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I have a simple apparatus: a box and inside it a test particle (small bullet). If the device is in inertial motion or affected only but (static) gravitational field of another massive body (in free fall to the Earth, on orbit around Sun etc.), the test particle doesn't move (or moves with constant velocity) relative to the box.

I assume that if a gravitational wave is passing through the box, I can't detect the wave with such a apparatus. In other words, the motion of the test particle relative the box is unaffected by the wave. Am I right?

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The condition for "free fall" is as follows:

1) estimate the size of the radius of curvature in your spacetime region. if we want a lazy geometric invariant that gives us a decent order of magnitude, we can compute $r_{c} = \left(R^{abcd}R_{abcd}\right)^{-\frac{1}{4}}$

2) now, our reference frame is a "free falling" reference frame only if the length scale $\ell$ of that reference frame is much smaller than $r_{c}$ AND we measure it for timescales less than $r_{c}/c$. Otherwise, Geodesic deviation will make our test particle paths diverge and no longer will they approximate special relativity.

Note that this makes your "free fall measurement time" pretty small if $r_{c}$ is at all macroscopic. So, for gravitational waves big enough to detect, the reference frame of the detector will not be "free falling" in this sense.

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  • $\begingroup$ Thank for the answer, though I don't understand it much. I thought that devices like LISA Pathfinder relies on the idea (motion of a small test body relative to the container is not affected by gravitational wave). $\endgroup$
    – Leos Ondra
    May 18, 2019 at 19:18

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