbl1065.53056

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