2.1 Asymptotic flatness

We will mostly be concerned with asymptotically-flat black holes. A spacetime (M, g) will be said to possess an asymptotically-flat end if M contains a spacelike hypersurface 𝒮 ext diffeomorphic to ℝn ∖ B (R ), where B (R) is an open coordinate ball of radius R, with the following properties: there exists a constant α > 0 such that, in local coordinates on 𝒮ext obtained from n ℝ ∖ B (R ), the metric γ induced by g on 𝒮ext, the extrinsic curvature tensor Kij of 𝒮ext, and the electromagnetic potential Aμ satisfy the fall-off conditions
γ − δ = O (r−α), K = O (r− 1− α), (2.1 ) ij ij d ij d− 1
and
Aμ = Od (r−α), (2.2 )
for some d > 1, where we write α f = Od (r ) if f satisfies
α−ℓ ∂i1 ...∂iℓf = O (r ), 0 ≤ ℓ ≤ d. (2.3 )

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