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):

2 1 g = − σ(dt + a) + --¯g. (8 ) σ
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 (8View Equation). The vacuum field equations are, therefore, equivalent to the three-dimensional Einstein-matter equations obtained from variations of the effective action
∫ ( 1 σ2 ) Seff = ¯∗ R¯ − ---2 ⟨dσ , d σ⟩ +--⟨da, da⟩ , (9 ) 2σ 2
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:

1 1 σ = − g(k,k ), ω ≡ --∗ (k ∧ dk) = −--σ2¯∗da , (10 ) 2 2
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.
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