5 Applications of the Coset 4 Stationary Space-Times4.4 Stationary Gauge Fields

4.5 The Stationary Einstein-Maxwell System 

In the Abelian case, both the off-diagonal Einstein equation (22Popup Equation) and the Maxwell equation (23Popup Equation) give rise to scalar potentials, (locally) defined by

  equation694

Like for the vacuum system, this enables one to apply the Lagrange multiplier method in order to express the effective action in terms of the scalar fields Y and tex2html_wrap_inline4151, rather than the one-forms a and tex2html_wrap_inline4039 . As one is often interested in the dimensional reduction of the EM system with respect to a space-like Killing field, we give here the general result for an arbitrary Killing field tex2html_wrap_inline3669 with norm N :

  equation699

where tex2html_wrap_inline4161, etc. The electro-magnetic potentials tex2html_wrap_inline4045 and tex2html_wrap_inline4151 and the gravitational scalars N and Y are obtained from the four-dimensional field strength F and the Killing field (one form) as follows: Popup Footnote

  equation712

  equation717

where tex2html_wrap_inline4179 . The inner product tex2html_wrap_inline3847 is taken with respect to the three-metric tex2html_wrap_inline3837, which becomes pseudo-Riemannian if tex2html_wrap_inline3669 is space-like. In the stationary and axisymmetric case, to be considered in Sect. 6, the Kaluza-Klein reduction will be performed with respect to the space-like Killing field. The additional stationary symmetry will then imply that the inner products in (25Popup Equation) have a fixed sign, despite the fact that tex2html_wrap_inline3837 is not a Riemannian metric in this case.

The action (25Popup Equation) describes a harmonic mapping into a four-dimensional target space, effectively coupled to three-dimensional gravity. In terms of the complex Ernst potentials, tex2html_wrap_inline4189 and tex2html_wrap_inline4191 [52Jump To The Next Citation Point In The Article], [53Jump To The Next Citation Point In The Article], the effective EM action becomes

  equation731

where tex2html_wrap_inline4193 . The field equations are obtained from variations with respect to the three-metric tex2html_wrap_inline3837 and the Ernst potentials. In particular, the equations for tex2html_wrap_inline3933 and tex2html_wrap_inline4199 become

  equation745

where tex2html_wrap_inline4201 . The isometries of the target manifold are obtained by solving the respective Killing equations [139] (see also [107], [108], [109], [110]). This reveals the coset structure of the target space and provides a parametrization of the latter in terms of the Ernst potentials. For vacuum gravity we have seen in Sect. 4.3 that the coset space, G / H, is SU (1,1)/ U (1), whereas one finds tex2html_wrap_inline4207 for the stationary EM equations. If the dimensional reduction is performed with respect to a space-like Killing field, then tex2html_wrap_inline4209 . The explicit representation of the coset manifold in terms of the above Ernst potentials, tex2html_wrap_inline3933 and tex2html_wrap_inline4199, is given by the hermitian matrix tex2html_wrap_inline4215, with components

  equation764

where tex2html_wrap_inline4219 is the Kinnersley vector [106], and tex2html_wrap_inline4221 . It is straightforward to verify that, in terms of tex2html_wrap_inline4215, the effective action (28Popup Equation) assumes the SU (2,1) invariant form

  equation777

where tex2html_wrap_inline4227 . The equations of motion following from the above action are the three-dimensional Einstein equations (obtained from variations with respect to tex2html_wrap_inline3837) and the tex2html_wrap_inline3625 -model equations (obtained from variations with respect to tex2html_wrap_inline4215):

  equation809

By virtue of the Bianchi identity, tex2html_wrap_inline4235, and the definition tex2html_wrap_inline4237, the tex2html_wrap_inline3625 -model equations are the integrability conditions for the three-dimensional Einstein equations.



5 Applications of the Coset 4 Stationary Space-Times4.4 Stationary Gauge Fields

image Stationary Black Holes: Uniqueness and Beyond
Markus Heusler
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