Relationship bewtween the principle of Galilean Relativity and absolute time

Question

The principle of Galilean Relativity is:

The laws of Mechanics are invariant in every inertial frame of reference.

I say "laws of Mechanics" specifically because I'm referring to the principle before Electrodynamics was included with Special Relativity.

Together with it there's also a structure of spacetime where we consider the notion of absolute time and relative space: although different observers on different reference frames sees space differently, they feel time the exact same way.

My question is: this structure of spacetime follows directly from the principle or it is one additional assumption based on observation people made at the time? I can't see anyway this follows from the principle, so for me it seems it is something additional, but still I'm not sure.

If it follows from the principle, what reasoning is used to conclude the structure of spacetime from Galilean Relativity?

In short, usually I see people saying that Galilean Relativity is bound to one certain structure of spacetime where space is relative and time is absolute. What really is the relation between such structure and the principle stated?

Another possibility is that the implication is the other way around: absolute time implies Galilean Relativity. But I also can't see how this could be.

Answer 1

In short, usually I see people saying that Galilean Relativity is bound to one certain structure of spacetime where space is relative and time is absolute. What really is the relation between such structure and the principle stated? Another possibility is that the implication is the other way around: absolute time implies Galilean Relativity. But I also can't see how this could be.

These people are wrong. Assuming only Galilean relativity and isotropy of the space you can get such space-time transformations. $$x' = \frac{x-vt}{\sqrt{1-kv^2}}, t' = \frac{t-kvx}{\sqrt{1-kv^2}}$$

Ofc setting $k=0$ you will get Galilean transformation and absolute time $t'=t$. But if we want the Maxwell equations work in the same way in every frame, we need to set $k=\frac{1}{c^2}$, which gives us Lorentz transformations and Special Relativity. Derivation:http://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations#From_group_postulates

Answer 2

There is no relation as stated, since Special Relativity restricted to mechanical processes is identical to Galilein Relativity. To make them different, you can include specifically the Galilean transform between inertial systems into the formulation of Galilein Relativity, but then it would be trivial to derive absolute time from this.