  We study the near-horizon spacetime for isolated and dynamical trapping
horizons (equivalently marginally outer trapped tubes). The metric is expanded
relative to an ingoing Gaussian null coordinate and the terms of that expansion
are explicitly calculated to second order. For the spacelike case, knowledge of
the intrinsic and extrinsic geometry of the (dynamical) horizon is sufficient
to determine the near-horizon spacetime, while for the null case (an isolated
horizon) more information is needed. In both cases spacetime is allowed to be
of arbitrary dimension and the formalism accomodates both general relativity as
well as more general field equations.
  The formalism is demonstrated for two applications. First, spacetime is
considered near an isolated horizon and the construction is both checked
against the Kerr-Newman solution and compared to the well-known near-horizon
limit for stationary extremal black hole spacetimes. Second, spacetime is
examined in the vicinity of a slowly evolving horizon and it is demonstrated
that there is always an event horizon candidate in this region. The geometry
and other properties of this null surface match those of the slowly evolving
horizon to leading order and in this approximation the candidate evolves in a
locally determined way. This generalizes known results for Vaidya as well as
certain spacetimes known from studies of the fluid-gravity correspondence.
