 Freely falling ring in transverse-traceless gauge

In a weak plus-polarized gravitational wave, initially stationary test particles have constant spatial coordinates in transverse-traceless gauge because ${\Gamma }^i{}_{00}=0$. Their physical separations nevertheless change. For a small transverse ring, $X=(1+h_{+}/2)x$ and $Y=(1-h_{+}/2)y$, so its principal semiaxes oscillate as $a(1\pm h_{+}/2)$. The radial response has the displayed angular dependence. These are local distance measurements in the long-wavelength limit; finite-arm light-travel times require the full detector response.