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| | Hi there. Let me start by introducing the author, her title is Sophia. It's not a common thing but what I like doing is to climb but I don't have the time lately. For years he's been living in Alaska and he doesn't plan on changing it. Invoicing is what I do for a living but I've usually wanted my personal company.<br><br>my web site online psychic reading - [http://www.indosfriends.com/profile-253/info/ http://www.indosfriends.com] - |
| {{Expert-subject|Physics|date=February 2009}}
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| {{Technical|date=August 2009}}
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| In [[general relativity]], '''geodesic deviation''' describes the tendency of objects to approach or recede from one another while moving under the influence of a spatially varying [[gravitational field]]. Put another way, if two objects are set in motion along two initially parallel trajectories, the presence of a [[tidal force|tidal gravitational force]] will cause the trajectories to bend towards or away from each other, producing a relative [[acceleration]] between the objects.<ref name="ohanian">{{cite book|last1=Ohanian|first1=Hans|title=Gravitation and Spacetime|edition=1st|year=1976|pages=271–6}}</ref>
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| Mathematically, the tidal force in general relativity is described by the [[Riemann curvature tensor]],<ref name="ohanian" /> and the trajectory of an object solely under the influence of gravity is called a ''[[geodesic]]''. The ''geodesic deviation equation'' relates the Riemann curvature tensor to the relative acceleration of two neighboring geodesics. In [[differential geometry]], the geodesic deviation equation is more commonly known as the [[Jacobi field|Jacobi equation]].
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| == Geodesic deviation equation ==
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| To quantify geodesic deviation, one begins by setting up a family closely spaced geodesics indexed by a continuous variable ''s'' and parametrized by an [[affine parameter]] ''t''. That is, for each fixed ''s'', the curve swept out by γ<sub>''s''</sub>(''t'') as ''t'' varies is a geodesic with affine parameter. If ''x''<sup>μ</sup>(''s'', ''t'') are the coordinates of the geodesic γ<sub>''s''</sub>(''t''), then the [[tangent vector]] of this geodesic is
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| :<math>T^\mu = \frac{\partial x^\mu(s, t)}{\partial t}.</math>
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| One can also define a ''deviation vector'', which is the displacement of two objects travelling along two infinitesimally separated geodesics:
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| :<math>X^\mu = \frac{\partial x^\mu(s, t)}{\partial s}.</math>
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| The ''relative acceleration'' ''A''<sup>μ</sup> of the two objects is defined, roughly, as the second derivative of the separation vector ''X''<sup>μ</sup> as the objects advance along their respective geodesics. Specifically, ''A''<sup>μ</sup> is found by taking the directional [[covariant derivative]] of ''X'' along ''T'' twice:
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| :<math> A^\mu = T^\alpha \nabla_\alpha (T^\beta \nabla_\beta X^\mu).</math>
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| The geodesic deviation equation relates ''A''<sup>μ</sup>, ''T''<sup>μ</sup>, ''X''<sup>μ</sup>, and the Riemann tensor ''R''<sup>μ</sup><sub>νρσ</sub>:<ref name="carroll">{{cite book|last=Carroll|first=Sean|title=Spacetime and Geometry|year=2004|pages=144–6}}</ref>
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| :<math> A^\mu = {R^\mu}_{\nu\rho\sigma} T^\nu T^\rho X^\sigma.</math>
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| An alternate notation for the directional covariant derivative <math>T^\alpha \nabla_\alpha</math> is <math>D/dt</math>, so the geodesic deviation equation may also be written as
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| :<math>\frac{D^2 X^\mu}{dt^2} = {R^\mu}_{\nu\rho\sigma} T^\nu T^\rho X^\sigma.</math>
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| The geodesic deviation equation can be derived from the [[second variation]] of the point particle [[Lagrangian]] along geodesics, or from the first variation of a combined Lagrangian.{{Clarify|date=September 2009}} The Lagrangian approach has two advantages. First it allows various formal approaches of [[Quantization (physics)|quantization]] to be applied to the geodesic deviation system. Second it allows deviation to be formulated for much more general objects than geodesics (any [[dynamical system]] which has a one [[spacetime]] indexed momentum appears to have a corresponding generalization of geodesic deviation).{{Citation needed|date=September 2009}}
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| ==See also==
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| *[[Bernhard Riemann]]
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| *[[Curvature]]
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| *[[Glossary of Riemannian and metric geometry]]
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| ==References==
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| {{reflist}}
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| *{{Citation|title=General relativity - an introduction to the theory of the gravitation field|first=Hans|last=Stephani|publisher=Cambridge University Press|year=1982|isbn=0-521-37066-3}}.
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| *{{Citation | last1=Wald | first1=Robert M. | author1-link=Robert Wald | title=[[General Relativity (book)|General Relativity]] | isbn=978-0-226-87033-5 | year=1984}}.
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| ==External links==
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| *[http://www.arXiv.org/abs/gr-qc/0404094 General Relativity and Quantum Cosmology]
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| *[http://www.mth.uct.ac.za/omei/gr/chap6/node11.html Tensors and Relativity: Geodesic deviation]
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| {{DEFAULTSORT:Geodesic Deviation Equation}}
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| [[Category:Geodesic (mathematics)]]
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| [[Category:Riemannian geometry]]
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| [[Category:Equations]]
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Hi there. Let me start by introducing the author, her title is Sophia. It's not a common thing but what I like doing is to climb but I don't have the time lately. For years he's been living in Alaska and he doesn't plan on changing it. Invoicing is what I do for a living but I've usually wanted my personal company.
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