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| | The author is known as Irwin. Bookkeeping is my occupation. California is exactly where I've always been living and I love every working day residing here. To perform baseball is the pastime he will never quit doing.<br><br>Here is my web site ... [http://www.reachoutsociety.info/blog/435 std home test] |
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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.
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| == Definitions ==
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| There are several equivalent definitions of global hyperbolicity. Let ''M'' be a smooth connected Lorentzian manifold without boundary. We make the following preliminary definitions:
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| * ''M'' is ''causal'' if it has no closed causal curves.
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| * Given any point ''p'' in ''M'', <math>J^+(p)</math> [resp. <math>J^-(p)</math>] is the collection of points which can be reached by a future-directed [resp. past-directed] continuous causal curve starting from ''p''.
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| * 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''.
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| * A subset ''S'' of ''M'' is ''achronal'' if no timelike curve intersects ''S'' more than once.
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| * A ''Cauchy surface'' for ''M'' is a closed achronal set whose domain of dependence is ''M''.
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| The following conditions are equivalent:
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| * The spacetime is causal, and for every pair of points ''p'' and ''q'' in ''M'', the space <math>J^-(p)\cap J^+(q)</math> is compact.
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| * 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.
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| * The spacetime has a [[Cauchy surface]].
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| 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.
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| == Remarks ==
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| In older literature, the condition of causality in the first two definitions of global hyperbolicity given above is replaced by the stronger condition of ''strong causality''. To be precise, a spacetime ''M'' is strongly causal if for any point ''p'' in ''M'' and any neighborhood ''U'' of ''p'', there is a neighborhood ''V'' of ''p'' contained in ''U'' such that any causal curve with endpoints in ''V'' is contained in ''U''. In 2007, Bernal and Sánchez<ref name="bernal_sanchez1">Antonio N. Bernal and Miguel Sánchez, "Globally hyperbolic spacetimes can be defined as 'causal' instead of 'strongly causal'", ''[[Classical and Quantum Gravity]]'' '''24''' (2007), no. 3, 745–749 [http://arxiv.org/abs/gr-qc/0611138]</ref> showed that the condition of strong causality can be replaced by causality. In particular, any globally hyperbolic manifold as defined in the previous section is strongly causal.
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| In 2003, Bernal and Sánchez<ref name="bernal_sanchez2">Antonio N. Bernal and Miguel Sánchez, " On smooth Cauchy hypersurfaces and Geroch's splitting theorem", ''[[Communications in Mathematical Physics]]'' '''243''' (2003), no. 3, 461–470 [http://arxiv.org/abs/gr-qc/0306108]</ref> showed that any globally hyperbolic manifold ''M'' has a smooth embedded three-dimensional Cauchy surface, and furthermore that any two Cauchy surfaces for ''M'' are diffeomorphic. In particular, ''M'' is diffeomorphic to the product of a Cauchy surface with <math>\mathbb{R}</math>. It was previously well-known that any Cauchy surface of a globally hyperbolic manifold is an embedded three-dimensional <math>C^0</math> submanifold, any two of which are homeomorphic, and such that the manifold splits topologically as the product of the Cauchy surface and <math>\mathbb{R}</math>. In particular, a globally hyperbolic manifold is foliated by Cauchy surfaces.
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| Global hyperbolicity, in the second form given above, was introduced by Leray<ref name="leray">Jean Leray, "Hyperbolic Differential Equations." Mimeographed notes, Princeton, 1952.</ref> in order to consider well-posedness of the Cauchy problem for the wave equation on the manifold. In 1970 Geroch<ref name="geroch">Robert P. Geroch, "Domain of dependence", ''[[Journal of Mathematical Physics]]'' '''11''', (1970) 437, 13pp</ref> proved the equivalence of the second and third definitions above. The first definition and its equivalence to the other two was given by Hawking and Ellis.<ref name="hawkingellis">Stephen Hawking and George Ellis, "The Large Scale Structure of Space-Time". Cambridge: Cambridge University Press (1973).</ref>
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| In view of the [[Initial value formulation (general relativity)|initial value formulation]] for Einstein's equations, global hyperbolicity is seen to be a very natural condition in the context of general relativity, in the sense that given arbitrary initial data, there is a unique maximal globally hyperbolic solution of Einstein's equations.
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| == See also ==
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| * [[Causality conditions]]
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| * [[Causal structure]]
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| * [[Light cone]]
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| == References ==
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| <references/>
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| * {{cite book | author=Hawking, Stephen; and Ellis, G. F. R. | title = The Large Scale Structure of Space-Time | location= Cambridge | publisher=Cambridge University Press | year=1973 |isbn = 0-521-09906-4}}
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| * {{cite book | author=Wald, Robert M.| title = General Relativity | location= Chicago | publisher=The University of Chicago Press | year=1984 |isbn = 0-226-87033-2}}
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| {{DEFAULTSORT:Globally Hyperbolic}}
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| [[Category:General relativity]]
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| [[Category:Mathematical methods in general relativity]]
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The author is known as Irwin. Bookkeeping is my occupation. California is exactly where I've always been living and I love every working day residing here. To perform baseball is the pastime he will never quit doing.
Here is my web site ... std home test