8.3 The Ernst equations

The circular σ-model equations (8.13View Equation) for the EM system, with target space SU (2,1)โˆ•S (U(2) × U (1)), are called Ernst equations. Here, again, we consider the dimensional reduction with respect to the axial Killing field. The fields can be parameterized in terms of the Ernst potentials Λ = − ฯ• + iψ and −2λ ¯ E = − e − ΛΛ + iY, where the four scalar potentials are obtained from Eqs. (6.21View Equation) and (6.22View Equation) with ξ = m. Instead of writing out the components of Eq. (8.13View Equation) in terms of Λ and E, it is more convenient to consider Eqs. (6.24View Equation), and to reduce them with respect to a static metric ¯g = − ρ2dt2 + &tidle;g (see Section 8.2). Introducing the complex potentials ๐œ€ and λ according to
1 − E 2Λ ๐œ€ = ------, λ = -----, (8.15 ) 1 + E 1 + E
one easily finds the two equations
dρ 2(¯๐œ€d๐œ€ + ¯λd λ) Δ δζ + โŸจd ζ,--+ ------2-----2โŸฉ = 0, (8.16 ) ρ 1 − |๐œ€| − |λ|
where ζ stands for either of the complex potentials ๐œ€ or λ. Here we have exploited the conformal invariance of the equations and used both the Laplacian Δ δ and the inner product with respect to a flat two-dimensional metric δ. Indeed, consider two black-hole solutions, then each black hole comes with its own metric &tidle;g. However, the equation is conformally covariant, and the (ρ,z) representation of the metric is manifestly conformally flat, with the same domain of coordinates for both black holes. This allows one to view the problem as that of two different Ernst maps defined on the same flat half-plane in (ρ,z)-coordinates.

8.3.1 A derivation of the Kerr–Newman metric

The Kerr–Newman metric is easily derived within this formalism. For this it is convenient to introduce, first, prolate spheroidal coordinates x and y, defined in terms of the Weyl coordinates ρ and z by

2 2 ( 2 2) ρ = μ x − 1)(1 − y , z = μxy, (8.17 )
where μ is a constant. The domain of outer communications, that is, the upper half-plane ρ ≥ 0, corresponds to the semi-strip ๐’ฎ = {(x,y)|x ≥ 1,|y| ≤ 1}. The boundary ρ = 0 consists of the horizon (x = 0) and the northern (y = 1) and southern (y = − 1) segments of the rotation axis. In terms of x and y, the metric &tidle;g becomes (x2 − 1 )− 1dx2 + (1 − y2 )− 1dy2, up to a conformal factor, which does not enter Eqs. (8.16View Equation). The Ernst equations finally assume the form (๐œ€x := ∂x๐œ€, etc.)
( 2 2){ 2 2 } 1 − |{๐œ€| − |λ|( ∂x (x − 1) )∂x + ∂y(1 − y( )∂y ζ = ) } − 2 (x2 − 1) ¯๐œ€๐œ€x + ¯λλx ∂x + (1 − y2) ¯๐œ€๐œ€y + ¯λλy ∂y ζ, (8.18 )
where ζ stand for ๐œ€ or λ. A particularly simple solution to those equations is
2 2 2 ๐œ€ = px + iqy, λ = λ0, wherep + q + λ0 = 1, (8.19 )
with real constants p, q and λ 0. The norm e−2λ, the twist potential Y and the electro-magnetic potentials ฯ• and ψ (all defined with respect to the axial Killing field) are obtained from the above solution by using Eqs. (8.15View Equation) and the expressions − 2λ 2 e = − Re (E) − |Λ |, Y = Im (E ), ฯ• = − Re(Λ ), ψ = Im(Λ ). The off-diagonal element of the metric, a = atdt, is obtained by integrating the twist expression (6.3View Equation), where the twist one-form is given in Eq. (6.22View Equation), and the Hodge dual in Eq. (6.3View Equation) now refers to the decomposition (8.7View Equation) with respect to the axial Killing field. Eventually, the metric function h is obtained from Eqs. (8.14View Equation) by quadratures.

The solution derived so far is the “conjugate” of the Kerr–Newman solution [56]. In order to obtain the Kerr–Newman metric itself, one has to perform a rotation in the tφ-plane: The spacetime metric is invariant under t → φ, φ → − t, if e −2λ, at and e2h are replaced by αe−2λ, α− 1at and αe2h, where 2 4λ 2 α := at − e ρ. This additional step in the derivation of the Kerr–Newman metric is necessary because the Ernst potentials were defined with respect to the axial Killing field ∂φ. If, on the other hand, one uses the stationary Killing field ∂t, then the Ernst equations are singular at the boundary of the ergoregion.

In terms of Boyer–Lindquist coordinates,

r = m(1 + px ), cos๐œ— = y, (8.20 )
one eventually finds the Kerr–Newman metric in the familiar form:
Δ [ ]2 sin2๐œ— [ ]2 [ 1 ] g = − -- dt − α sin2๐œ—d φ + ------ (r2 + α2)dφ − αdt + Ξ --dr2 + d๐œ—2 , (8.21 ) Ξ Ξ Δ
where the constant α is defined by at := α sin2 ๐œ—. The expressions for Δ, Ξ and the electro-magnetic vector potential A show that the Kerr–Newman solution is characterized by the total mass M, the electric charge Q, and the angular momentum J = αM:
2 2 2 2 2 2 Δ = r − 2M r + α + Q , Ξ = r + α cos ๐œ—. (8.22 )
Q- [ 2 ] A = Ξ r dt − α sin ๐œ—d φ . (8.23 )

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