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Conformal evolution of spacetime solutions of Einstein’s equations. (English) Zbl 1168.53035
Authors’ abstract: In this paper we study a spacially compact space-time $\left(M,g\right)$ 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.
##### MSC:
 53C50 Lorentz manifolds, manifolds with indefinite metrics 53C80 Applications of global differential geometry to physics 83C15 Closed form solutions of equations in general relativity 83C40 Gravitational energy and conservation laws; groups of motions
##### Keywords:
space-time; Killing vector; mean curvature