4.2 Stationary toroidal Kaluza–Klein black holes

The four-dimensional vacuum Einstein equations simplify considerably in the stationary and axisymmetric setting by reducing to a harmonic map into the hyperbolic plane (see Sections 8 and 3.2.3). A similar such reduction in (n + 1 )-dimensions works when the isometry group includes โ„ × ๐•‹n− 2, i.e., besides the stationary vector there exist n − 2 commuting axial Killing vectors.

Since the center ๐•‹s of SO (n) has dimension

⌊ n⌋ s = 2 ,
in the asymptotically flat case the existence of such a group of isometries is only possible for n = 3 or n = 4. However, one can move beyond the usual asymptotic-flatness and consider instead KK-asymptotically-flat spacetimes, in the sense of Section 2.2, with asymptotic ends ๐’ฎext ≈ (โ„N โˆ– B) × ๐•‹s, satisfying N = 3,4 and N + s = n, with the isometry group containing n−2 โ„ × ๐•‹, n ≥ 3. Here one takes ( N ¯ ) s ๐’ฎext ≈ โ„ โˆ–B (R ) × ๐•‹, with the reference metric of the form หšg = δ ⊕ หšh, where หšh is the flat s-torus metric. Finally, the action of โ„ × ๐•‹n−2 on (M, g ) by isometries is assumed in the exterior region Mext ≈ โ„ × ๐’ฎext to take the form
โ„ × ๐•‹n−2 ∋ (τ,g) : (t,p) โ†ฆ→ (t + τ,g ⋅ p). (4.1 )
Such metrics will be referred to as stationary toroidal Kaluza–Klein metrics.
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