6.2 The coset structure of vacuum gravity

For many applications, in particular for the black-hole uniqueness theorems, it is convenient to replace the one-form a by a function, namely the twist potential. We have already pointed out that a, parameterizing the non-static part of the metric, enters the effective action (6.2View Equation) only via the field strength, f = da. For this reason, the variational equation for a (that is, the off-diagonal Einstein equation) takes in vacuum the form of a source-free Maxwell equation:
( ) ( ) d¯∗ V2da = 0 = ⇒ dY = − ¯∗ V 2da . (6.4 )
By virtue of Eq. (6.3View Equation), the (locally-defined) function Y is a potential for the twist one-form, dY = 2ω. In order to write the effective action (6.2View Equation) in terms of the twist potential Y, rather than the one-form a, one considers f 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 = − ¯∗(V −2dY ), which is used to eliminate f in favor of Y. One finds 1 2 1 −2 2V f ∧ ¯∗f − Y df → − 2V dY ∧ ¯∗dY. Thus, the action (6.2View Equation) becomes
∫ ( ) Seff = ¯∗ ¯R − ⟨dV--,dV-⟩ +-⟨dY-,dY-⟩ , (6.5 ) 2V 2
where we recall that ⟨,⟩ is the inner product with respect to the three-metric ¯g defined in Eq. (6.1View Equation).

The action (6.5View Equation) describes a harmonic map into a two-dimensional target space, effectively coupled to three-dimensional gravity. In terms of the complex Ernst potential E [102Jump To The Next Citation Point, 103Jump To The Next Citation Point], one has

∫ ( ¯ ) S = ¯∗ ¯R − 2 ⟨dE,d-E⟩- , E := V + iY . (6.6 ) eff (E + ¯E)2
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, parameterized by the Ernst potential E, is a Kähler manifold with metric GE ¯E = ∂E∂E¯ln(V ) (see [115] for details). By virtue of the mapping

E ↦→ z = 1-−-E-, (6.7 ) 1 + E
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, P S2 = {(x0,x1, x2) ∈ ℝ3 | − (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) (see, e.g., [196Jump To The Next Citation Point] or [26Jump To The Next Citation Point] for the general theory of symmetric spaces). Using the universal covering SU (1,1) of SO (2,1), one can parameterize PS2 ≈ SU (1,1)∕U (1) in terms of a positive hermitian matrix Φ (x ), defined by
( 0 1 2) ( 2 ) Φ (x) = 1 x 2 x + 0ix = ---1---- 1 + |z| 2z 2 . (6.8 ) x − ix x 1 − |z|2 2¯z 1 + |z|
Hence, the effective action for stationary vacuum gravity becomes the standard action for a σ-model coupled to three-dimensional gravity [250Jump To The Next Citation Point],
∫ ( ) 𝒮 = ¯∗ R¯ − 1-Trace⟨𝒥 ,𝒥 ⟩ , (6.9 ) eff 4
where
Trace⟨𝒥 ,𝒥 ⟩ ≡ ⟨𝒥BA,𝒥 BA ⟩ := ¯gij(𝒥i )AB (𝒥j)BA, (6.10 )
and the currents 𝒥i are defined as
− 1¯ 𝒥i := Φ ∇i Φ. (6.11 )

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

2 2 2 ¯g = dr + ρ (r)dΩ , (6.12 )
one immediately obtains the equations − 4 ρ′′∕ρ = (V ′∕V )2 and (ρ2V ′∕V )′ = 0, the solution of which is the Schwarzschild metric in the usual parametrization: V = 1 − 2M ∕r, 2 2 ρ = V (r)r.
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