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Zbl 1168.53035
Duggal, K.L.; Sharma, R.
Conformal evolution of spacetime solutions of Einstein's equations.
(English)
[J] Commun. Appl. Anal. 11, No. 1, 17-25 (2007). ISSN 1083-2564

Authors' abstract: In this paper we study a spacially compact space-time $(M, g)$ evolved through a conformal Killing vector (CKV) field $\xi$ such that: (a) the normal component of $\xi$ is constant on each space-like slice $\Sigma$ and each $\Sigma$ has constant mean curvature; (b) the stress energy tensor obeys the mixed energy condition; (c) the conformal scalar function is non-decreasing along the evolution CKV field $\xi$. We prove that: (i) $\xi$ is homothetic and orthogonal to $\Sigma$; (ii) $\Sigma$ is hyperbolic and totally umbilical in $M$; and (iii) $M$ is a vacuum space-time. We also discuss a physically important case of Killing horizon when $\xi$ is a null Killing vector field and $\Sigma$ degenerates to a null hypersurface.
[Constantin Udrişte (Bucureşti)]
MSC 2000:
*53C50 Lorentz manifolds, manifolds with indefinite metrics
53C80 Appl. of global differential geometry to physics
83C15 Closed form solutions of equations in general relativity
83C40 Groups of motions, etc.

Keywords: space-time; Killing vector; mean curvature

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