Stability of geodesic incompleteness for Robertson-Walker space-times
General Relativity and Gravitation(1981)
摘要
Let (M, g) be a Lorentzian warped product space-timeM=(a, b)×H, g = −dt2 ⊕fh, where −∞⩽ab⩽+∞, (H, h) is a Riemannian manifold andf: (a, b)→(0, ∞) is a smooth function. We show that ifa>−∞ and (H, h) is homogeneous, then the past incompleteness of every timelike geodesic of (M,g) is stable under smallC0 perturbations in the space Lor(M) of Lorentzian metrics forM. Also we show that if (H,h) is isotropic and (M,g) contains a past-inextendible, past-incomplete null geodesic, then the past incompleteness of all null geodesics is stable under smallC1 perturbations in Lor(M). Given either the isotropy or homogeneity of the Riemannian factor, the background space-time (M,g) is globally hyperbolic. The results of this paper, in particular, answer a question raised by D. Lerner for big bang Robertson-Walker cosmological models affirmatively.
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关键词
space time,big bang
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