Reduction of the Einstein–Hilbert action

4.2 Reduction of the Einstein–Hilbert action

By definition, a stationary spacetime $(M, g)$ admits an asymptotically time-like Killing field, that is, a vector field $k$ with $Lkg = 0$ , $Lk$ denoting the Lie derivative with respect to $k$ . At least locally, $M$ has the structure $Σ × G$ , where $G ≈ IR$ denotes the one-dimensional group generated by the Killing symmetry, and $Σ$ is the three-dimensional quotient space $M ∕G$ . A stationary spacetime is called static, if the integral trajectories of $k$ are orthogonal to $Σ$ .

With respect to the adapted time coordinate $t$ , defined by $k ≡ ∂t$ , the metric of a stationary spacetime is parametrized in terms of a three-dimensional (Riemannian) metric $g¯≡ ¯g dxidxj ij$ , a one-form $i a ≡ aidx$ , and a scalar field $σ$ , where stationarity implies that $¯gij$ , $ai$ and $σ$ are functions on $(Σ, ¯g)$ :

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 metric³⁹ . The result shows that the Einstein–Hilbert action of a stationary spacetime reduces to the action for a scalar field $σ$ 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 $¯g$ 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 $¯gij$ , $σ$ and $a$ . (Here and in the following $¯R$ and $⟨ , ⟩$ denote the Ricci scalar and the inner product⁴⁰ with respect to $¯g$ .)

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

where $∗$ and $¯∗$ denote the Hodge dual with respect to $g$ and $¯g$ , respectively⁴¹ . Since $a$ is the connection of a fiber bundle with base space $Σ$ and fiber $G$ , it behaves like an Abelian gauge potential under coordinate transformations of the form $t → t + φ (xi)$ . Hence, it enters the effective action in a gauge-invariant way, that is, only via the “Abelian field strength”, $f ≡ da$ .