### 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 can be replaced by a function (twist potential). We have already pointed out that , parametrizing the non-static part of the metric, enters the effective action (9) only via the field strength, . For this reason, the variational equation for (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 is a potential for the twist one-form, . In order to write the effective action (9) in terms of the twist potential , rather than the one-form , one considers as a fundamental field and imposes the constraint with the Lagrange multiplier . The variational equation with respect to then yields , which is used to eliminate in favor of . One finds . Thus, the action (9) becomes
where we recall that is the inner product with respect to the three-metric 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  [5253], one has

The stationary vacuum equations are obtained from variations with respect to the three-metric [-equations] and the Ernst potential [-equations]. One easily finds and , where is the Laplacian with respect to .

The target space for stationary vacuum gravity, parametrized by the Ernst potential , is a Kähler manifold with metric (see [62] for details). By virtue of the mapping

the semi-plane where the Killing field is time-like, , is mapped into the interior of the complex unit disc, , with standard metric . By virtue of the stereographic projection, , , the unit disc is isometric to the pseudo-sphere, . As the three-dimensional Lorentz group, , acts transitively and isometrically on the pseudo-sphere with isotropy group , the target space is the coset . Using the universal covering of , one can parametrize in terms of a positive hermitian matrix , defined by
Hence, the effective action for stationary vacuum gravity becomes the standard action for a -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 one has and . With respect to the general spherically symmetric ansatz

one immediately obtains the equations and , the solution of which is the Schwarzschild metric in the usual parametrization: , .