= Freely falling ring in transverse-traceless gauge
{title2=$\delta\ell/a=\tfrac12h_+\cos(2\varphi)$}
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.
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