### 8.3 The Ernst equations

The circular -model equations (8.13) for the EM system, with target space ,
are called Ernst equations. Here, again, we consider the dimensional reduction with respect to the axial
Killing field. The fields can be parameterized in terms of the Ernst potentials and
, where the four scalar potentials are obtained from Eqs. (6.21) and (6.22) with
. Instead of writing out the components of Eq. (8.13) in terms of and , it is
more convenient to consider Eqs. (6.24), and to reduce them with respect to a static metric
(see Section 8.2). Introducing the complex potentials and according to
one easily finds the two equations
where stands for either of the complex potentials or . Here we have exploited the conformal
invariance of the equations and used both the Laplacian and the inner product with respect to a flat
two-dimensional metric . Indeed, consider two black-hole solutions, then each black hole comes with its
own metric . However, the equation is conformally covariant, and the representation of the
metric is manifestly conformally flat, with the same domain of coordinates for both black holes. This allows
one to view the problem as that of two different Ernst maps defined on the same flat half-plane in
-coordinates.

#### 8.3.1 A derivation of the Kerr–Newman metric

The Kerr–Newman metric is easily derived within this formalism. For this it is convenient to introduce,
first, prolate spheroidal coordinates and , defined in terms of the Weyl coordinates and by

where is a constant. The domain of outer communications, that is, the upper half-plane ,
corresponds to the semi-strip . The boundary consists of the horizon
() and the northern () and southern () segments of the rotation axis. In terms of
and , the metric becomes , up to a conformal factor, which
does not enter Eqs. (8.16). The Ernst equations finally assume the form (, etc.)
where stand for or . A particularly simple solution to those equations is
with real constants , and . The norm , the twist potential and the electro-magnetic
potentials and (all defined with respect to the axial Killing field) are obtained from the above
solution by using Eqs. (8.15) and the expressions , , ,
. The off-diagonal element of the metric, , is obtained by integrating the twist
expression (6.3), where the twist one-form is given in Eq. (6.22), and the Hodge dual in Eq. (6.3) now
refers to the decomposition (8.7) with respect to the axial Killing field. Eventually, the metric function
is obtained from Eqs. (8.14) by quadratures.
The solution derived so far is the “conjugate” of the Kerr–Newman solution [56]. In order to obtain the
Kerr–Newman metric itself, one has to perform a rotation in the -plane: The spacetime metric is
invariant under , , if , and are replaced by , and ,
where . This additional step in the derivation of the Kerr–Newman metric is necessary
because the Ernst potentials were defined with respect to the axial Killing field . If, on the other hand,
one uses the stationary Killing field , then the Ernst equations are singular at the boundary of the
ergoregion.

In terms of Boyer–Lindquist coordinates,

one eventually finds the Kerr–Newman metric in the familiar form:
where the constant is defined by . The expressions for , and the
electro-magnetic vector potential show that the Kerr–Newman solution is characterized by the total
mass , the electric charge , and the angular momentum :