2.4 The Kerr Space-Time
39
giving the components g ab of the metric tensor in coordinates x a = (θ, φ, r, u).
When a = 0 this reduces to the Eddington–Finkelstein form of the Schwarzschild
line element:
ds
2
= −r
2 (dθ
2
+ sin
2 θ dφ
2 ) + 2 du dr +
1 −
2 m
r
du
2 .
(2.147)
To apply to (2.146) the transformation (see [14, 15]) x a → X i = (x, y, z, t) given
by
x + i y = (r + i a) e
i φ sin θ , z = r cos θ , u = t − r ,
(2.148)
we first note that r is a function of x, y, z given by
x 2 + y 2
r 2 + a 2 +
z 2
r 2 = 1 ,
(2.149)
and hence
dr − a sin
2 θ dφ =
r x + a y
r 2 + a 2
dx +
r y − a x
r 2 + a 2
dy +
z
r
dz ,
(2.150)
from which it follows that
du + a sin
2 θ dφ = dt − (dr − a sin
2 θ dφ)
= dt −
r x + a y
r 2 + a 2
dx −
r y − a x
r 2 + a 2
dy −
z
r
dz
= k i dX
i (say) .
(2.151)
Also we have
dx
2
+ dy
2
+ dz
2
= (dr − a sin
2 θ dφ)
2
+ (r
2
+ a
2 cos
2 θ )(dθ
2
+ sin
2 θ dφ
2 ) ,
(2.152)
and so, using (2.151),
− dx 2 −dy
2
− dz
2
+ dt
2
= −(r
2
+ a
2 cos
2 θ )(dθ
2
+ sin
2 θ dφ
2 )
+ 2 (du + a sin
2 θ dφ)
dr − a sin
2 θ dφ +
1
2
(du + a sin
2 θ dφ)
.
(2.153)