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Revista mexicana de física
versión impresa ISSN 0035-001X
Rev. mex. fis. vol.53 supl.2 México feb. 2007
From conformal Killing vector fields to boostrotational symmetry
J. EstevezDelgado* and T. Zannias**
* Facultad de Ciencias FísicoMatemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mich, MÉXICO, email: joaquin@fismat.umich.mx
** Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Apartado Postal 82, 58040 Morelia, Mich, MÉXICO, email: zannias@ginette.ifm.umich.mx
Recibido el 18 de julio de 2005
Aceptado el 14 de marzo de 2005
Abstract
We discuss a connection between threedimensional Riemannian manifolds (Σ,) admitting a special conformal Killing vector field ξ and static vacuum or nonvacuum spacetimes. Any such (Σ,) generates a vacuum spacetime (M,g) but it also generates a spacetime (M, g, Φ), where (g, Φ) satisfies the EinsteinKleinGordon massless minimally coupled gravity equations, or the EinsteinConformal scalar field equations. The resulting spacetimes either admit four Killing vector fields or possess boost and rotational symmetry. We argue that this connection goes beyond the vacuum or Einsteinscalar field system and it should be viewed as a mechanism of generating solutions for the Einstein equations, admitting a hypersurface orthogonal Killing vector field.
Keywords: General relativity; conformal Killing vector field; Einstein equations.
Resumen
Se discute la conexión entre variedades Riemannianas (Σ,) de dimension tres que admiten un campo vectorial de Killing conforme ξ y espacios tiempo estáticos asociados a sistemas en el vacío o novacío. Cualquiera de estas variedades (Σ,) generan un espaciotiempo (M, g) e igual generan un espaciotiempo (M, g,Φ), donde (g, Φ) satisfacen las ecuaciones para el campo escalar asociadas a los sistemas de EinsteinKleinGordon con acoplamiento mínimo o conforme. Los espaciostiempo asociados resultantes admiten cuatro campos vectoriales de Killing o una simetría de "boost" y rotacional. Se argumenta como esta conexión va mas allá de los sistemas en el vacío o de los sistemas de campos escalares y esto puede ser visto como un mecanismo para generar soluciones de las ecuaciones de Einstein, que admitan un campo vectorial de Killing ortogonal a una hipersuperficie.
Descriptores: Relatividad general; campo vectorial de Killing conforme; ecuaciones de Einstein.
PACS: 04.20.jb; 04.20.q
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Acknowledgments
The present work was sparked after Prof. Alberto Garcia pointed out to us the relevance of Ref. 13 to a generalized family of Cmetrics constructed in Ref. 3. Our thanks to him and also to U.Nucamendi for discussions regarding the issues raised in this work. The research of TZ was partially supported by grant of Coordinación Científica UMSNH while JED would like to acknowledge financial support through PROMEP via a grant: PTC74.
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