### 7.1 The Mazur identity

In the presence of a second Killing field, the EM equations (6.26) 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 [230], which is the key to the uniqueness theorem for the Kerr–Newman metric [228, 229], is a consequence of the coset structure of the field equations. Note, however, that while the derivation of the general form of this identity only requires one Killing vector, its application to the uniqueness argument uses two; we will return to this issue shortly.

In order to obtain the Mazur identity, one considers two arbitrary Hermitian matrices, and . The aim is to compute the Laplacian with respect to a metric (which in the application of interest will be flat) 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 . Using , the Laplacian of becomes

where, as before, the inner product is taken with respect to the three-metric and also involves a matrix multiplication. For hermitian matrices one has and , which can be used to combine the trace of the first two terms on the right-hand side of the above expression. One easily finds

The above expression is an identity for the relative difference of two arbitrary Hermitian matrices, with all operations taken with respect to (recall (6.10)). If the latter are solutions of a non-linear -model with action , then their currents are conserved [see Eq. (6.26)], implying that the second term on the right-hand side vanishes. Moreover, if the -model describes a mapping with coset space , then this is parameterized by positive Hermitian matrices of the form . (We refer to [196, 95], and [26] for the theory of symmetric spaces.) 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 right-hand side 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 right-hand side of Eq. (7.3) 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. (7.3) with respect to the second Killing field and the integration of the resulting expression will be discussed in Section 8.