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David,
Being a good professor, you’ve given me exercises to perform. As you say, they “are far from being as trivial as [my] other thought experiments.†The most important point remains, though, that the behavior of the test particles in both exercises are distinguishable from test particles in a box being accelerated by a force. That, after all, remains the issue between us.
In the first exercise, I (we) are asked to “consider a box in orbit around the Earth. Let's start by considering the box to be orbiting in such a way as to maintain one side always facing the Earth, so we have the "floor" being the side facing the Earth, the "ceiling" facing away from the Earth, the "front" facing the direction of "travel" of the orbiting box, in opposition we have the "back", with the "left" side being on the left and the "right" side on the right of an astronaut facing the "front". The question, then, is what will be the motions of free neutral test particles within this box? Especially, what will be the motions of the subset of such particles that don't hit any of the sides of the box?â€
Test particles that are placed in the box with relative motion will of course bounce around the box. Test particles that are placed in the box without relative motion will begin to separate slightly toward the walls, depending on their placement. Those that are closer to the “floor†will tend more toward the “floor†and the “frontâ€, as their geodesics have greater curvature than the box as a whole, and their velocity is greater than is required for a stable orbit at their relative elevation from the earth. The influence on particles closer to the “ceiling†will be opposite. The behavior of particles in the spherical container will have a similar separation, and the characteristics of their separation can indicate the direction of the earth.
In all cases, the tendency to separation will be different from particles in a box or sphere being accelerated by a force – the latter will have the same apparent acceleration toward the walls no matter where they are placed (although they may separate or converge if the force is centrifugal or centripetal). Can we return to that point?
Jim