### 6.3 The Ernst equations

The Ernst equations [52, 53] – being the key to the Kerr–Newman metric – are the explicit
form of the circular -model equations (61) for the EM system, that is, for the coset
.
The latter is parametrized in terms of the Ernst potentials and ,
where the four scalar potentials are obtained from Eqs. (26) and (27) with . Instead of writing out
the components of Eq. (61) in terms of and , it is more convenient to consider Eqs. (29), and to
reduce them with respect to the static metric (see Section 6.2). Introducing the complex
potentials and according to
one easily finds the two equations
where stands for either of the complex potentials or , and where the Laplacian and the inner
product refer to the two-dimensional metric .
In order to control the boundary conditions for black holes, it is convenient to introduce 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 Riemannian metric becomes , up to a conformal
factor which does not enter Eqs. (64). The Ernst equations finally assume the form (,
etc.)

where stand for or . A particularly simple solution to the Ernst 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. (63) and the expressions ,
, , . The off-diagonal element of the metric, , is
obtained by integrating the twist expression (10), where the twist one-form is given in
Eq. (27).
Eventually, the metric function is obtained from Eqs. (62) by quadrature.
The solution derived in this way is the “conjugate” of the Kerr–Newman solution [37]. 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 ergo-region.

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 :