4.4 Stationary Gauge Fields4 Stationary Space-Times4.2 Reduction of the Einstein-Hilbert

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 (9Popup Equation) only via the field strength, tex2html_wrap_inline3891 . For this reason, the variational equation for a (that is, the off-diagonal Einstein equation) assumes the form of a source-free Maxwell equation,

  equation506

By virtue of Eq. (10Popup Equation), the (locally defined) function Y is a potential for the twist one-form, tex2html_wrap_inline3903 . In order to write the effective action (9Popup Equation) in terms of the twist potential Y, rather than the one-form a, one considers tex2html_wrap_inline3891 as a fundamental field and imposes the constraint tex2html_wrap_inline3911 with the Lagrange multiplier Y . The variational equation with respect to f then yields tex2html_wrap_inline3917, which is used to eliminate f in favor of Y . One finds tex2html_wrap_inline3923 tex2html_wrap_inline3925 tex2html_wrap_inline3927 . Thus, the action (9Popup Equation) becomes

  equation520

where we recall that tex2html_wrap_inline3847 is the inner product with respect to the three-metric tex2html_wrap_inline3837 defined in Eq. (8Popup Equation).

The action (12Popup Equation) describes a harmonic mapping into a two-dimensional target space, effectively coupled to three-dimensional gravity. In terms of the complex Ernst potential tex2html_wrap_inline3933 [52Jump To The Next Citation Point In The Article], [53Jump To The Next Citation Point In The Article], one has

  equation535

The stationary vacuum equations are obtained from variations with respect to the three-metric tex2html_wrap_inline3837 [(ij)-equations] and the Ernst potential tex2html_wrap_inline3933 [tex2html_wrap_inline3941 -equations]. One easily finds tex2html_wrap_inline3943 and tex2html_wrap_inline3945, where tex2html_wrap_inline3947 is the Laplacian with respect to tex2html_wrap_inline3837 .

The target space for stationary vacuum gravity, parametrized by the Ernst potential tex2html_wrap_inline3933, is a Kähler manifold with metric tex2html_wrap_inline3953 (see [60] for details). By virtue of the mapping

  equation557

the semi-plane where the Killing field is time-like, tex2html_wrap_inline3955, is mapped into the interior of the complex unit disc, tex2html_wrap_inline3957, with standard metric tex2html_wrap_inline3959 . By virtue of the stereographic projection, tex2html_wrap_inline3961, tex2html_wrap_inline3963, the unit disc D is isometric to the pseudo-sphere, tex2html_wrap_inline3967 . 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 tex2html_wrap_inline3973 Popup Footnote . Using the universal covering SU (1,1) of SO (2,1), one can parametrize tex2html_wrap_inline3979 in terms of a positive hermitian matrix tex2html_wrap_inline3981, defined by

  equation572

Hence, the effective action for stationary vacuum gravity becomes the standard action for a tex2html_wrap_inline3625 -model coupled to three-dimensional gravity [139Jump To The Next Citation Point In The Article],

  equation585

The simplest nontrivial solution to the vacuum Einstein equations is obtained in the static, spherically symmetric case: For tex2html_wrap_inline3985 one has tex2html_wrap_inline3987 and tex2html_wrap_inline3989 . With respect to the general spherically symmetric ansatz

  equation596

one immediately obtains the equations tex2html_wrap_inline3993 and tex2html_wrap_inline3995, the solution of which is the Schwarzschild metric in the usual parametrization: tex2html_wrap_inline3997, tex2html_wrap_inline3999 .



4.4 Stationary Gauge Fields4 Stationary Space-Times4.2 Reduction of the Einstein-Hilbert

image Stationary Black Holes: Uniqueness and Beyond
Markus Heusler
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