126
9 Spherically Symmetric Gravitational Fields
This is simply the metric of flat space time, that is the Minkowski space of special
relativity, expressed in spherical space coordinates. To describe the gravitational field
it must be modified. We already know approximately what the modification must be,
for the relation between the metric and the Newtonian field in (7.23) tells us that for
a body of mass M
g 00 = 1 +
2φ
c 2 = 1 −
2G M
c 2 r
.
(9.3)
Thus it is clear that we should allow g 00 to be a function of the radial coordinate.
Moreover we may guess that since the field is spherically symmetric we must allow
g 11 to also be a function of r. We thus look for a solution of the form
ds
2
= e
ν(r ) c
2 dt
2
− e
λ(r ) dr
2
− r
2
dθ
2
+ sin
2
θ dϕ
2
.
(9.4)
The use of exponential functions is entirely for future mathematical convenience.
Thus we have guessed a simple form of metric with only two unknown functions of
r, which must be chosen to satisfy the field equations. Note that we have allowed the
angular part of the line element to remain in exactly the same form as the flat space
line element. The simplicity of (9.4) is the key to the exact solution.
The next step in solving the field equations is to calculate the connections needed
to write out the Ricci tensor. This is straight-forward if we use the shortcut discussed
in Chap. 5, and is left as an exercise for the reader. Most of the connections are zero,
and the task is correspondingly easy, giving
0
10 =
0
01 =
ν
2
, ,
1
00 =
ν
2
e
ν−λ
, ,
1
11 =
λ
2
,
1
22 = −e
−λ r, ,
1
33 = −e
−λ r sin
2
θ, ,
2
12 =
2
21 =
1
r
,
2
33 = − sin θ cos θ, ,
3
13 =
3
31 =
1
r
, ,
3
23 =
3
32 = cos θ.
(9.5)
Here the prime denotes a derivative with respect to r. Next we obtain the metric
determinant, which is needed to calculate the contracted connection using (6.28),
and find
log
|g| =
ν + λ
2
+ 2 log r + log |sin θ |.
(9.6)
(Recall that |g| is taken as the absolute value of the determinant as we noted in
Chap. 4.)
We are now ready to write out the field equations in terms of the coordinates and
the unknown functions ν and λ.
First we consider the Einstein equation for μ = ν = 0. From the Ricci tensor in
(9.1) and the contracted connection in (6.18) we have
9 Spherically Symmetric Gravitational Fields
This is simply the metric of flat space time, that is the Minkowski space of special
relativity, expressed in spherical space coordinates. To describe the gravitational field
it must be modified. We already know approximately what the modification must be,
for the relation between the metric and the Newtonian field in (7.23) tells us that for
a body of mass M
g 00 = 1 +
2φ
c 2 = 1 −
2G M
c 2 r
.
(9.3)
Thus it is clear that we should allow g 00 to be a function of the radial coordinate.
Moreover we may guess that since the field is spherically symmetric we must allow
g 11 to also be a function of r. We thus look for a solution of the form
ds
2
= e
ν(r ) c
2 dt
2
− e
λ(r ) dr
2
− r
2
dθ
2
+ sin
2
θ dϕ
2
.
(9.4)
The use of exponential functions is entirely for future mathematical convenience.
Thus we have guessed a simple form of metric with only two unknown functions of
r, which must be chosen to satisfy the field equations. Note that we have allowed the
angular part of the line element to remain in exactly the same form as the flat space
line element. The simplicity of (9.4) is the key to the exact solution.
The next step in solving the field equations is to calculate the connections needed
to write out the Ricci tensor. This is straight-forward if we use the shortcut discussed
in Chap. 5, and is left as an exercise for the reader. Most of the connections are zero,
and the task is correspondingly easy, giving
0
10 =
0
01 =
ν
2
, ,
1
00 =
ν
2
e
ν−λ
, ,
1
11 =
λ
2
,
1
22 = −e
−λ r, ,
1
33 = −e
−λ r sin
2
θ, ,
2
12 =
2
21 =
1
r
,
2
33 = − sin θ cos θ, ,
3
13 =
3
31 =
1
r
, ,
3
23 =
3
32 = cos θ.
(9.5)
Here the prime denotes a derivative with respect to r. Next we obtain the metric
determinant, which is needed to calculate the contracted connection using (6.28),
and find
log
|g| =
ν + λ
2
+ 2 log r + log |sin θ |.
(9.6)
(Recall that |g| is taken as the absolute value of the determinant as we noted in
Chap. 4.)
We are now ready to write out the field equations in terms of the coordinates and
the unknown functions ν and λ.
First we consider the Einstein equation for μ = ν = 0. From the Ricci tensor in
(9.1) and the contracted connection in (6.18) we have