The relative motion of two test masses in a gravitational wave spacetime can be derived in the long wave-length limit, e.g., from the geodesic deviation equation, the result (for plus mode) is of the form $\ddot x\approx \frac{1}{2}\ddot h(t) x$, where x is the two-body seperation. What if the two bodies are connected with a spring (with spring constant $k$)? Naively $\ddot x\approx (-k^2+\frac{1}{2}\ddot h(t)) x$, and there can be parametric resonance if $h(t)\propto \sin t$. But this is incorrect since the test masses no longer follow the geodesic equation. What is the correct equation of motion? Would parametric resonance be possible?
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$\begingroup$ How did you arrive at this equation? If you consider two particles with the seperation $\delta x$ between them and the first at $x$, then the geodesic equation for $x$ and $x+\delta x$ yield the approximation (for example $\delta x$ is obmitted in second order) $\ddot{\delta x}^\sigma+2\Gamma_{\mu\nu}^\sigma(x)\dot{\delta x}^\mu x^\nu=0$. How did you get your equation out of this equation? Seems a bit odd to me as the geodesic equation relates $\ddot{\delta x}$ with $\dot{\delta x}$, while yours relates $\ddot{\delta x}$ with $\delta x$. $\endgroup$– Samuel Adrian AntzCommented Mar 4, 2023 at 16:00
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