c+v and c-v are not Galilean transformation?
A common misunderstanding related to Special Relativity, is that c+v and c-v are not Galilean transformation.
In elementary school, we solved chasing problems and meeting problems using addition/subtraction of speeds.
Although we had never heard of relativity motion, we all knew they are relative speeds.
As a great scientist, Galileo's contribution is not the velocity addition rule, but that he pointed out that nothing is at absolute rest.
And that's why the velocity addition rule under Newtonian system is named after him.
Under Newtonian system, all velocity addition, whether on straight or curved lines,
whether by simple algebraic addition or vector addition, whether in inertial frames or non-inertial frames, are all called Galilean addition.
This rule applies everywhere, as under Newtonian system, all velocity related calculation are based on these definitions and assumptions:
evenly distributed time(t1+t2=t3), evenly distributed space(d1+d2=d3), and speed definition (v=d/t). Further more,
any distance or time interval always has the same measurement in all refrence frames.
So, under Newtonian system, we have only one velocity addition rule, which is called the Galilean addition rule.
Similar to the above, in a relativistic system, there is an unique rule for velocity addition,
which is based on these definitons and assumptions:
relativistic time (t1 * t2 = t3）, relativistic space（d1 # d2 = d3），and speed definition（v = d / t).
Also, the measurements for distance and time interval are dependant on the selection of reference frames.
Here * and # represent the time addition rule and space addition rule respectively.
As we have only one set of definitions and assumptions, the velocity addition rule must be unique under Special Relativity.
Further more, as these definitions and assumptions under Newtonian system and Relative system are different, their velocity addition rules must be different and cannot be mixed.