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
: