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 ,
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 findssolutions 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  for the theory of symmetric spaces.) Hence, the “on-shell” restriction of the Mazur identity to -models with coset becomes
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.
Living Rev. Relativity 15, (2012), 7
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