6.1 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 ≈ ℝ 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 an adapted coordinate t, so that k := ∂t, the metric of a stationary spacetime can be parameterized in terms of a three-dimensional (Riemannian) metric ¯g := ¯gijdxidxj, a one-form i a := aidx, and a scalar field V, where stationarity implies that ¯gij, ai and V are functions on (Σ, ¯g):

2 1- g = − V (dt + a) + V ¯g. (6.1 )

The notation t suggests that t is a time coordinate, g(∇t, ∇t) < 0, but this restriction does not play any role in the local form of the equations that we are about to derive. Similarly the local calculations that follow remain valid regardless of the causal character of k, provided that k is not null everywhere, and then one only considers the region where g(k, k) ≡ − V does not change sign. On any connected component of this region k is either spacelike or timelike, as determined by the sign of V, and then the metric ¯g is Lorentzian, respectively Riemannian, there. In any case, both the parameterization of the metric and the equations become singular at places where V has zeros, so special care is required wherever this occurs.

Using Cartan’s structure equations (see, e.g., [300]), it is a straightforward task to compute the Ricci scalar for the above decomposition of the spacetime metric; see, e.g., [155] for the details of the derivation. The result is that the Einstein–Hilbert action of a stationary spacetime reduces to the action for a scalar field V and a 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 (6.1View Equation). The vacuum field equations are thus seen to be equivalent to the three-dimensional Einstein-matter equations obtained from variations of the effective action

∫ ( 2 ) ¯ ¯ -1-- V-- Seff = ∗ R − 2V 2⟨dV ,dV ⟩ + 2 ⟨da, da⟩ , (6.2 )
with respect to ¯gij, V and a. Here and in the following ¯R denotes the Ricci scalar of ¯g, while for p-forms α and β, their inner product is defined by ¯∗ ⟨α,β⟩ := α ∧ ¯∗β, where ¯∗ is the Hodge dual with respect to ¯g.

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

V = − g(k, k), ω := 1-∗ (k ∧ dk ) = − 1V 2¯∗da, (6.3 ) 2 2
where ∗ and ¯∗ denote the Hodge dual with respect to g and ¯g, respectively. Here and in the following we use the symbol k for both the Killing field ∂t and the corresponding one-form − V (dt + a). One can view a as a connection on a principal bundle with base space Σ and fiber ℝ, since it behaves like an Abelian gauge potential under coordinate transformations of the form t → t + φ (xi). Not surprisingly, it enters the effective action in a gauge-invariant way, that is, only via the “Abelian field strength”, f := da.
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