The coset structure of vacuum gravity

4.3 The coset structure of vacuum gravity

For many applications, in particular for the black hole uniqueness theorems, it is of crucial importance that the one-form $a$ can be replaced by a function (twist potential). We have already pointed out that $a$ , parametrizing the non-static part of the metric, enters the effective action (9

) only via the field strength, $f ≡ da$ . For this reason, the variational equation for $a$ (that is, the off-diagonal Einstein equation) assumes the form of a source-free Maxwell equation,

By virtue of Eq. (10

), the (locally defined) function $Y$ is a potential for the twist one-form, $dY = 2ω$ . In order to write the effective action (9

) in terms of the twist potential $Y$ , rather than the one-form $a$ , one considers $f ≡ da$ as a fundamental field and imposes the constraint $df = 0$ with the Lagrange multiplier $Y$ . The variational equation with respect to $f$ then yields $f = − ¯∗(σ−2dY )$ , which is used to eliminate $f$ in favor of $Y$ . One finds $1 2 1 −2 2σ f ∧ ¯∗f − Ydf → − 2σ dY ∧ ¯∗dY$ . Thus, the action (9

) becomes

where we recall that $⟨ , ⟩$ is the inner product with respect to the three-metric $¯g$ defined in Eq. (8

The action (12 ) describes a harmonic mapping into a two-dimensional target space, effectively coupled to three-dimensional gravity. In terms of the complex Ernst potential $E$ [52 , 53 ], one has

The stationary vacuum equations are obtained from variations with respect to the three-metric $¯g$ [ $(ij)$ -equations] and the Ernst potential $E$ [ $(0μ)$ -equations]. One easily finds $R¯ij = 2(E + ¯E)−2E,iE¯,j$ and $¯ΔE = 2(E + ¯E )−1⟨dE , dE ⟩$ , where $¯Δ$ is the Laplacian with respect to $¯g$ .

The target space for stationary vacuum gravity, parametrized by the Ernst potential $E$ , is a Kähler manifold with metric $GE ¯E = ∂E ∂¯Eln(σ)$ (see [62 ] for details). By virtue of the mapping

the semi-plane where the Killing field is time-like, $Re (E) > 0$ , is mapped into the interior of the complex unit disc, $D = {z ∈ ℂ | |z| < 1}$ , with standard metric $2 −2 (1 − |z| ) ⟨dz, d¯z⟩$ . By virtue of the stereographic projection, $1 0 −1 Re (z) = x (x + 1)$ , $2 0 −1 Im (z) = x (x + 1)$ , the unit disc $D$ is isometric to the pseudo-sphere, $3 P S2 = { (x0, x1,x2) ∈ IR | − (x0)2 + (x1)2 + (x2)2 = − 1}$ . As the three-dimensional Lorentz group, $SO (2,1)$ , acts transitively and isometrically on the pseudo-sphere with isotropy group $SO (2)$ , the target space is the coset $P S2 ≈ SO (2,1)∕SO (2)$ ⁴² . Using the universal covering $SU (1,1)$ of $SO (2, 1)$ , one can parametrize $2 PS ≈ SU (1,1)∕U (1)$ in terms of a positive hermitian matrix $Φ(x)$ , defined by

Hence, the effective action for stationary vacuum gravity becomes the standard action for a $σ$ -model coupled to three-dimensional gravity [139

The simplest nontrivial solution to the vacuum Einstein equations is obtained in the static, spherically symmetric case: For $E = σ(r)$ one has $2R¯rr = (σ ′∕σ )2$ and $¯Δ ln(σ) = 0$ . With respect to the general spherically symmetric ansatz

one immediately obtains the equations $− 4 ρ′′∕ρ = (σ′∕σ)2$ and $(ρ2σ′∕σ)′ = 0$ , the solution of which is the Schwarzschild metric in the usual parametrization: $σ = 1 − 2M ∕r$ , $ρ2 = σ (r)r2$ .