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| {{Merge|Causal structure|date=July 2010}}
| | I'm Veda and I live in a seaside city in northern United Kingdom, West Stoke. I'm 34 and I'm will soon finish my study at Occupational Therapy.<br><br>Here is my website; [http://plainjoephotoblog.com/uggs.asp?p=7 Ugg Kids Bailey Bow Red Boots on Sale] |
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| {{About|the classification of Lorentzian manifolds according to the types of causal structures they admit|a basic treatment of the possible causal relationships among points in a Lorentzian manifold, including the definitions of terms used in this article|Causal structure}}
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| In the study of [[Lorentzian manifold]] [[spacetime]]s there exists a hierarchy of '''causality conditions''' which are important in proving mathematical theorems about the global structure of such manifolds. These conditions were collected during the late 1970s.<ref name="CausalLadder">E. Minguzzi and M. Sanchez, ''The causal hierarchy of spacetimes'' in H. Baum and D. Alekseevsky (eds.), vol. Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys., (Eur. Math. Soc. Publ. House, Zurich, 2008), pp. 299–358, ISBN=978-3-03719-051-7, [[arXiv:gr-qc/0609119]]</ref>
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| The weaker the causality condition on a spacetime, the more ''unphysical'' the spacetime is. Spacetimes with [[closed timelike curve]]s, for example, present severe interpretational difficulties. See the [[grandfather paradox]].
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| It is reasonable to believe that any physical spacetime will satisfy the strongest causality condition: [[global hyperbolicity]]. For such spacetimes the equations in [[general relativity]] can be posed as an [[initial value problem]] on a [[Cauchy surface]].
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| == The hierarchy ==
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| There is a hierarchy of causality conditions, each one of which is strictly stronger than the previous. This is sometimes called the '''causal ladder'''. The conditions, from weakest to strongest, are:
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| * Non-totally vicious
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| * Chronological
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| * Causal
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| * Distinguishing
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| * Strongly causal
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| * Stably causal
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| * Causally continuous
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| * Causally simple
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| * Globally hyperbolic
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| Given are the definitions of these causality conditions for a [[Lorentzian manifold]] <math>(M,g)</math>. Where two or more are given they are equivalent.
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| '''Notation''':
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| * <math>p \ll q</math> denotes the [[chronological relation]].
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| * <math>p \prec q</math> denotes the [[causal relation]].
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| (See [[Causal structure#Causal relations|causal structure]] for definitions of <math>\,I^+(x)</math>, <math>\,I^-(x)</math> and <math>\,J^+(x)</math>, <math>\,J^-(x)</math>.)
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| == Non-totally vicious ==
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| * For some points <math>p \in M</math> we have <math>p \not\ll p</math>.
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| == Chronological ==
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| * There are no closed chronological (timelike) curves.
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| * The [[chronological relation]] is [[irreflexive]]: <math>p \not\ll p</math> for all <math> p \in M </math>.
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| == Causal ==
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| * There are no closed causal (non-spacelike) curves.
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| * If both <math>p \prec q</math> and <math>q \prec p</math> then <math>p = q</math>
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| == Distinguishing ==
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| === Past-distinguishing ===
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| * Two points <math>p, q \in M</math> which share the same chronological past are the same point:
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| :: <math>I^-(p) = I^-(q) \implies p = q </math>
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| * For any neighborhood <math>U</math> of <math>p \in M</math> there exists a neighborhood <math>V \subset U, p \in V</math> such that no past-directed non-spacelike curve from <math>p</math> intersects <math>V</math> more than once.
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| === Future-distinguishing ===
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| * Two points <math>p, q \in M</math> which share the same chronological future are the same point:
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| <math>I^+(p) = I^+(q) \implies p = q </math> | |
| * For any neighborhood <math>U</math> of <math>p \in M</math> there exists a neighborhood <math>V \subset U, p \in V</math> such that no future-directed non-spacelike curve from <math>p</math> intersects <math>V</math> more than once.
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| == Strongly causal ==
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| * For any <math>p \in M</math> there exists a neighborhood <math>U</math> of <math>p</math> such that there exists no timelike curve that passes through <math>U</math> more than once.
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| * For any neighborhood <math>U</math> of <math>p \in M</math> there exists a neighborhood <math>V \subset U, p \in V</math> such that <math>V</math> is causally convex in <math>M</math> (and thus in <math>U</math>).
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| * The [[Alexandrov topology]] agrees with the manifold topology.
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| == Stably causal ==
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| A manifold satisfying any of the weaker causality conditions defined above may fail to do so if the metric is given a small [[perturbation theory|perturbation]]. A spacetime is stably causal if it cannot be made to contain closed [[causal curve]]s by arbitrarily small perturbations of the metric. [[Stephen Hawking]] showed<ref name="StablyCausal">S.W. Hawking, [http://links.jstor.org/sici?sici=0080-4630%2819690114%29308%3A1494%3C433%3ATEOCTF%3E2.0.CO%3B2-U ''The existence of cosmic time functions''] Proc. R. Soc. Lond. (1969), '''A308''', 433</ref> that this is equivalent to:
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| * There exists a ''global time function'' on <math>M</math>. This is a [[scalar (physics)|scalar]] field <math>t</math> on <math>M</math> whose [[Gradient#The gradient on Riemannian manifolds|gradient]] <math>\nabla^a t</math> is everywhere timelike and future-directed. This ''global time function'' gives us a stable way to distinguish between future and past for each point of the spacetime (and so we have no causal violations).
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| == Globally hyperbolic ==
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| * <math>\,M</math> is [[Causality_conditions#Strongly_causal|strongly causal]] and every set <math>J^+(x) \cap J^-(y)</math> (for points <math>x,y \in M</math>) is [[Compact space|compact]].
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| [[Robert Geroch]] showed<ref name="GloballyHyperbolic">R. Geroch, [http://link.aip.org/link/?JMAPAQ/11/437/1 ''Domain of Dependence''] J. Math. Phys. (1970) '''11''', 437–449</ref> that a spacetime is globally hyperbolic [[if and only if]] there exists a [[Cauchy surface]] for <math>M</math>. This means that:
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| * <math>M</math> is topologically equivalent to <math>\mathbb{R} \times\!\, S</math> for some [[Cauchy surface]] <math>S</math> (Here <math>\mathbb{R}</math> denotes the [[real line]]).
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| == See also ==
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| * [[Spacetime]]
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| * [[Lorentzian manifold]]
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| * [[Causal structure]]
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| * [[Globally hyperbolic manifold]]
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| * [[Closed timelike curve]]
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| == References ==
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| <references/>
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| *{{cite book | author=[[Stephen Hawking|S.W. Hawking]], [[George Francis Rayner Ellis|G.F.R. Ellis]], | title=[[The Large Scale Structure of Space-Time]] | location=Cambridge | publisher=Cambridge University Press | year=1973 | id=ISBN 0-521-20016-4}}
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| *{{cite book | author = [[Stephen Hawking|S.W. Hawking]], [[Werner Israel|W. Israel]], | title = [[General Relativity, an Einstein Centenary Survey]]| publisher = Cambridge University Press | year =1979 | id=ISBN 0-521-22285-0}}
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| [[Category:Lorentzian manifolds]]
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| [[Category:Theory of relativity]]
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| [[Category:General relativity]]
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| [[Category:Theoretical physics]]
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I'm Veda and I live in a seaside city in northern United Kingdom, West Stoke. I'm 34 and I'm will soon finish my study at Occupational Therapy.
Here is my website; Ugg Kids Bailey Bow Red Boots on Sale