  In an external electric or magnetic field, a gravitational wave (GW) may be
converted into electromagnetic radiation. We present a coordinate-invariant
framework to describe the GW signal in a detector that is based on this effect,
such as cavities for axion searches. In this framework, we pay special
attention to the definition of manifestly coordinate-independent expressions
for the electromagnetic fields that an external observer would detect. A
careful assessment of the detector's perceived motion allows us to treat both
its mechanical and its electromagnetic response to the GW consistently. We
further introduce well-defined approximations for which this motion may be
neglected, and hence provide suggestions on which coordinate frame is suitable
to characterise the GW signal in practice. We illustrate our findings in two
examples, an infinitesimally thin rod and a spherical electromagnetic cavity.
