physics - 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 a b = η a b + εγ a b

where $η a b$ are the Minkowski metric components, $γ$ is the deviation from the Minkowski metric and $ε$ is taken to be a non-zero real constant.

We can obtain a relation between the Newtonian gravitational potential $Φ$ and the deviation term above. Calculating the Christoffel symbols $\Gamma ^a_{44}$ , we get (upon ignoring terms of order higher than $ε$ ):

$\Gamma ^a_{44}=-\frac{\epsilon}{2}g^{ad} \gamma_{44,d}$

From this last equation, we find that:

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

$\Gamma ^i_{44}=-\frac{\epsilon}{2}\gamma_{44,i}$ (

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 $Φ$ 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.

References

Ronald Adler, Maurice Bazin, and Menahem Schiffer, Introduction to General Relativity (New York: McGraw-Hill Book Company, 1965). ISBN 0070004234

Categories: General relativity