Galilean-Newtonian Relativity

A common view today is that Newtonian Mechanics requires the notion of absolute space, which however cannot be identified, and so is used to motivate a departure into relativistic mechanics in the form of Einstein's Special Theory of Relativity SR without gravitation and General Theory of Relativity GR with gravitation.

But Newtonian Mechanics is already relativistic in the sense that it takes the same form in all Euclidean coordinate systems moving with constant velocity with respect to each other (so called inertial systems) and so it is impossible by experiment to identify a special system to act as absolute space. This is because both Newton's 2nd Law N2 and Newton's Law of Gravitation NG take the same form in all inertial systems. This was well understood already by Galileo. 

We have seen in posts on New View on Gravitation that NG can according to Laplace be formulated as the local assignment

• $\rho (x)=\Delta\phi (x)$ for all $x$       (NG)  

where $\rho (x)$ is mass density and $\phi (x)$ gravitational potential depending on a Euclidean space variable $x$ and $\Delta$ is the Laplacian differential operator. Viewing (NG) as assignment means that somehow matter is endowed with mass through that action of the Laplacian on the gravitational potential.

Since the Laplacian is invariant under both translation and rotation, it follows that so is NG and thus the generation of mass from gravitational potential does not depend on choice of coordinate system, which is a natural requirement. 

On the other hand, N2 is not invariant under rotation since centripetal forces from acceleration arise. It is thus possible to detect rotation by experiment such as Newton's Bucket. 

Conclusion: Newtonian Mechanics is relativistic in the Galilean sense of taking the same form in all inertial systems, and so does not require the notion of absolute space. Thus there does not seem to be any compelling reason to replace Newtonian Mechanics by SR/GR mechanics. Adding propagation of light to the mechanics picture motivates a modification on celestial scales as shown in Many-Minds Relativity. 

Remark:In Newton's Bucket experiment it can be decided that it is the bucket which rotates vs the fixed stars (and not the other way around) by considering the evolution in time from an initial state with the bucket at rest with respect to the fixed stars. With this point of view a Euclidean coordinate system determined by the fixed stars, may serve as an absolute space, but it is also possible to connect coordinate system to the Sun, Earth, or your living room.