Weak-field approximation - Revision history

Radagast83: completed merge

2006-12-05T21:49:20Z

completed merge

← Previous revision		Revision as of 21:49, 5 December 2006
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			#REDIRECT [[Linearized gravity]]
	{{expert}}
	{{Mergeto\|Linearized gravity\|date=October 2006}}

	The '''weak-field approximation''' in [[general relativity]] is used to describe the gravitational field very far from the source of [[gravity]].

	In this approximation, we assume the metric for [[spacetime]] (<math>g \ </math>) be written in coordinates as:

	:<math>g_{ab}=\eta_{ab} +\epsilon \gamma_{ab} \ </math>

	where <math>\eta_{ab} \ </math> is the [[Minkowski metric]], <math>\gamma \ </math> is the deviation from the Minkowski metric and <math>\epsilon \ </math> is taken to be a non-zero real constant.

	A relation between the Newtonian gravitational potential <math>\Phi \ </math> and the deviation term above can be obtained by calculating the [[Christoffel symbols]] <math> \Gamma ^a {}_{44} \ </math> (upon ignoring terms of order higher than <math>\epsilon \ </math>):

	:<math> \Gamma ^a {}_{00}=-\frac{\epsilon}{2}g^{ad} \gamma_{00,d} \ </math>

	from which follows:

	:<math>\Gamma ^0 {}_{00}=0 \ </math>

	:<math>\Gamma ^i {}_{00}=-\frac{\epsilon}{2}\gamma_{00,i} \ </math> (<math>i=1, 2, 3 </math>)

	The [[geodesic equation]] becomes

	:<math>\frac {d^2 x^i}{dt^2} =-\Gamma^i {}_{00} = \frac{\epsilon }{2} \gamma_{00,i} =-\nabla \Phi \ </math>

	where <math>\Phi \ </math> is the Newtonian gravitational potential and <math>c \ </math> is the [[speed of light]]. Thus:

	:<math>\Phi=-\frac{\epsilon}{2}\gamma_{00} \ </math>

	As we know that

	:<math>\Phi=-\frac{Gm}{r} \ </math>

	where <math>G \ </math> is the [[gravitational constant]], <math>m \ </math> is the mass of the gravitating body and <math>r \ </math> is the radial distance from the centre of this body, we find that:

	:<math>g_{00} = -c^2 + \frac{2Gm}{r} \ </math>

	The weak-field approximation is useful in finding the values of certain constants, for example in the [[Einstein field equations]] and in the [[Schwarzschild metric]].

	==See also==

	*[[Linearised Einstein field equations]]
	*[[Gravitoelectromagnetism]]

	==References==

	* {{cite book \| author=Stephani, Hans \| title=General Relativity: An Introduction to the Theory of the Gravitational Field, \| location=Cambridge \| publisher=Cambridge University Press \| year=1990 \| id=ISBN 0-521-37941-5}}

	* {{cite book \| author=Adler, Ronald; Bazin, Maurice' & Schiffer, Menahem \| title=Introduction to General Relativity \| location=New York \| publisher=McGraw-Hill \| year=1965 \| id=ISBN 0-07-000423-4}}

	[[Category:Mathematical methods in general relativity]]

	[[fr:Approximation des champs faibles]]


	{{relativity-stub}}

SmackBot: ISBN formatting/gen fixes using AWB

2006-11-18T15:28:03Z

ISBN formatting/gen fixes using AWB

← Previous revision		Revision as of 15:28, 18 November 2006
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	{{Mergeto\|Linearized gravity\|date=October 2006}}		{{Mergeto\|Linearized gravity\|date=October 2006}}

	The [[weak-field approximation]] in [[general relativity]] is used to describe the gravitational field very far from the source of [[gravity]].		The '''weak-field approximation''' in [[general relativity]] is used to describe the gravitational field very far from the source of [[gravity]].

