  Electromagnetic fields which solve the vacuum Maxwell equations in one
spacetime are well-known to also be solutions in all spacetimes with
conformally-related metrics. This provides a sense in which electromagnetism
alone cannot be used to measure certain aspects of geometry. We show that there
is actually much more which cannot be so measured; relatively little of a
spacetime's geometry is in fact imprinted in any particular electromagnetic
field. This is demonstrated by finding a much larger class of metric
transformations---involving five free functions---which preserve Maxwell
solutions both in vacuum, without local currents, and also for the force-free
electrodynamics associated with a tenuous plasma. One consequence of this is
that many of the exact force-free fields which have previously been found
around Schwarzschild and Kerr black holes are also solutions in
appropriately-identified flat backgrounds. As a more direct application, we use
our metric transformations to write down a large class of electromagnetic waves
which remain unchanged by a large class of gravitational waves propagating "in
the same direction."
