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}{}_{00}=-{\frac {\epsilon }{2}}g^{ad}\gamma _{00,d}\$

from which follows:

$\Gamma ^{0}{}_{00}=0\$

$\Gamma ^{i}{}_{00}=-{\frac {\epsilon }{2}}\gamma _{00,i}\$ ($i=1,2,3$)

The geodesic equation becomes

${\frac {d^{2}x^{i}}{dt^{2}}}=-\Gamma ^{i}{}_{00}={\frac {\epsilon }{2}}\gamma _{00,i}=-\nabla \Phi \$

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

$\Phi =-{\frac {\epsilon }{2}}\gamma _{00}\$

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_{00}=-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 Einstein field equations
• Gravitoelectromagnetism

References

• Stephani, Hans (1990). General Relativity: An Introduction to the Theory of the Gravitational Field,. Cambridge: Cambridge University Press. ISBN 0-521-37941-5.{{cite book}}: CS1 maint: publisher location (link)

• Adler, Ronald; Bazin, Maurice' & Schiffer, Menahem (1965). Introduction to General Relativity. New York: McGraw-Hill. ISBN 0-070-00423-4.{{cite book}}: CS1 maint: multiple names: authors list (link)