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 (9View Equation) 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,
( ) ( ) d¯∗ σ2da = 0 = ⇒ dY ≡ − ¯∗ σ2da . (11 )
By virtue of Eq. (10View Equation), the (locally defined) function Y is a potential for the twist one-form, dY = 2ω. In order to write the effective action (9View Equation) 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 (9View Equation) becomes
∫ ( ) Seff = ¯∗ R¯− ⟨dσ-, dσ-⟩ +-⟨dY-, dY-⟩ , (12 ) 2σ2
where we recall that ⟨ , ⟩ is the inner product with respect to the three-metric ¯g defined in Eq. (8View Equation).

The action (12View Equation) describes a harmonic mapping into a two-dimensional target space, effectively coupled to three-dimensional gravity. In terms of the complex Ernst potential E [52Jump To The Next Citation Point53Jump To The Next Citation Point], one has

∫ ( ¯ ) S = ¯∗ R¯ − 2⟨dE-,-dE⟩- , E ≡ σ + iY . (13 ) 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, 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

E ↦→ z = 1-−-E-, (14 ) 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, 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
( ) ( ) x0 x1 + ix2 ---1---- 1 + |z|2 2z Φ (x) = x1 − ix2 x0 = 1 − |z|2 2z¯ 1 + |z|2 . (15 )
Hence, the effective action for stationary vacuum gravity becomes the standard action for a σ-model coupled to three-dimensional gravity [139Jump To The Next Citation Point],
∫ ( ) S = ¯∗ R¯− 1-Trace⟨Φ −1dΦ , Φ −1dΦ ⟩ . (16 ) eff 4

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

¯g = dr2 + ρ2(r)dΩ2, (17 )
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.
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