8.1 Robinson–Trautman Solutions
189
with R (a)(b) the components of the Ricci tensor on the half null tetrad calculated
with the metric given via the line element (8.1), result in [1]
c = K − 2 H r −
2 M
r
+
e 2
r 2 ,
(8.10)
with
K = log p , M = m(u) − 2 e
2 w ,
(8.11)
and
1
4
= ˙
M − 3 H M + e
2 p
2 (w
2
x + w
2
y ) .
(8.12)
For the Reissner–Nordström spherically symmetric and static solution e, m are
constants,
w = 0 , p = p 0 = 1 +
1
4
(x
2
+ y
2 ) , H = 0 , K = 1 ,
(8.13)
and so the Maxwell 2-form reads
F = −
e 2
r 2 ϑ
(3)
∧ ϑ
(4) ,
(8.14)
with e the charge on the source, and
ds
2
= −r
2 p
−2
0 (dx
2
+ dy
2 ) + 2 du dr +
1 −
2 m
r
+
e 2
r 2
du
2 ,
(8.15)
with m the mass of the source. Hence it is possible to view the space-time as a
perturbation of Minkowskian space-time with
ds
2
= ds
2
0 + γ ij dx
i dx
j with x
i
= (x, y, r, u) ,
(8.16)
where
ds
2
0 = −r
2 p
−2
0 (dx
2
+ dy
2 ) + 2 du dr + du
2 ,
(8.17)
is the line element of Minkowskian space-time and r = 0 is a time-like geodesic in
this space-time with u proper-time or arc length along it, and
γ ij dx
i dx
j
=
−
2 m
r
+
e 2
r 2
du
2 ,
(8.18)