2.2 Kaluza–Klein asymptotic flatness

There exists a generalization of the notion of asymptotic flatness, which is relevant to both four- and higher-dimensional gravitation. We shall say that 𝒮ext is a Kaluza–Klein asymptotic end if 𝒮ext is diffeomorphic to ( ) ℝN ∖ B¯(R) × Q, where B¯(R) is a closed coordinate ball of radius R and Q is a compact manifold of dimension s ≥ 0; a spacetime containing such an end is said to have N + 1 asymptotically-large dimensions. Let ˚ h be a fixed Riemaniann metric on Q, and let ˚ ˚g = δ ⊕ h, where δ is the Euclidean metric on N ℝ. A spacetime (M, g) containing such an end will be said to be Kaluza–Klein asymptotically flat, or KK-asymptotically flat if, for some α > 0, the metric γ induced by g on 𝒮ext and the extrinsic curvature tensor Kij of 𝒮ext, satisfy the fall-off conditions
−α− l − 1− α−l γij − ˚gij = Od (r ), Kij = Od− 1(r ), (2.4 )
where, in this context, r is the radius in ℝN and we write f = Od (rα) if f satisfies
D˚i ...˚Di f = O(rα− l), 0 ≤ ℓ ≤ d, (2.5 ) 1 l
with D˚ the Levi-Civita connection of ˚g.
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