	In this approximation, we assume the metric for [[spacetime]] (<math>g \ </math>) be written in coordinates as:		In this approximation, we assume the metric for [[spacetime]] (<math>g \ </math>) be written in coordinates as:
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	* {{cite book \| author=Stephani, Hans \| title=General Relativity: An Introduction to the Theory of the Gravitational Field, \| location=Cambridge \| publisher=Cambridge University Press \| year=1990 \| id=ISBN 0-521-37941-5}}		* {{cite book \| author=Stephani, Hans \| title=General Relativity: An Introduction to the Theory of the Gravitational Field, \| location=Cambridge \| publisher=Cambridge University Press \| year=1990 \| id=ISBN 0-521-37941-5}}

	* {{cite book \| author=Adler, Ronald; Bazin, Maurice' & Schiffer, Menahem \| title=Introduction to General Relativity \| location=New York \| publisher=McGraw-Hill \| year=1965 \| id=ISBN 0-~~070~~-~~00423~~-4}}		* {{cite book \| author=Adler, Ronald; Bazin, Maurice' & Schiffer, Menahem \| title=Introduction to General Relativity \| location=New York \| publisher=McGraw-Hill \| year=1965 \| id=ISBN 0-07-000423-4}}

⚫	{{relativity-stub}}

	[[Category:Mathematical methods in general relativity]]		[[Category:Mathematical methods in general relativity]]

	[[fr:Approximation des champs faibles]]		[[fr:Approximation des champs faibles]]


		⚫	{{relativity-stub}}

Alphachimpbot: BOT - updating merge tag

2006-10-07T13:26:20Z

BOT - updating merge tag

← Previous revision		Revision as of 13:26, 7 October 2006
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	{{expert}}		{{expert}}
	{{~~mergeto~~\|Linearized gravity}}		{{Mergeto\|Linearized gravity\|date=October 2006}}

	The [[weak-field approximation]] in [[general relativity]] is used to describe the gravitational field very far from the source of [[gravity]].		The [[weak-field approximation]] in [[general relativity]] is used to describe the gravitational field very far from the source of [[gravity]].

Mpatel: Revert to revision 51494079 dated 2006-05-04 07:35:43 by Ligulem using popups

2006-08-28T17:15:01Z

Revert to revision 51494079 dated 2006-05-04 07:35:43 by Ligulem using popups

190.49.8.35 at 03:34, 28 August 2006

2006-08-28T03:34:40Z

Ligulem: extracting location on {{cite book}}

2006-05-04T07:35:43Z

extracting location on {{cite book}}

← Previous revision		Revision as of 07:35, 4 May 2006
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	==References==		==References==

	* {{cite book \| author=Stephani, Hans \| title=General Relativity: An Introduction to the Theory of the Gravitational Field, \| ~~publisher~~=Cambridge: Cambridge University Press \| year=1990 \| id=ISBN 0-521-37941-5}}		* {{cite book \| author=Stephani, Hans \| title=General Relativity: An Introduction to the Theory of the Gravitational Field, \| location=Cambridge \| publisher=Cambridge University Press \| year=1990 \| id=ISBN 0-521-37941-5}}

	* {{cite book \| author=Adler, Ronald; Bazin, Maurice' & Schiffer, Menahem \| title=Introduction to General Relativity \| ~~publisher~~=New York: McGraw-Hill \| year=1965 \| id=ISBN 0-070-00423-4}}		* {{cite book \| author=Adler, Ronald; Bazin, Maurice' & Schiffer, Menahem \| title=Introduction to General Relativity \| location=New York \| publisher=McGraw-Hill \| year=1965 \| id=ISBN 0-070-00423-4}}

	{{relativity-stub}}		{{relativity-stub}}

That Guy, From That Show!: migrate {{book reference}} to {{cite book}} using AWB

2006-02-26T08:00:48Z

migrate {{book reference}} to {{cite book}} using AWB

← Previous revision		Revision as of 08:00, 26 February 2006
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	==References==		==References==

	* {{~~Book~~ ~~reference~~ \| ~~Author~~=Stephani, Hans \| ~~Title~~=General Relativity: An Introduction to the Theory of the Gravitational Field, \| ~~Publisher~~=Cambridge: Cambridge University Press \| ~~Year~~=1990 \| ID=ISBN 0-521-37941-5}}		* {{cite book \| author=Stephani, Hans \| title=General Relativity: An Introduction to the Theory of the Gravitational Field, \| publisher=Cambridge: Cambridge University Press \| year=1990 \| id=ISBN 0-521-37941-5}}

