Galilean Transformations and Postulates

[RedX]

I read that the form of a galilean transformation on the point (t,x) is the following:

constant velocity transform by velocity v: (t,x) ---> (t,x+vt)
translation transform by (t₀,x₀): (t,x)--->(t+t₀,x+x₀)
rotation transformation by rotation matrix R: (t,x)--->(t,Rx)

and that it is based on the following postulates: two observers observe the same time between events, and two observers measure the same distance for simultaneous events.

My question is, based on those two postulates, couldn't you modify the first transform to

(t,x)--->(t,x+vf(t))

This transformation preserves the time between events (it doesn't change time at all). It also preserves simultaneous distance:

|(x₁+vf(t))-(x₂+vf(t))|=|x₁-x₂|

 

[darkSun]

Yeah, doesn't that just correspond to moving with an arbitrary velocity v(t)=vf(t)?

 

[RedX]

Yeah, doesn't that just correspond to moving with an arbitrary velocity v(t)=vf(t)?

Yeah, but unfortunately the book I'm using (Mathematical Methods of Classical Mechanics, V. Arnold) emphasizes that v is uniform.

If you take the simplest case of a uniformily accelerated reference frame, then the transformation would look like:

(t,x)--->(t,x+.5*g*t²)

This transformation preserves the time between two events, and the distance between simultaneous events. So two observers, one uniformily accelerating with respect to the other, observe the same time between events and the same distance between simultaneous events.

So according to Arnold, such a transformation is a gallilean transformation.

Bringing in relativity, things are much different. Instead of preserving time between events and distance between simultaneous events separately, you consider transformations that preserve:

(ct)^2-(x)^2

Here you can show that the Poincare group (as opposed to the gallilean group) is a 10-parameter group consisting of uniform boosts, rotations, and translations. The boosts have to be uniform and can't be a function of spacetime.

But it seems in classical mechanics, according to Arnold, you can have boosts that depend on time (so reference frames can be accelerated), although not space.