Approximation of the weak fields

The approximation of the weak fields in General relativity is used to describe the gravitational fields far from the source of the Gravité.

It makes it possible to find the laws of the gravitation of Newton.

Mathematical description

In this approximation, one supposes that one can write the Métrique Espace-temps ( $g \$ ) in the form

g_ {\ driven \ naked} = \ eta_ {\ driven \ naked} + \ naked epsilon \ gamma_ {\ driven \} \

where $\ eta_ {\ driven \ naked} \$ is the metric of Minkowski, $\ gamma_ {\ driven \ naked} \$ is the deviation (weak) compared to the latter and $\ epsilon \$ a nonnull real constant.

A relation between the Newtonian potential of gravity $\ Phi \$ and the term of deviation quoted above can be obtained by calculating the Symboles of Christoffel $\ Gamma ^ \ driven {} _ {44} \$ , by being unaware of the terms of a nature more important than $\ epsilon \$ :

$\ Gamma ^ \ driven {} _ {00} = \ frac {\ naked epsilon} {2} g^ {\ driven \} \ gamma_ {00, \ naked} \$

and one from of deduced:

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

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

Geodetic

The equation of the Géodésique becomes

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

where $\ Phi \$ is the Newtonian potential of gravitation and $c \$ speed of light.

One has as follows:

$\ Phi=- \ frac {\ epsilon} {2} \ gamma_ {00} \$

As in addition it is known that:

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

where $G \$ is the gravitational Constante, $m \$ is the mass of the attractile body and $r \$ the radial distance to the center of this body, one finds that:

$g_ {00} = - c^2 + \ frac {2Gm} {R} \$

The approximation of the weak fields is useful to find the values of certain constants, for example in the equation of Einstein and the Métrique of Schwarzschild

See too

Tensor metric

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