Abstract
Embeddings into higher dimensions are very important in the study of higher-dimensional theories of our Universe and in high-energy physics. Theorems which have been developed recently guarantee the existence of embeddings of pseudo-Riemannian manifolds into Einstein spaces and more general pseudo-Riemannian spaces. These results provide a technique that can be used to determine solutions for such embeddings. Here we consider local isometric embeddings of four-dimensional spherically symmetric spacetimes into five-dimensional Einstein manifolds. Difficulties in solving the five-dimensional equations for given four-dimensional spaces motivate us to investigate embedded spaces that admit bulks of a specific type. We show that the general Schwarzschild–de Sitter spacetime and Einstein Universe are the only spherically symmetric spacetimes that can be embedded into an Einstein space of a particular form, and we discuss their five-dimensional solutions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M Janet, Ann. Soc. Polon. Math. 5, 38 (1926) E Cartan, Ann. Soc. Polon. Math. 6, 1 (1927) A Friedman, J. Math. Mech. 10, 625 (1961) J Nash, Ann. Math. 60, 383 (1954) J Nash, Ann. Math. 63, 20 (1956) C J S Clarke, Proc. R. Soc. London A314, 417 (1970) R E Greene, Mem. Amer. Math. Soc. 97, 1 (1970) R E Greene and H Jacobowitz, Ann. Math. 93, 189 (1971) M Gunther, Ann. Global Anal. Geom. 7, 69 (1989)
J E Campbell, A course of differential geometry (Clarendon Press, Oxford, 1926) L Magaard, Zur einbettung riemannscher Raume in Einstein-Raume und konformeuclidische Raume, Ph.D. thesis (Kiel, 1963)
J E Lidsey, C Romero, R Tavakol and S Rippl, Class. Quant. Grav. 14, 865 (1997)
E Anderson and J E Lidsey, Class. Quant. Grav. 18, 4831 (2001)
F Dahia and C Romero, J. Math. Phys. 43, 5804 (2002)
F Dahia and C Romero, J. Math. Phys. 43, 3097 (2002)
E Anderson, F Dahia, J E Lidsey and C Romero, J. Math. Phys. 44, 5108 (2003)
N I Katzourakis, The global embedding problem of semi-Riemannian into Einstein manifolds (2005), arXiv:math-ph/0407067v4
J Moodley and G Amery, Global embeddings of pseudo-Riemannian spaces (2010), in preparation
M B Green, J H Schwarz and E Witten, Superstring theory (Cambridge University Press, Cambridge, 1987) K Becker, M Becker and J H Schwarz, String theory and M-theory: A modern introduction (Cambridge University Press, Cambridge, 2007)
P Horava and E Witten, Nucl. Phys. B460, 506 (1996)
J Polchinski, Phys. Rev. Lett. 75, 4724 (1995) J Polchinski, in: Fields, strings and duality: TASI 96 edited by C Efthimiou and B Greene (World Scientific, Singapore, 1997) p. 293 C V Johnson, D-Branes (Cambridge University Press, Cambridge, 2003)
N Arkani-Hamed, S Dimopoulos and G Dvali, Phys. Lett. B429, 263 (1998) N Arkani-Hamed, S Dimopoulos and G Dvali, Phys. Rev. D59, 086004 (1999) G Dvali, G Gabadadze and M Porrati, Phys. Lett. B485, 208 (2000)
L Randall and R Sundrum, Phys. Rev. Lett. 83, 3370 (1999) L Randall and R Sundrum, Phys. Rev. Lett. 83, 4690 (1999)
P S Wesson and J Ponce de Leon, J. Math. Phys. 33, 3883 (1992) P S Wesson et al, Int. J. Mod. Phys. A11, 3247 (1996) J M Overduin and P S Wesson, Phys. Rep. 283, 303 (1997) P S Wesson, Space–time–matter (World Scientific, Singapore, 1999)
N Deruelle and J Katz, Phys. Rev. D64, 083515 (2001)
J P Londal, Embedding spherically symmetric spacetimes in higher dimensions, M.Sc. thesis (University of KwaZulu-Natal, South Africa, 2005) G Amery, J Moodley and J P Londal, Isometric embeddings in cosmology and astrophysics (2011), Proceedings of the South African Mathematical Society 53rd Annual Congress, Pramana – J. Phys. 77(3), 415 (2011)
J Ponce de Leon, Gen. Relativ. Gravit. 20, 539 (1988) P S Wesson, Astrophys. J. 394, 19 (1992) P S Wesson, Astrophys. J. 436, 547 (1994)
J Moodley and G Amery, Gravitational field of a four-dimensional global monopole embedded in a five-dimensional vacuum (2011), in preparation
G T Horowitz and A Strominger, Nucl. Phys. B360, 197 (1991) R Gregory and R Laflamme, Phys. Rev. Lett. 70, 2837 (1993) R Gregory and R Laflamme, Nucl. Phys. B428, 399 (1994)
T Wiseman, Phys. Rev. D65, 124007 (2002)
H F Goenner, in: General relativity and gravitation: 100 years after the birth of Albert Einstein edited by A Held (Plenum Press, New York, 1980)
F Dahia and C Romero, Class. Quant. Grav. 22, 5005 (2005) F Dahia and C Romero, Braz. J. Phys. 35, 1140 (2005)
H Stephani, Relativity: An introduction to special and general relativity, 3rd edn (Cambridge University Press, Cambridge, 2004)
M Okelola, K S Govinder, J Moodley and G Amery (2011), in preparation
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
MOODLEY, J., AMERY, G. An investigation of embeddings for spherically symmetric spacetimes into Einstein manifolds. Pramana - J Phys 77, 533–543 (2011). https://doi.org/10.1007/s12043-011-0173-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12043-011-0173-5