But, for an object on the Earth's surface, the Earth's surface is a proper frame while the falling elevator is not a proper frame.
The use of a proper frame is essential for the investigation of physical laws within the framework of general relativity.
As our first example of a proper frame, one uses the following frame to find the Earth: The Earth is situated in the center with respect to the observer (or our point of reference) of our next example, the Sun is at the bottom.
𝜕 can be thought of as the set of sets (including proper frames) of all possible motions of a given object, such that a proper frame always results.
𝜕 can be thought of as the unique set of frames that are conserved under gravity, allowing that the particles of gravitation do not collapse on an object after the initial contact (for example, they remain in the frame they have been suspended in).