Weak-field approximation: Difference between revisions
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#REDIRECT [[Linearized gravity]] |
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{{expert}} |
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{{Mergeto|Linearized gravity|date=October 2006}} |
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The '''weak-field approximation''' in [[general relativity]] is used to describe the gravitational field very far from the source of [[gravity]]. |
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In this approximation, we assume the metric for [[spacetime]] (<math>g \ </math>) be written in coordinates as: |
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:<math>g_{ab}=\eta_{ab} +\epsilon \gamma_{ab} \ </math> |
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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. |
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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>): |
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:<math> \Gamma ^a {}_{00}=-\frac{\epsilon}{2}g^{ad} \gamma_{00,d} \ </math> |
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from which follows: |
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:<math>\Gamma ^0 {}_{00}=0 \ </math> |
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:<math>\Gamma ^i {}_{00}=-\frac{\epsilon}{2}\gamma_{00,i} \ </math> (<math>i=1, 2, 3 </math>) |
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The [[geodesic equation]] becomes |
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:<math>\frac {d^2 x^i}{dt^2} =-\Gamma^i {}_{00} = \frac{\epsilon }{2} \gamma_{00,i} =-\nabla \Phi \ </math> |
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where <math>\Phi \ </math> is the Newtonian gravitational potential and <math>c \ </math> is the [[speed of light]]. Thus: |
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:<math>\Phi=-\frac{\epsilon}{2}\gamma_{00} \ </math> |
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As we know that |
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:<math>\Phi=-\frac{Gm}{r} \ </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: |
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:<math>g_{00} = -c^2 + \frac{2Gm}{r} \ </math> |
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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]]. |
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==See also== |
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*[[Linearised Einstein field equations]] |
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*[[Gravitoelectromagnetism]] |
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==References== |
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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}} |
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* {{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}} |
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[[Category:Mathematical methods in general relativity]] |
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[[fr:Approximation des champs faibles]] |
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{{relativity-stub}} |
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Latest revision as of 21:49, 5 December 2006
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