Introduction to Cosmology

Weak Field Limit 61

does not have a source, it is its own source. The only source of energy in the Einstein
Equation (3.26) is matter in the stress-energy tensor. In the absence of matter퐺휇휈=0.
But the question then arises how to include the energy of the gravitational field itself.
Since gravitational waves are expected to cause physical effects at some distance from
the source, the Einstein equation is clearly lacking something.
This lack arises because the Einstein equation describes a linearized theory, it is
formulated using the Ricci tensor which is also linearized and which vanishes in the
absence of a stress-energy tensor. The remedy is to introduce higher-order corrections
in Equation (3.26) [4]. One replaces the gravity field푔휇휈by푔휇휈(퐼)+푔휇휈(II)where (I)
and (II) refer to first and second order terms, and where the background metric푔휇휈(퐼)
is not regarded as static, but as responding to the gravity waves. Thus in the absence
of matter,
퐺휇휈=퐺휇휈(퐼)+퐺휇휈(II)= 0 , (3.32)

one has퐺휇휈(퐼)=−퐺휇휈(II). At large scale where one can neglect the wavelengths of the
gravity waves the first order description is adequate, at shorter scales the influence of
gravity waves is described by퐺휇휈(II)and higher order terms.
Carrying out this argument in full detail leads to replacing the Einstein tensor by
a pseudotensor [4].

3.4 Weak Field Limit

The starting point is Newton’s law of gravitation, because this has to be true anyway
in the limit of very weak fields. From Equation (1.27), the gravitational force experi-
enced by a unit mass at distance푟from a body of mass푀and density휌is a vector in
three-space

r̈=F=−GMr

푟^3

,

in component form (푖= 1 , 2 ,3)

d^2 푥푖

d푡^2

=퐹푖=−GM푥

푖

푟^3

. (3.33)

Let us define a scalargravitational potential휙by

휕휙

휕푥푖

=−퐹푖.

This can be written more compactly as

∇휙=−F. (3.34)

Integrating the flux of the forceFthrough a spherical surface surrounding푀and
using Stokes’s theorem, one can show that the potential휙obeys Poisson’s equation

∇^2 휙= 4 휋퐺휌. (3.35)
Let us next turn to the relativistic equation of motion (3.14). In the limit of
weak and slowly varying fields for which all time derivatives of푔휇휈vanish and the