The Ernst equations

6.3 The Ernst equations

The Ernst equations [52 , 53 ] – being the key to the Kerr–Newman metric – are the explicit form of the circular $σ$ -model equations (61

) 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. (26

) and (27

) with $ξ = m$ . Instead of writing out the components of Eq. (61

) in terms of $Λ$ and $E$ , it is more convenient to consider Eqs. (29

), 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

one easily finds the two equations

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

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. (64

). 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

where $ζ$ stand for $𝜀$ or $λ$ . A particularly simple solution to the Ernst equations is

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. (63

) 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 (10

), where the twist one-form is given in Eq. (27

).⁶⁷ Eventually, the metric function $h$ is obtained from Eqs. (62

) 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,

one eventually finds the Kerr–Newman metric in the familiar form:

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 $J = αM$ :