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Tidal locking, when the spin rate of a body matches the orbital rate so that it always faces the other body with the same side, usually occurs for just one of the bodies in orbit. However, there are also systems in which both bodies are tidally locked to one another (e.g. Charon around Pluto), see animation.

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Consider an universe where these two bodies are the only bodies, so that there are no external reference points. If an observer is on one of the bodies, the system will appear frozen in its reference frame (i.e. the other body is always in the same spot in the sky showing the same face).

Is there a way the observer can deduce that the bodies are in orbit without simply arguing "gravity exists, we are not falling towards each other, hence we must be in orbit" * I can imagine a civilization in such a scenario would have a harder time understanding gravity in the first place ;)

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There would be Coriolis and centrifugal forces due to the rotation of the reference frame. These could be detected. For example, a Foucault pendulum with its vertical axis perpendicular to the plane of the orbit would precess once per “year”.

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