At the beginning of the developing of special relativity the following principles are assumed true:
- Principle 1: every physical law is equal in form in every inertial frame.
- Principle 2: there is a maximum finite velocity equal in all the inertial frames of reference that is equal to the speed of light.
- Definition: an inertial frame is a frame in which an object not subjected to a force moves with uniform rectilinear motion.
However, in the developing of the theory it seems to me that a slightly different version of the first principle is used, namely:
- Principle $ \alpha$: the full symmetry group is given by translation in space and time, rotations and boosts of velocity. Namely, space and time are homogenous, space is isotropic and that laws of physics don't change after boosts of velocity.
To me it looks that principle $ \alpha$ can't be derived from principle 1 because principle 1 doesn't give any information about which are the transformation between an inertial frame to another
On the other side principle 1 can be derived from principle $ \alpha $ , for example: from principle $ \alpha $ we know that after a boost the form of the physical laws doesn't change which implies that the dynamics remains unchanged. It follows that a uniform rectilinear motion will remain a uniform rectilinear motion and for this reason boosts transform an inertial frame in another inertial frame without changing the form of the equations. The same can be said for translation in space and time and rotations.
So, question: is principle 1 equivalent to principle $ \alpha $ and consequently enogh to develop special relativity or do we need to restate it in a more complete form (that is principle $ \alpha $)