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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 (M,g) evolved through a conformal Killing vector (CKV) field ξ such that: (a) the normal component of ξ is constant on each space-like slice Σ and each Σ 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 ξ. We prove that: (i) ξ is homothetic and orthogonal to Σ; (ii) Σ 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 ξ is a null Killing vector field and Σ degenerates to a null hypersurface.
53C50Lorentz manifolds, manifolds with indefinite metrics
53C80Applications of global differential geometry to physics
83C15Closed form solutions of equations in general relativity
83C40Gravitational energy and conservation laws; groups of motions