5.1 The Mazur identity

In the presence of a second Killing field, the EM equations (32View Equation) 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 [133Jump To The Next Citation Point], which is the key to the uniqueness theorem for the Kerr–Newman metric [131Jump To The Next Citation Point132], is a consequence of the coset structure of the field equations, which only requires the existence of one Killing field.46

In order to obtain the Mazur identity, one considers two arbitrary hermitian matrices, Φ 1 and Φ 2. The aim is to compute the Laplacian (with respect to an arbitrary metric ¯g) of the relative difference Ψ, say, between Φ2 and Φ1,

Ψ ≡ Φ Φ −1 − 𝟙. (33 ) 2 1
It turns out to be convenient to introduce the current matrices 𝒥1 = Φ−1 1¯∇Φ1 and 𝒥2 = Φ−2 1∇¯Φ2, and their difference 𝒥△ = 𝒥2 − 𝒥1, where ∇¯ denotes the covariant derivative with respect to the metric under consideration. Using ∇¯Ψ = Φ 𝒥 Φ −1 2 △ 1, the Laplacian of Ψ becomes
Δ¯Ψ = ⟨¯∇ Φ2, 𝒥△ ⟩Φ −1 + Φ2 ⟨𝒥△ ,∇¯Φ −1⟩ + Φ2 (∇¯𝒥 △) Φ− 1. 1 1 1

For hermitian matrices one has ¯ † ∇ Φ2 = 𝒥 2Φ2 and ¯ − 1 −1 † ∇ Φ1 = − Φ 1 𝒥 1, which can be used to combine the trace of the first two terms on the RHS of the above expression. One easily finds

Trace{ ¯Δ Ψ} = Trace{ ⟨Φ −1𝒥 † , Φ 𝒥 ⟩ + Φ (∇¯𝒥 )Φ −1} . (34 ) 1 △ 2 △ 2 △ 1

The above expression is an identity for the relative difference of two arbitrary hermitian matrices. If the latter are solutions of a non-linear σ-model with action ∫ Trace{𝒥 ∧ ¯∗𝒥 }, then their currents are conserved [see Eq. (32View Equation)], implying that the second term on the RHS vanishes. Moreover, if the σ-model describes a mapping with coset space SU (p,q)∕S (U (p) × U (q )), then this is parametrized by positive hermitian matrices of the form Φ = gg†.47 Hence, the “on-shell” restriction of the Mazur identity to σ-models with coset SU (p,q)∕S (U (p ) × U (q)) becomes

{ ¯ } † Trace Δ Ψ = Trace ⟨ℳ , ℳ ⟩, (35 )
where ℳ ≡ g−1𝒥 †g 1 △ 2.

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 Φ be represented in the form gg † forces the reduction of the EM system with respect to a space-like Killing field; otherwise the coset is SU (2,1)∕S (U (1,1) × U (1)), which is not of the desired form. As a consequence of the space-like reduction, the three-metric ¯g is not Riemannian, and the RHS of Eq. (35View Equation) is indefinite, unless the matrix valued one-form ℳ is space-like. This is the case if there exists a time-like Killing field with LkΦ = 0, implying that the currents are orthogonal to k: 𝒥 (k) = ikΦ− 1dΦ = Φ −1Lk Φ = 0. The reduction of Eq. (35View Equation) with respect to the second Killing field and the integration of the resulting expression will be discussed in Section 6.

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