3.4 Kerr Black Hole and Light-Like Matter
61
3.4
Part II: A Kerr Black Hole and Light-Like Matter
In this part of the current chapter we describe a generalisation of the Kerr space-time
in the spirit of the Vaidya generalisation of the Schwarzschild space-time. We begin
by describing the latter in a form convenient for our purposes. Using Eqs. (2.151)
and (2.154) of Chap. 2 we can write the Schwarzschild line element in the form
ds
2
= dT
2
− dX
2
− dY
2
− dZ
2
−
2 m
r
(k i dX
i )
2
= g ij dX
i dX
j ,
(3.65)
in coordinates X i = (T , X, Y, Z) with i = 0, 1, 2, 3. The parameter m is the
constant mass of the source,
r
2
= X
2
+ Y
2
+ Z
2 ,
(3.66)
and the 1-form k i dX i is given by
k i dX
i
= dt −
1
r
(X dX + Y dY + Z dZ) .
(3.67)
Clearly from (3.65) we have the Kerr–Schild form of metric tensor components
g ij = η ij −
2 m
r
k i k j with η ij = diag(1, −1, −1, −1) .
(3.68)
We see from (3.66) and (3.67) that η ij k i k j = 0 so that k i = η ij k j is a null vector
field with respect to the Minkowskian metric η ij . It is also a null vector field with
respect to the metric g ij given via (3.65) since
g
ij
= η
ij
+
2 m
r
k
i k
j with k
i
= η
ij k j = g
ij k j .
(3.69)
We make the simple observation that
u ≡ g ij k
i X
j
= η ij k
i X
j
= T − r ,
(3.70)
using (3.66). If with this variable u we make the generalisation of (3.65)
ds
2
= dT
2
− dX
2
− dY
2
− dZ
2
−
2 m(u)
r
(k i dX
i )
2 ,
(3.71)
by replacing the constant m in (3.65) by the function m(u) we have arrived at the
Vaidya [7] generalisation of the Schwarzschild space-time. For the space-time with
line element (3.71) the Ricci tensor components R ij are given by
R ij =
2 ˙
m
r 2 k i k j with ˙
m =
dm
du
.
(3.72)