= Solution
Take the real physical wave and write $h(t,z)=A\cos[k(z-ct)]$. The specified amplitude has zero trace, so its <metric perturbation> equals its <trace-reversed metric perturbation>. It is in <transverse-traceless gauge>:
$$
ds^2=-c^2dt^2+[1+h(t,z)]dx^2+[1-h(t,z)]dy^2+dz^2
$$
to first order. Assume the freely falling particles are initially at rest in these coordinates. Since $g_{00}=-1$ and $g_{0i}=0$ in coordinates $(ct,x,y,z)$,
$$
\Gamma^i{}_{00}
=\frac12g^{ij}(2\partial_0g_{0j}-\partial_jg_{00})=0.
$$
The <geodesic equations> therefore admit
$$
\boxed{x=x_0,\qquad y=y_0,\qquad z=z_0,\qquad t=\tau.}
$$
Uniqueness preserves these initial conditions. Thus the coordinate locations remain fixed; this conclusion does not apply to arbitrary initially moving particles.
Coordinate position is not measured separation. In the central observer's local <orthonormal frame in spacetime>, and in the long-wavelength limit appropriate to a small detector, the transverse physical separations are
$$
X=\left(1+\frac h2\right)x,\qquad
Y=\left(1-\frac h2\right)y
$$
to first order. In particular the two principal semiaxes of the ring are
$$
\boxed{
a_x=a\left(1+\frac h2\right),\qquad
a_y=a\left(1-\frac h2\right).
}
$$
For a point at angular position $\varphi$ the radial displacement is
$$
\delta\ell(\varphi,t)=\frac{a h(t,z_0)}2\cos(2\varphi).
$$
The <freely falling ring in transverse-traceless gauge> alternates between ellipses elongated in the $x$ and $y$ directions. Its area is unchanged to first order. This is the plus <gravitational wave polarization>; the axes exchange their roles after half a period.
\Image[/past-exam-of-the-general-relativity-and-cosmology-course-of-the-university-of-oxford/2017-gravitational-wave-ring.png]
{title=Physical distances in a freely falling ring under plus polarization, with exaggerated strain}
An observer at the center detects these changing distances, for example by exchanging light signals with the particles. Equivalently <geodesic deviation> gives the local <tidal acceleration>
$$
\ddot X=\frac12\ddot h\,X,\qquad
\ddot Y=-\frac12\ddot h\,Y
$$
to the same order. There is physical relative motion and nonzero <Riemann curvature tensor>, even though the selected coordinate positions are constant. For a detector comparable to a wavelength, its light-travel-time response must be included rather than treating all distances as simultaneously measured local lengths.
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