Is clause "distance doesn't depend on frame of reference" an axiom in Newtonian Mechanics? Consider 2 object is 1 and 2, at time t1: 1 has position is C and 2 has position is A.

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*In frame of reference 1 (1 is stand still), from time t1 to time t2, 2 moves from A to B

*In frame of reference 2 (2 is stand still), from time t1 to time t2, 1 moves from C to D.

My textbook said that, the distance CB (in frame 1) equals to the distance DA (in frame 2).
So, distance doesn't depend on frame of reference in Newtonian Mechanics. My question is : Is this an axiom or it can be proved ???

 A: It can be proved. A change of reference frame in classical mechanics is mathematically described by a Galilean transformation. One can show that Newtons Laws are invariant under such transformations. Galilean transformations form a group. Every element of that group can be written as a product of a rotation, translation and a uniform motion. It is easy to show, and also intuitive, that all of these operations leave distances invariant.
In special relativity, the Galilean group is replaced by the Poincare group, which does not leave distances (at least the distances in the three spatial dimensions) invariant. That is why in special relativity distance depends on the frame of reference.
A: The preservation of length between reference frames that are moving at a constant speed relative to one another may be proven from the Galilean transformation, which is the classical way to translate coordinates from one inertial frame to another.  But the Galilean transformation is itself an assumption of the theory.
Ultimately, proof in physics relies on experiment.  There is good experimental evidence that length is not precisely preserved when the constant relative speed of the two frames is not zero (see length contraction), although in very many cases of practical interest the difference is negligible.
