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| | Hi there. Let me begin by introducing the author, her name is Sophia Boon but she by no means truly liked that title. To climb is something she would by no means give up. Mississippi is exactly where his house is. Invoicing is what I do.<br><br>Also visit my website tarot readings; [http://mybrandcp.com/xe/board_XmDx25/107997 http://mybrandcp.com/xe/board_XmDx25/107997], |
| '''Post-Newtonian expansions''' in [[general relativity]] are used for finding an approximate solution of the [[Einstein field equations]] for the [[metric tensor (general relativity)|metric tensor]].
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| == Expansion in 1/''c''<sup>2</sup> ==
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| The '''post-Newtonian approximations''' are expansions in a small parameter, which is the ratio of the velocity of matter, forming the gravitational field, to the [[speed of light]], which in this case is better called the [[speed of gravity]].<ref>{{cite journal | author=Kopeikin, S.|authorlink = Sergei Kopeikin|title=The speed of gravity in general relativity and theoretical interpretation of the Jovian deflection experiment|url=http://iopscience.iop.org/0264-9381/21/13/010/ |journal=Classical and Quantum Gravity|year=2004|volume= 21| issue= 13|pages= 3251–3286}}</ref>
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| In the limit, when the fundamental speed of gravity becomes infinite, the post-Newtonian expansion reduces to [[Isaac Newton|Newton]]'s law of gravity.<ref>{{cite book|author=Kopeikin, S., Efroimsky, M., Kaplan, G.|year=2011|title=Relativistic Celestial Mechanics of the Solar System|url=http://www.wiley-vch.de/publish/en/books/ISBN978-3-527-40856-6/authorinformation/?sID=kuqbnspc0lafd8qhvtg2i2bru0|pages=860|publisher=[[Wiley-VCH]]|isbn=978-3-527-40856-6}}</ref>
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| == Expansion in ''h'' ==
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| Another approach is to expand the equations of general relativity in a power series in the deviation of the metric from its [[Minkowski metric|value in the absence of gravity]]
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| :<math>h_{\alpha \beta} = g_{\alpha \beta} - \eta_{\alpha \beta} \,.</math>
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| To this end, one must choose a coordinate system in which the [[eigenvalue]]s of <math>h_{\alpha \beta} \eta^{\beta \gamma} \,</math> all have absolute values less than 1.
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| For example, if one goes one step beyond [[linearized gravity]] to get the expansion to the second order in ''h'':
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| :<math> g^{\mu \nu} \approx \eta^{\mu \nu} - \eta^{\mu \alpha} h_{\alpha \beta} \eta^{\beta \nu} + \eta^{\mu \alpha} h_{\alpha \beta} \eta^{\beta \gamma} h_{\gamma \delta} \eta^{\delta \nu} \,.</math>
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| :<math> \sqrt{- g} \approx 1 + \tfrac12 h_{\alpha \beta} \eta^{\beta \alpha} + \tfrac18 h_{\alpha \beta} \eta^{\beta \alpha} h_{\gamma \delta} \eta^{\delta \gamma} - \tfrac14 h_{\alpha \beta} \eta^{\beta \gamma} h_{\gamma \delta} \eta^{\delta \alpha} \,.</math>
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| == Hybrid expansion ==
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| Sometimes, as with the [[Parameterized post-Newtonian formalism]], a hybrid approach is used in which both the reciprocal of the speed of gravity and masses are assumed to be small.
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| ==See also==
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| *[[Coordinate conditions]]
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| *[[Einstein–Infeld–Hoffmann equations]]
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| *[[Linearized gravity]]
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| *[[Parameterized post-Newtonian formalism]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| *[http://www.math.ca/cjm/v1/p209 "On the Motion of Particles in General Relativity Theory" by A.Einstein and L.Infeld]
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| {{DEFAULTSORT:Post-Newtonian Expansion}}
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| [[Category:General relativity]]
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| {{relativity-stub}}
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Hi there. Let me begin by introducing the author, her name is Sophia Boon but she by no means truly liked that title. To climb is something she would by no means give up. Mississippi is exactly where his house is. Invoicing is what I do.
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