Frames of rigid objects

By definition, every reference frame is fixed to a rigid object. That is, the particles of the rigid object are at rest relative to the reference frame.

Note: to lend "fixed" a meaning for non-inertial frames too, we can assume that time is "passing" at every single location of any reference frame.

In special relativity, the scope is limited to inertial frames. So let K denote an inertial frame. Then, what was said above means that for every inertial frame, there exists an underlying rigid object that is moving uniformly relative to K. To round off this relationship, it is reasonable to assume the converse too:

(Assumption) To every rigid object O that is moving uniformly relative to K, there exists an inertial frame relative to which O is at rest.

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