4.3 The Coset Structure of 4 Stationary Space-Times4.1 Killing Horizons

4.2 Reduction of the Einstein-Hilbert Action 

By definition, a stationary spacetime tex2html_wrap_inline3665 admits an asymptotically time-like Killing field, that is, a vector field k with tex2html_wrap_inline3791, tex2html_wrap_inline3793 denoting the Lie derivative with respect to k . At least locally, M has the structure tex2html_wrap_inline3799, where tex2html_wrap_inline3801 denotes the one-dimensional group generated by the Killing symmetry, and tex2html_wrap_inline3803 is the three-dimensional quotient space M / G . A stationary spacetime is called static, if the integral trajectories of k are orthogonal to tex2html_wrap_inline3803 .

With respect to the adapted time coordinate t, defined by tex2html_wrap_inline3813, the metric of a stationary spacetime is parametrized in terms of a three-dimensional (Riemannian) metric tex2html_wrap_inline3815, a one-form tex2html_wrap_inline3817, and a scalar field tex2html_wrap_inline3625, where stationarity implies that tex2html_wrap_inline3821, tex2html_wrap_inline3823 and tex2html_wrap_inline3625 are functions on tex2html_wrap_inline3827 :

  equation467

Using Cartan's structure equations (see, e.g. [165]), it is a straightforward task to compute the Ricci scalar for the above decomposition of the spacetime metric Popup Footnote . The result shows that the Einstein-Hilbert action of a stationary spacetime reduces to the action for a scalar field tex2html_wrap_inline3625 and an Abelian vector field a, which are coupled to three-dimensional gravity. The fact that this coupling is minimal is a consequence of the particular choice of the conformal factor in front of the three-metric tex2html_wrap_inline3837 in the decomposition (8Popup Equation). The vacuum field equations are, therefore, equivalent to the three-dimensional Einstein-matter equations obtained from variations of the effective action

  equation476

with respect to tex2html_wrap_inline3821, tex2html_wrap_inline3625 and a . (Here and in the following tex2html_wrap_inline3845 and tex2html_wrap_inline3847 denote the Ricci scalar and the inner product Popup Footnote with respect to tex2html_wrap_inline3837 .)

It is worth noting that the quantities tex2html_wrap_inline3625 and a are related to the norm and the twist of the Killing field as follows:

  equation495

where tex2html_wrap_inline3869 and tex2html_wrap_inline3857 denote the Hodge dual with respect to tex2html_wrap_inline3873 and tex2html_wrap_inline3837, respectively Popup Footnote . Since a is the connection of a fiber bundle with base space tex2html_wrap_inline3803 and fiber G, it behaves like an Abelian gauge potential under coordinate transformations of the form tex2html_wrap_inline3889 . Hence, it enters the effective action in a gauge-invariant way, that is, only via the ``Abelian field strength'', tex2html_wrap_inline3891 .



4.3 The Coset Structure of 4 Stationary Space-Times4.1 Killing Horizons

image Stationary Black Holes: Uniqueness and Beyond
Markus Heusler
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