2.1 Asymptotic flatness
We will mostly be concerned with asymptotically-flat black holes. A spacetime
will be
said to possess an asymptotically-flat end if
contains a spacelike hypersurface
diffeomorphic to
, where
is an open coordinate ball of radius
, with the
following properties: there exists a constant
such that, in local coordinates on
obtained from
, the metric
induced by
on
, the extrinsic curvature
tensor
of
, and the electromagnetic potential
satisfy the fall-off conditions
and
for some
, where we write
if
satisfies