Some of the assumptions and properties of Newton's relativity theory are:
1) The existence of infinitely many inertial frames. Each frame is of infinite size (covers the entire universe). Any two frames are in relative uniform motion. (The relativistic nature of mechanics derived above shows that the absolute space assumption is not necessary.)
2) The inertial frames move in all possible relative uniform motion.
3) There is a universal, or absolute, time.
4) Two inertial frames are related by a Galilean transformation.
5) In all inertial frames, Newton's laws, and gravity, hold.
In comparison, the corresponding statements from special relativity theory by Einstein are:
1) Same as the Newtonian assumption.
2) Rather than allowing all relative uniform motion, the relative velocity between two inertial frames is bounded above by the speed of light.
3) Instead of universal time, each inertial frame has its own time.
4) The Galilean transformations are replaced by Lorentz transformations.
5) In all inertial frames, all laws of physics are the same (this leads to the invariance of the speed of light).
Notice both theories assume the existence of inertial frames. In practice, the size of the frames in which they remain valid differ greatly, depending on gravitational tidal forces.
In the appropriate context, a local Newtonian inertial frame, where Newton's theory remains a good model, extends to, roughly, 107 light years.
In special relativity, one considers Einstein's cabins, cabins that fall freely in a gravitational field. According to Einstein's thought experiment, a man in such a cabin experiences (to a good approximation) no gravity and therefore the cabin is an approximate inertial frame. However, one has to assume that the size of the cabin is sufficiently small so that the gravitational field is approximately parallel in its interior. This can greatly reduce the sizes of such approximate frames, in comparison to Newtonian frames. For example, an artificial satellite orbiting around earth can be viewed as a cabin. However, reasonably sensitive instruments would detect "microgravity" in such a situation because the "lines of force" of the earth's gravitational field converge.
In general, the convergence of gravitational fields in the universe dictates the scale at which one might consider such (local) inertial frames. For example, a spaceship falling into a black hole or neutron star would be subjected to tidal forces so strong that it would be crushed. In comparison, however, such forces might only be uncomfortable for the astronauts inside (compressing their joints, making it difficult to extend their limbs in any direction perpendicular to the gravity field of the star). Reducing the scale further, it might have almost no effects at all on a mouse. This illustrates the idea that all freely falling frames are locally inertial (acceleration and gravity-free) if the scale is chosen correctly.
Maxwell's equations governing electromagnetism possess a different symmetry, Lorentz invariance, under which lengths and times are affected by a change in velocity, which is then described mathematically by a Lorentz transformation.
Albert Einstein's central insight in formulating special relativity was that, for full consistency with electromagnetism, mechanics must also be revised such that Lorentz invariance replaces Galilean invariance. At the low relative velocities characteristic of everyday life, Lorentz invariance and Galilean invariance are nearly the same, but for relative velocities close to that of light they are very different.
In the sophisticated approach to the Galilean plane, say that taken by Isaak Yaglom or V.V. Kisil, there is a study of parabolas called cycles which generalize the concept of a circle to the peculiarities of Galilean geometry, particularly its theory of angle.