JHEP01(2020)123
The first step of the Ernst procedure consists in identifying the f, ω, γ functions ap-
pearing in the Lewis-Weyl-Papapetrou ansatz (2.4) for the seed metric, which in this case
is the Kerr-Newman one (3.1). At this purpose the coordinates transformation
ρ(r, θ) := sin θ
p
∆(r) , z(r, θ) := cos θ(r − m) (3.4)
is applied to (2.4) to get a better suited LWP line element for the seed coordinates
ds
2
= −f(r, θ) [dt+ω(r, θ)dϕ]
2
+
1
f(r, θ)
e
2γ(r,θ)
(r−m)
2
−κ
2
cos
2
θ
dr
2
∆(r)
+dθ
2
+sin
2
θ ∆(r)dϕ
2
.
(3.5)
The constant κ, specifically for the Kerr-Newman spacetime, takes the value κ =
p
m
2
− a
2
− q
2
− p
2
.
By comparing the metrics (3.1) and (3.5) it is possible to determinate the structure
functions of the Kerr-Newman metric. We will present them in more ergonomic coordinates
(r, x := cos θ):
f
0
(r, x) = 1 +
q
2
+ p
2
− 2mr
r
2
+ a
2
x
2
(3.6)
ω
0
(r, x) =
a(1 − x
2
)(2mr − q
2
− p
2
)
2(r
2
− 2mr + q
2
+ p
2
+ a
2
x
2
)
(3.7)
e
2γ
0
(r, x) =
r
2
− 2mr + q
2
+ p
2
+ a
2
x
2
(r − m)
2
− κ
2
x
2
. (3.8)
The differential operators
−→
∇ and ∇
2
in terms of the coordinates (r, x) becomes
−→
∇φ(r, x) =
1
p
(r − m)
2
− κ
2
x
2
−→
e
r
p
∆(r)
∂φ(r, x)
∂r
+
−→
e
x
p
1 − x
2
∂φ(r, x)
∂x
(3.9)
∇
2
φ(r, x) =
1
(r − m)
2
− κ
2
x
2
∂
∂r
∆(r)
∂φ(r, x)
∂r
+
∂
∂x
(1 − x
2
)
∂φ(r, x)
∂x
. (3.10)
The (r, x) coordinates are closely related with the prolate spherical ones (y, x). To obtain
these latter is sufficient to define y := (r − m)/κ.
Then in order to identify the electromagnetic seed Ernst potential Φ
0
,
7
as defined
in (2.9), for the Kerr-Newman gauge field we need to derive
˜
A
ϕ
from eq. (2.10) and taking
into account eqs. (3.2), (3.6)–(3.9). For the seed under consideration we have
˜
A
ϕ0
(r, x) =
aqx − pr
r
2
+ a
2
x
2
=⇒ Φ
0
(r, x) = −
q + ip
r + iax
. (3.11)
While to obtain the gravitational Ernst potential for the seed Kerr-Newman metric E
0
we
have first to integrate (2.11) to get
h
0
(r, x) =
2mx
r
2
+ a
2
x
2
=⇒ E
0
(r, x) = 1 −
2m
r + iax
. (3.12)
7
The zero subscript in Φ
0
point out that we refer to the seed fields.
– 6 –
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