  Gravitational wave detectors are typically described as responding to
gravitational wave metric perturbations, which are gauge-dependent and ---
correspondingly --- unphysical quantities. This is particularly true for
ground-based interferometric detectors, like LIGO, space-based detectors, like
LISA and its derivatives, spacecraft doppler tracking detectors, and pulsar
timing arrays detectors. The description of gravitational waves, and a
gravitational wave detector's response, to the unphysical metric perturbation
has lead to a proliferation of false analogies and descriptions regarding how
these detectors function, and true misunderstandings of the physical character
of gravitational waves.
  Here we provide a fully physical and gauge invariant description of the
response of a wide class of gravitational wave detectors in terms of the
Riemann curvature, the physical quantity that describes gravitational phenomena
in general relativity. In the limit of high frequency gravitational waves, the
Riemann curvature separates into two independent gauge invariant quantities: a
"background" curvature contribution and a "wave" curvature contribution. In
this limit the gravitational wave contribution to the detector response reduces
to an integral of the gravitational wave contribution of the curvature along
the unperturbed photon path between components of the detector. The description
presented here provides an unambiguous physical description of what a
gravitational wave detector measures and how it operates, a simple means of
computing corrections to a detectors response owing to general detector motion,
a straightforward way of connecting the results of numerical relativity
simulations to gravitational wave detection, and a basis for a general and
fully relativistic pulsar timing formula.
