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In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). This is relevant to Einstein's theory of general relativity, and potentially to other metric gravitational theories. == Definitions == There are several equivalent definitions of global hyperbolicity. Let ''M'' be a smooth connected Lorentzian manifold without boundary. We make the following preliminary definitions: * ''M'' is ''causal'' if it has no closed causal curves. * Given any point ''p'' in ''M'', ( ) is the collection of points which can be reached by a future-directed (past-directed ) continuous causal curve starting from ''p''. * Given a subset ''S'' of ''M'', the ''domain of dependence'' of ''S'' is the set of all points ''p'' in ''M'' such that every inextendible causal curve through ''p'' intersects ''S''. * A subset ''S'' of ''M'' is ''achronal'' if no timelike curve intersects ''S'' more than once. * A ''Cauchy surface'' for ''M'' is a closed achronal set whose domain of dependence is ''M''. The following conditions are equivalent: * The spacetime is causal, and for every pair of points ''p'' and ''q'' in ''M'', the space is compact. * The spacetime is causal, and for every pair of points ''p'' and ''q'' in ''M'', the space of continuous future directed causal curves from ''p'' to ''q'' is compact. * The spacetime has a Cauchy surface. If any of these conditions are satisfied, we say ''M'' is ''globally hyperbolic''. If ''M'' is a smooth connected Lorentzian manifold with boundary, we say it is globally hyperbolic if its interior is globally hyperbolic. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Globally hyperbolic manifold」の詳細全文を読む スポンサード リンク
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