In Newtonian mechanics, by the following two assumptions:

(i) The time is absolute.

(ii) The length is absolute.

it is easy find the relations betweem two coordinate systems with uniform motion respect to each other (galilean transformations). Then is not complicated to prove that two inertial frames (i.e. two coordinate systems where holds the law of inertia) must move with constant velocity relative to each other.

In order to find the real relations between two inertial frames, in special relativity should assume the principle of constancy of the velocity of light. My question is:

Why two inertial frames of reference (in special relativity) moves with constant velocity relative to each other?

We only prove it in the context of Newtonian mechanics!

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    $\begingroup$ I think you're essentially asking why the Lorentz transformations are linear. $\endgroup$ – ACuriousMind May 1 '16 at 11:41
  • $\begingroup$ I am essentially asking if *inertial frames of reference (in special relativity) moves with constant velocity relative to each * is a previous assumption. Thanks for help $\endgroup$ – FUUNK1000 May 1 '16 at 11:46
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    $\begingroup$ No, it is not a previous assumuption - have a look at the question I linked, it proves that Lorentz transformations must be linear from the assumption they preserve the spacetime interval. What remains is that you convince yourself that linear transformations on the coordinates cannot be transformations between frames moving with time-varying velocity to each other. $\endgroup$ – ACuriousMind May 1 '16 at 11:48
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    $\begingroup$ It is possible to prove that if two reference frames assign constant velocity to every isolated point (in every direction and starting from any point of the space at any time) and furthermore there is a velocity which has the same value for both observers, then their relative motion is translational with constant velocity. An important step in this proof is establishing that the transformation of coordinates is linear non-homogeneous. The proof is a bit long and technical however. (If you understand Italian you can find it in the first pages of my lecture notes on special relativity.) $\endgroup$ – Valter Moretti May 1 '16 at 11:57