9.1 The Schwarzschild Solution
127
R 00 =
log
|g|
,0,0
−
α
00,α +
β
τ 0
τ
β0 −
τ
00
log
|g|
,τ
= 0.
(9.7)
Many of the terms in this component are zero. From (9.5) and (9.6) we find it reduces
to
R 00 = −
1
00,1 + 2
1
00
0
01 −
1
00
log
|g|
,1
= −
1
2
ν
e
ν−λ
+
1
2
ν
2 e
ν−λ
−
1
2
ν
e
ν−λ
ν
+ λ
2
+
2
r
= −
e
ν−λ
2
ν
+
1
2
ν
2
−
1
2
λ
ν
+
2
r
ν
= 0
(9.8)
Thus the μ = ν = 0 equation distills down to
ν
+
1
2
ν
2
−
1
2
λ
ν
+
2
r
ν
= 0.
(9.9)
In the same manner we work out the μ = ν = 1 term of the field equations and
find
R 11 =
log
|g|
,1,1
− α
11,α +
β
τ 1 τ
β1 − τ
11
log
|g|
,1
=
log
|g|
,1,1
− 1
11,1 + 0
01 0
01 + 1
11 1
11 + 2
21 2
21 + 3
31 3
31
− 1
11
log
|g|
,1
=
1
2
ν +
1
2
ν 2 −
1
2
λ ν −
2
r
λ
= 0,
so ν +
1
2
ν 2 −
1
2
λ ν −
2
r
λ = 0
(9.10)
Equations (9.9) and (9.10) suffice to find the unknown functions. They are ordinary
second order differential equations.
Subtracting (9.10) from (9.9) we see that
(ν + λ)
= 0, ν + λ = const.
(9.11)
Differential equations generally require boundary conditions. In this problem the
appropriate boundary condition is quite obvious: we ask that the metric be that of
gravity-free flat space at a large distance from the origin; that is the line element
should approach (9.2). This in turn means that the two unknown functions ν and λ
must both approach zero. Thus the constant in (9.11) must be zero, and
λ = −ν.
(9.12)
We now substitute this into (9.10) and find the following equation for ν
ν
+ ν
2
+
2
r
ν
= 0.
(9.13)