The Mazur identity

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.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 $Φ1$ ,

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

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. (32 )], 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

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. (35 ) 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. (35 ) with respect to the second Killing field and the integration of the resulting expression will be discussed in Section 6.