	* {{~~Book~~ ~~reference~~ \| ~~Author~~=Adler, Ronald; Bazin, Maurice' & Schiffer, Menahem \| ~~Title~~=Introduction to General Relativity \| ~~Publisher~~=New York: McGraw-Hill \| ~~Year~~=1965 \| ID=ISBN 0-070-00423-4}}		* {{cite book \| author=Adler, Ronald; Bazin, Maurice' & Schiffer, Menahem \| title=Introduction to General Relativity \| publisher=New York: McGraw-Hill \| year=1965 \| id=ISBN 0-070-00423-4}}

	{{relativity-stub}}		{{relativity-stub}}

Zejames at 13:51, 14 February 2006

2006-02-14T13:51:36Z

← Previous revision		Revision as of 13:51, 14 February 2006
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	[[Category:Mathematical methods in general relativity]]		[[Category:Mathematical methods in general relativity]]

			[[fr:Approximation des champs faibles]]

Mpatel: recat

2006-02-06T17:31:35Z

recat

← Previous revision		Revision as of 17:31, 6 February 2006
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	{{relativity-stub}}		{{relativity-stub}}

	[[Category:~~General~~ relativity]]		[[Category:Mathematical methods in general relativity]]

Hillman: /* See also */ +lnk

2006-02-02T05:40:44Z

← Previous revision		Revision as of 17:15, 28 August 2006
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	In this approximation, we assume the metric for [[spacetime]] (<math>g \ </math>) be written in coordinates as:		In this approximation, we assume the metric for [[spacetime]] (<math>g \ </math>) be written in coordinates as:

	:<math>g_{~~\mu \nu~~}=\eta_{~~\mu \nu~~} +\epsilon \gamma_{~~\mu \nu~~} \ </math>		:<math>g_{ab}=\eta_{ab} +\epsilon \gamma_{ab} \ </math>

	where <math>\eta_{~~\mu \nu~~} \ </math> is the [[Minkowski metric]], <math>\gamma \ </math> is the deviation from the Minkowski metric and <math>\epsilon \ </math> is taken to be a non-zero real constant.		where <math>\eta_{ab} \ </math> is the [[Minkowski metric]], <math>\gamma \ </math> is the deviation from the Minkowski metric and <math>\epsilon \ </math> is taken to be a non-zero real constant.

	A relation between the Newtonian gravitational potential <math>\Phi \ </math> and the deviation term above can be obtained by calculating the [[Christoffel symbols]] <math> \Gamma ^a {}_{00} \ </math> (upon ignoring terms of order higher than <math>\epsilon \ </math>):		A relation between the Newtonian gravitational potential <math>\Phi \ </math> and the deviation term above can be obtained by calculating the [[Christoffel symbols]] <math> \Gamma ^a {}_{44} \ </math> (upon ignoring terms of order higher than <math>\epsilon \ </math>):

	:<math> \Gamma ^a {}_{00}=-\frac{\epsilon}{2}g^{ad} \gamma_{00,d} \ </math>		:<math> \Gamma ^a {}_{00}=-\frac{\epsilon}{2}g^{ad} \gamma_{00,d} \ </math>
Line 24:		Line 24:
	:<math>\frac {d^2 x^i}{dt^2} =-\Gamma^i {}_{00} = \frac{\epsilon }{2} \gamma_{00,i} =-\nabla \Phi \ </math>		:<math>\frac {d^2 x^i}{dt^2} =-\Gamma^i {}_{00} = \frac{\epsilon }{2} \gamma_{00,i} =-\nabla \Phi \ </math>

	where <math>\Phi \ </math> is the Newtonian gravitational potential. Thus:		where <math>\Phi \ </math> is the Newtonian gravitational potential and <math>c \ </math> is the [[speed of light]]. Thus:

	:<math>\Phi=-\frac{\epsilon}{2}\gamma_{00} \ </math>		:<math>\Phi=-\frac{\epsilon}{2}\gamma_{00} \ </math>
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	where <math>G \ </math> is the [[gravitational constant]], <math>m \ </math> is the mass of the gravitating body and <math>r \ </math> is the radial distance from the centre of this body, we find that:		where <math>G \ </math> is the [[gravitational constant]], <math>m \ </math> is the mass of the gravitating body and <math>r \ </math> is the radial distance from the centre of this body, we find that:

	:<math>g_{00} = 1 + \frac{2Gm}{r} \ </math>		:<math>g_{00} = -c^2 + \frac{2Gm}{r} \ </math>

	The weak-field approximation is useful in finding the values of certain constants, for example in the [[Einstein field equations]] and in the [[Schwarzschild metric]].		The weak-field approximation is useful in finding the values of certain constants, for example in the [[Einstein field equations]] and in the [[Schwarzschild metric]].

