### 5.1 The Mazur identity

In the presence of a second Killing field, the EM equations (32) experience further,
considerable simplifications, which will be discussed later. In this section we will not yet
require the existence of an additional Killing symmetry. The Mazur identity [133], which is
the key to the uniqueness theorem for the Kerr–Newman metric [131, 132], is a consequence
of the coset structure of the field equations, which only requires the existence of one Killing
field.
In order to obtain the Mazur identity, one considers two arbitrary hermitian matrices, and .
The aim is to compute the Laplacian (with respect to an arbitrary metric ) of the relative difference
, say, between and ,

It turns out to be convenient to introduce the current matrices and , and
their difference , where denotes the covariant derivative with respect to the metric
under consideration. Using , the Laplacian of becomes
For hermitian matrices one has and , which can be used to
combine the trace of the first two terms on the RHS of the above expression. One easily finds

The above expression is an identity for the relative difference of two arbitrary hermitian matrices. If the latter
are solutions of a non-linear -model with action , then their currents are conserved [see
Eq. (32)], implying that the second term on the RHS vanishes. Moreover, if the -model describes a mapping
with coset space , then this is parametrized by positive hermitian matrices of the form
.
Hence, the “on-shell” restriction of the Mazur identity to -models with coset
becomes

where .
Of decisive importance to the uniqueness proof for the Kerr–Newman metric is the fact that the RHS of
the above relation is non-negative. In order to achieve this one needs two Killing fields: The requirement
that be represented in the form forces the reduction of the EM system with respect to a
space-like Killing field; otherwise the coset is , which is not of the desired
form. As a consequence of the space-like reduction, the three-metric is not Riemannian, and the
RHS of Eq. (35) is indefinite, unless the matrix valued one-form is space-like. This is the
case if there exists a time-like Killing field with , implying that the currents are
orthogonal to : . The reduction of Eq. (35) with respect
to the second Killing field and the integration of the resulting expression will be discussed in
Section 6.