  In this work we analyse asymptotically flat, spherically symmetric spacetimes
in which an event horizon is present without any trapped surfaces. We identify
two types of such spacetimes, each related to the asymptotic behaviour (in
time) of one of the two degrees of freedom of the metric. We study the causal
structure of both types, showing that one almost always has a Cauchy horizon
beyond which it is extendable, while the other is inextendable but has two
separate future null infinity regions on either side of the horizon. We also
study what energy conditions can be satisfied by the matter around the horizon.
Some of these spacetimes were first introduced in an earlier work in which
semiclassical effects near black-hole horizons were analysed. Here we
generalise this analysis to a larger family of geometries.
