  Typical applications of gravitational lensing use the properties of
electromagnetic or gravitational waves to infer the geometry through which
those waves propagate. Nevertheless, the optical fields themselves - as opposed
to their interactions with material bodies - encode very little of that
geometry: It is shown here that any given configuration is compatible with a
very large variety of spacetime metrics. For scalar fields in geometric optics,
or observables which are not sensitive to the detailed polarization content of
electromagnetic or gravitational waves, seven of the ten metric components are
essentially irrelevant. With polarization, five components are irrelevant. In
the former case, this result together with diffeomorphism invariance allows
essentially any geometric-optics configuration associated with a particular
spacetime to be embedded into any other spacetime, at least in finite regions.
Going beyond the geometric-optics approximation breaks some of this degeneracy,
although much remains even then. Overall, high-frequency wave propagation is
shown to be insensitive to compositions of certain conformal, Kerr-Schild, and
related transformations of the background metric. One application is that new
solutions for scalar, electromagnetic, and gravitational waves may be generated
from old ones. In one example described here, the high-frequency scattering of
a plane wave by a point mass is computed by transforming a plane wave in flat
spacetime.
