62 General Relativity
(spatial) velocity components d푥푖∕d휏are negligible compared with d푥^0 ∕d휏=푐d푡∕d휏,
Equation (3.14) reduces to
d^2 푥휇
d휏^2+푐^2 훤 00 휇
(
d푡
d휏) 2
= 0. (3.36)
From Equation (3.13) these components of the affine connection are
훤 00 휇=−^1
2푔휇휌
휕푔 00
휕푥휌
,
where푔 00 is the time–time component of푔휇휈and the sum over휌is implied.
In a weak static field the metric is almost that of flat space-time, so we can approx-
imate푔휇휈by
푔휇휈=휂휇휈+ℎ휇휈,
whereℎ휇휈is a small increment to휂휇휈. To lowest order inℎ휇휈we can then write
훤 00 휇=−
1
2
휂휇휌
휕ℎ 00
휕푥휌
. (3.37)
Inserting this expression into Equation (3.36), the equations of motion become
d^2 x
d휏^2=−^1
2
(
d푡
d휏) 2
푐^2 ∇ℎ 00 , (3.38)
d^2 푡
d휏^2= 0. (3.39)
Dividing Equation (3.38) by(d푡∕d휏)^2 we obtain
d^2 x
d푡^2=−^1
2
푐^2 ∇ℎ 00. (3.40)
Comparing this with the Newtonian equation of motion (3.31) in the푥푖direction we
obtain the value of the time–time component ofℎ휇휈,
ℎ 00 = 2휙
푐^2
,
from which it follows that
푔 00 = 1 + 2휙
푐^2
= 1 −
2 GM
푐^2 푟
. (3.41)
We can now put several things together: replacing휌in the field equation (3.35)
푇 00 ∕푐^2 and substituting휙from Equation (3.41) we obtain a field equation for weak
static fields generated by nonrelativistic matter:
∇^2 푔 00 =^8 휋퐺
푐^4푇 00. (3.42)
Let us now assume with Einstein that the right-hand side could describe the source
term of a relativistic field equation of gravitation if we made it generally covariant.
This suggests replacing푇 00 with푇휇휈. In a matter-dominated universe where the grav-
itational field is produced by massive stars, and where the pressure between stars is
negligible, the only component of importance is then푇 00.