### 4.2 Reduction of the Einstein–Hilbert action

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

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

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. The result shows that the Einstein–Hilbert action of a stationary spacetime reduces to the action for a scalar field and an Abelian vector field , 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 in the decomposition (8). The vacuum field equations are, therefore, equivalent to the three-dimensional Einstein-matter equations obtained from variations of the effective action
with respect to , and . (Here and in the following and denote the Ricci scalar and the inner product with respect to .)

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

where and denote the Hodge dual with respect to and , respectively. Since is the connection of a fiber bundle with base space and fiber , it behaves like an Abelian gauge potential under coordinate transformations of the form . Hence, it enters the effective action in a gauge-invariant way, that is, only via the “Abelian field strength”, .