Weak-field approximation

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 ($g\$) be written in coordinates as:

$g_{ab}=\eta _{ab}+\epsilon \gamma _{ab}\$

where $\eta _{ab}\$ is the Minkowski metric, $\gamma \$ is the deviation from the Minkowski metric and $\epsilon \$ is taken to be a non-zero real constant.

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

$\Gamma ^{a}{}_{44}=-{\frac {\epsilon }{2}}g^{ad}\gamma _{44,d}\$

from which follows:

$\Gamma ^{4}{}_{44}=0\$

$\Gamma ^{i}{}_{44}=-{\frac {\epsilon }{2}}\gamma _{44,i}\$ ($i=1,2,3$)

The geodesic equation becomes

${\frac {d^{2}x^{i}}{dt^{2}}}=-\Gamma ^{i}{}_{44}={\frac {\epsilon }{2}}\gamma _{44,i}=-\nabla \Phi \$

where $\Phi \$ is the Newtonian gravitational potential and $c\$ is the speed of light. Thus:

$\Phi =-{\frac {\epsilon }{2}}\gamma _{44}\$

As we know that

$\Phi =-{\frac {Gm}{r}}\$

where $G\$ is the gravitational constant, $m\$ is the mass of the gravitating body and $r\$ is the radial distance from the centre of this body, we find that:

$g_{44}=-c^{2}+{\frac {2Gm}{r}}\$

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 gravity
• Linearised Einstein field equations

References

• . ISBN 0-521-37941-5. {{cite book}}: Missing or empty |title= (help); Unknown parameter |Author= ignored (|author= suggested) (help); Unknown parameter |Publisher= ignored (|publisher= suggested) (help); Unknown parameter |Title= ignored (|title= suggested) (help); Unknown parameter |Year= ignored (|year= suggested) (help)CS1 maint: extra punctuation (link)

• . ISBN 0-070-00423-4. {{cite book}}: Missing or empty |title= (help); Unknown parameter |Author= ignored (|author= suggested) (help); Unknown parameter |Publisher= ignored (|publisher= suggested) (help); Unknown parameter |Title= ignored (|title= suggested) (help); Unknown parameter |Year= ignored (|year= suggested) (help)