### 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 by a function, namely the twist potential. We have already pointed out that , parameterizing the non-static part of the metric, enters the effective action (6.2) only via the field strength, . For this reason, the variational equation for (that is, the off-diagonal Einstein equation) takes in vacuum the form of a source-free Maxwell equation:
By virtue of Eq. (6.3), the (locally-defined) function is a potential for the twist one-form, . In order to write the effective action (6.2) 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 (6.2) becomes
where we recall that is the inner product with respect to the three-metric defined in Eq. (6.1).

The action (6.5) describes a harmonic map into a two-dimensional target space, effectively coupled to three-dimensional gravity. In terms of the complex Ernst potential  [102, 103], 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, parameterized by the Ernst potential , is a Kähler manifold with metric (see [115] 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 (see, e.g., [196] or [26] for the general theory of symmetric spaces). Using the universal covering of , one can parameterize 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 [250],
where
and the currents are defined as

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: , .