  We consider local energy decay estimates for solutions to scalar wave
equations on nontrapping asymptotically flat space-times. Our goals are
two-fold. First we consider the stationary case, where we can provide a full
spectral characterization of local energy decay bounds; this characterization
simplifies in the stationary symmetric case. Then we consider the almost
stationary, almost symmetric case. There we establish two main results: The
first is a "two point" local energy decay estimate which is valid for a general
class of (non-symmetric) almost stationary wave equations which satisfy a
certain nonresonance property at zero frequency. The second result, which also
requires the almost symmetry condition, is to establish an exponential
trichotomy in the energy space via finite dimensional time dependent stable and
unstable sub-spaces, with an infinite dimensional complement on which solutions
disperse via the usual local energy decay estimate.
