Weak-field approximation: Difference between revisions
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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: |
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:<math>g_{ |
:<math>g_{ab}=\eta_{ab} +\epsilon \gamma_{ab} \ </math> |
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where <math>\eta_{ |
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 {}_{ |
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> |
:<math> \Gamma ^a {}_{00}=-\frac{\epsilon}{2}g^{ad} \gamma_{00,d} \ </math> |
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:<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> |
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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: |
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:<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: |
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:<math>g_{00} = |
:<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]]. |
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]]. |
Revision as of 17:15, 28 August 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.
In this approximation, we assume the metric for spacetime () be written in coordinates as:
where is the Minkowski metric, is the deviation from the Minkowski metric and is taken to be a non-zero real constant.
A relation between the Newtonian gravitational potential and the deviation term above can be obtained by calculating the Christoffel symbols (upon ignoring terms of order higher than ):
from which follows:
- ()
The geodesic equation becomes
where is the Newtonian gravitational potential and is the speed of light. Thus:
As we know that
where is the gravitational constant, is the mass of the gravitating body and is the radial distance from the centre of this body, we find that:
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
References
- Stephani, Hans (1990). General Relativity: An Introduction to the Theory of the Gravitational Field,. Cambridge: Cambridge University Press. ISBN 0-521-37941-5.