7.1 The Mazur identity

In the presence of a second Killing field, the EM equations (6.26View 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 [230Jump To The Next Citation Point], which is the key to the uniqueness theorem for the Kerr–Newman metric [228Jump To The Next Citation Point, 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, Φ1 and Φ2. The aim is to compute the Laplacian with respect to a metric ¯g (which in the application of interest will be flat) of the relative difference Ψ, say, between Φ 2 and Φ 1,

Ψ := Φ Φ− 1− πŸ™. (7.1 ) 2 1
It turns out to be convenient to introduce the current matrices π’₯1 = Φ−1 1¯∇Φ1 and π’₯2 = Φ−2 1∇¯Φ2, and their difference π’₯β–³ = π’₯2 − π’₯1, where ∇¯ denotes the covariant derivative with respect to g¯. Using ∇¯Ψ = Φ π’₯ Φ−1 2 β–³ 1, the Laplacian of Ψ becomes
Δ¯Ψ = ⟨¯∇ Φ ,π’₯ βŸ©Φ −1 + Φ βŸ¨π’₯ ,∇¯Φ −1⟩ + Φ (∇¯π’₯ )Φ −1, 2 β–³ 1 2 β–³ 1 2 β–³ 1

where, as before, the inner product ⟨⋅,⋅⟩ is taken with respect to the three-metric ¯g and also involves a matrix multiplication. For hermitian matrices one has ¯∇Φ2 = π’₯ †2Φ2 and ¯∇Φ −11 = − Φ −11π’₯1†, 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

( ) Trace (Δ¯Ψ ) = Trace βŸ¨Φ −1π’₯ †,Φ π’₯ ⟩ + Φ (¯∇π’₯ )Φ −1 . (7.2 ) 1 β–³ 2 β–³ 2 β–³ 1
The above expression is an identity for the relative difference of two arbitrary Hermitian matrices, with all operations taken with respect to ¯g (recall (6.10View Equation)). If the latter are solutions of a non-linear σ-model with action ∫ Trace (π’₯ ∧ ¯∗π’₯ ), then their currents are conserved [see Eq. (6.26View Equation)], implying that the second term on the right-hand side vanishes. Moreover, if the σ-model describes a mapping with coset space SU (p, q)βˆ•S (U (p) × U (q)), then this is parameterized by positive Hermitian matrices of the form Φ = gg†. (We refer to [196, 95], and [26] for the theory of symmetric spaces.) Hence, the “on-shell” restriction of the Mazur identity to σ-models with coset SU (p,q )βˆ•S (U (p) × U (q)) becomes
( ) † Trace ¯Δ Ψ = TraceβŸ¨β„³, β„³ ⟩, (7.3 )
where − 1 † β„³ := g1 π’₯ β–³g2.

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 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)βˆ•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 right-hand side of Eq. (7.3View Equation) is indefinite, unless the matrix valued one-form β„³ 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: π’₯ (k) = ikΦ− 1dΦ = Φ −1Lk Φ = 0. The reduction of Eq. (7.3View Equation) with respect to the second Killing field and the integration of the resulting expression will be discussed in Section 8.

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