6.3 The Ernst equations

The Ernst equations [5253] – being the key to the Kerr–Newman metric – are the explicit form of the circular σ-model equations (61View Equation) for the EM system, that is, for the coset SU (2, 1)∕S (U (2) × U (1)).66 The latter is parametrized in terms of the Ernst potentials Λ = − ϕ + iψ and E = − X − Λ ¯Λ + iY, where the four scalar potentials are obtained from Eqs. (26View Equation) and (27View Equation) with ξ = m. Instead of writing out the components of Eq. (61View Equation) in terms of Λ and E, it is more convenient to consider Eqs. (29View Equation), and to reduce them with respect to the static metric ¯g = − ρ2dt2 + &tidle;g (see Section 6.2). Introducing the complex potentials 𝜀 and λ according to
1 − E 2Λ ðœ€ = 1-+-E-, λ = 1 +-E-, (63 )
one easily finds the two equations
dρ 2(¯ðœ€d𝜀 + ¯λd λ) &tidle;Δ ζ + ⟨dζ ,---+ -------------⟩ = 0, (64 ) ρ 1 − |𝜀|2 − |λ|2
where ζ stands for either of the complex potentials 𝜀 or λ, and where the Laplacian and the inner product refer to the two-dimensional metric &tidle;g.

In order to control the boundary conditions for black holes, it is convenient to introduce prolate spheroidal coordinates x and y, defined in terms of the Weyl coordinates ρ and z by

( ) ρ2 = μ2 x2 − 1)(1 − y2 , z = μ xy , (65 )
where μ is a constant. The domain of outer communications, that is, the upper half-plane ρ ≥ 0, corresponds to the semi-strip 𝒮 = {(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 Riemannian metric &tidle;g becomes (x2 − 1)−1dx2 + (1 − y2)−1dy2, up to a conformal factor which does not enter Eqs. (64View Equation). The Ernst equations finally assume the form (𝜀x ≡ ∂x𝜀, etc.)
( 2 2){ 2 2 } 1 − |𝜀| − |λ| ∂x (x − 1 )∂x + ∂y (1 − y )∂y ζ

{ 2 ( ¯ ) 2 ( ¯ ) } = − 2 (x − 1) ¯ðœ€ðœ€x + λλx ∂x + (1 − y ) ¯ðœ€ðœ€y + λλy ∂y ζ, (66 )
where ζ stand for 𝜀 or λ. A particularly simple solution to the Ernst equations is
𝜀 = px + i qy, λ = λ , where p2 + q2 + λ2 = 1, (67 ) 0 0
with real constants p, q and λ0. The norm X, the twist potential Y and the electro-magnetic potentials ϕ and ψ (all defined with respect to the axial Killing field) are obtained from the above solution by using Eqs. (63View Equation) and the expressions X = − Re (E) − |Λ|2, Y = Im (E), ϕ = − Re(Λ ), ψ = Im(Λ ). The off-diagonal element of the metric, a = atdt, is obtained by integrating the twist expression (10View Equation), where the twist one-form is given in Eq. (27View Equation).67 Eventually, the metric function h is obtained from Eqs. (62View Equation) by quadrature.

The solution derived in this way is the “conjugate” of the Kerr–Newman solution [37]. 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 X, at and e2h are replaced by kX, k−1at and ke2h, where k ≡ a2t − X −2ρ2. 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 ergo-region.

In terms of Boyer–Lindquist coordinates,

r = m (1 + px), cos𝜗 = y, (68 )
one eventually finds the Kerr–Newman metric in the familiar form:
[ ] 2 [ ] [ ] g = − Δ- dt − α sin2𝜗dφ 2 + sin-𝜗- (r2 + α2)dφ − αdt 2 + Ξ -1 dr2 + d𝜗2 , (69 ) Ξ Ξ Δ
where the constant α is defined by at ≡ α sin2𝜗. 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 𝜗. (70 )
Q- [ 2 ] A = Ξ r dt − α sin 𝜗d φ . (71 )

  Go to previous page Go up Go to next page