Hilbert's nineteenth problem - Revision history

en>John of Reading: /* References */Typo fixing, replaced: traslated → translated using AWB

2015-01-04T21:07:02Z

References: Typo fixing, replaced: traslated → translated using AWB

← Older revision		Revision as of 23:07, 4 January 2015
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	~~{{multiple issues\|~~		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}}~~
	~~{{Technical\|date=August 2009}}~~
	}}

	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>~~

	~~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]]~~.

	~~== Geodesic deviation equation ==~~

	~~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~~

	~~:<math>T^\mu = \frac{\partial x^\mu(s, t)}{\partial t}.</math>~~

	~~One can also define a ''deviation vector'', which is the displacement of two objects travelling along two infinitesimally separated geodesics:~~

	~~:<math>X^\mu = \frac{\partial x^\mu(s, t)}{\partial s}.</math>~~

	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:
	~~:<math> A^\mu = T^\alpha \nabla_\alpha (T^\beta \nabla_\beta X^\mu).</math>~~

	~~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>
	~~:<math> A^\mu = {R^\mu}_{\nu\rho\sigma} T^\nu T^\rho X^\sigma~~.<~~/math>~~

	~~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~~
	~~:<math>\frac{D^2 X^\mu}{dt^2} = {R^\mu}_{\nu\rho\sigma} T^\nu T^\rho X^\sigma.</math>~~

	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}}

	~~==See also==~~
	*[[Bernhard Riemann]]
	*[[Curvature]]
	*[[Glossary of Riemannian and metric geometry]]

	~~==References==~~
	~~{{reflist}}~~

	*{{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}}.~~
	*{{Citation \| last1=Wald \| first1=Robert M. \| author1-link=Robert Wald \| title=[[General Relativity (book)\|General Relativity]] \| isbn=978-0-226-87033-5 \| year=1984}}.

	~~==External links==~~
	*[http://www.~~arXiv~~.~~org~~/~~abs~~/~~gr-qc~~/~~0404094 General Relativity and Quantum Cosmology]~~
	*[http://www.~~mth.uct.ac.za/omei/gr/chap6/node11~~.~~html Tensors and Relativity: Geodesic deviation]~~

	~~{{DEFAULTSORT:Geodesic Deviation Equation}}~~
	~~[[Category:Geodesic (mathematics)]]~~
	~~[[Category:Riemannian geometry]]~~
	~~[[Category:Equations]~~]

en>Monkbot: /* References */Fix CS1 deprecated date parameter errors

2014-01-21T01:14:35Z

References: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 03:14, 21 January 2014
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	~~They call me Emilia~~. ~~Years ago we moved~~ to ~~North Dakota~~ and ~~I adore every day living right here~~. ~~He used~~ to ~~be unemployed but now he~~ is a ~~meter reader~~. ~~Body developing~~ is ~~one~~ of the ~~things I love most~~.<br><br>~~Feel free~~ to ~~visit my site~~: [http://www.~~breda~~.nl/~~users~~/~~noeliadfebdftijfsdnt std testing at home~~]		{{multiple issues\|
			{{Expert-subject\|Physics\|date=February 2009}}
			{{Technical\|date=August 2009}}
			}}

			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>

			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]].

			== Geodesic deviation equation ==

			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

			:<math>T^\mu = \frac{\partial x^\mu(s, t)}{\partial t}.</math>

			One can also define a ''deviation vector'', which is the displacement of two objects travelling along two infinitesimally separated geodesics:

			:<math>X^\mu = \frac{\partial x^\mu(s, t)}{\partial s}.</math>

			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:
			:<math> A^\mu = T^\alpha \nabla_\alpha (T^\beta \nabla_\beta X^\mu).</math>

			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>
			:<math> A^\mu = {R^\mu}_{\nu\rho\sigma} T^\nu T^\rho X^\sigma.</math>

			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
			:<math>\frac{D^2 X^\mu}{dt^2} = {R^\mu}_{\nu\rho\sigma} T^\nu T^\rho X^\sigma.</math>

			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}}

			==See also==
			*[[Bernhard Riemann]]
			*[[Curvature]]
			*[[Glossary of Riemannian and metric geometry]]

			==References==
			{{reflist}}

			*{{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}}.
			*{{Citation \| last1=Wald \| first1=Robert M. \| author1-link=Robert Wald \| title=[[General Relativity (book)\|General Relativity]] \| isbn=978-0-226-87033-5 \| year=1984}}.

			==External links==
			*[http://www.arXiv.org/abs/gr-qc/0404094 General Relativity and Quantum Cosmology]
			*[http://www.mth.uct.ac.za/omei/gr/chap6/node11.html Tensors and Relativity: Geodesic deviation]

			{{DEFAULTSORT:Geodesic Deviation Equation}}
			[[Category:Geodesic (mathematics)]]
			[[Category:Riemannian geometry]]
			[[Category:Equations]]

en>Kilom691: ref corrected

2012-02-11T08:46:04Z

ref corrected

New page

They call me Emilia. Years ago we moved to North Dakota and I adore every day living right here. He used to be unemployed but now he is a meter reader. Body developing is one of the things I love most.<br><br>Feel free to visit my site: [http://www.breda.nl/users/noeliadfebdftijfsdnt std testing at home]