← Previous revision		Revision as of 03:34, 28 August 2006
Line 6:		Line 6:
	In this approximation, we assume the metric for [[spacetime]] (<math>g \ </math>) be written in coordinates as:		In this approximation, we assume the metric for [[spacetime]] (<math>g \ </math>) be written in coordinates as:

	:<math>g_{ab}=\eta_{ab} +\epsilon \gamma_{ab} \ </math>		:<math>g_{\mu \nu}=\eta_{\mu \nu} +\epsilon \gamma_{\mu \nu} \ </math>

	where <math>\eta_{ab} \ </math> is the [[Minkowski metric]], <math>\gamma \ </math> is the deviation from the Minkowski metric and <math>\epsilon \ </math> is taken to be a non-zero real constant.		where <math>\eta_{\mu \nu} \ </math> is the [[Minkowski metric]], <math>\gamma \ </math> is the deviation from the Minkowski metric and <math>\epsilon \ </math> is taken to be a non-zero real constant.

	A relation between the Newtonian gravitational potential <math>\Phi \ </math> and the deviation term above can be obtained by calculating the [[Christoffel symbols]] <math> \Gamma ^a {}_{44} \ </math> (upon ignoring terms of order higher than <math>\epsilon \ </math>):		A relation between the Newtonian gravitational potential <math>\Phi \ </math> and the deviation term above can be obtained by calculating the [[Christoffel symbols]] <math> \Gamma ^a {}_{00} \ </math> (upon ignoring terms of order higher than <math>\epsilon \ </math>):

	:<math> \Gamma ^a {}_{00}=-\frac{\epsilon}{2}g^{ad} \gamma_{00,d} \ </math>		:<math> \Gamma ^a {}_{00}=-\frac{\epsilon}{2}g^{ad} \gamma_{00,d} \ </math>
Line 24:		Line 24:
	:<math>\frac {d^2 x^i}{dt^2} =-\Gamma^i {}_{00} = \frac{\epsilon }{2} \gamma_{00,i} =-\nabla \Phi \ </math>		:<math>\frac {d^2 x^i}{dt^2} =-\Gamma^i {}_{00} = \frac{\epsilon }{2} \gamma_{00,i} =-\nabla \Phi \ </math>

	where <math>\Phi \ </math> is the Newtonian gravitational potential ~~and <math>c \ </math> is the [[speed of light]]~~. Thus:		where <math>\Phi \ </math> is the Newtonian gravitational potential. Thus:

	:<math>\Phi=-\frac{\epsilon}{2}\gamma_{00} \ </math>		:<math>\Phi=-\frac{\epsilon}{2}\gamma_{00} \ </math>
Line 34:		Line 34:
	where <math>G \ </math> is the [[gravitational constant]], <math>m \ </math> is the mass of the gravitating body and <math>r \ </math> is the radial distance from the centre of this body, we find that:		where <math>G \ </math> is the [[gravitational constant]], <math>m \ </math> is the mass of the gravitating body and <math>r \ </math> is the radial distance from the centre of this body, we find that:

	:<math>g_{00} = ~~-c^2~~ + \frac{2Gm}{r} \ </math>		:<math>g_{00} = 1 + \frac{2Gm}{r} \ </math>

	The weak-field approximation is useful in finding the values of certain constants, for example in the [[Einstein field equations]] and in the [[Schwarzschild metric]].		The weak-field approximation is useful in finding the values of certain constants, for example in the [[Einstein field equations]] and in the [[Schwarzschild metric]].

← Previous revision		Revision as of 05:40, 2 February 2006
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	*[[Linearised Einstein field equations]]		*[[Linearised Einstein field equations]]
			*[[Gravitoelectromagnetism]]

	==References==		==References==