# Elements of the Galilean group

The definition I have of a Galilean transformation is one that preserves intervals of time and the distance between simultaneous events.

But now I can consider a transformation that takes $$(t,\mathbf{x})$$ to $$(t,\mathbf{x}+\mathbf{v}t^2)$$, which certainly preserves times, and also preserves the relative distance between simulatenous events, as the $$\mathbf{v}t^2$$ cancels out in the subtraction.

I am certain that this is wrong, so is there another constraint on what a Galilean transformation should be (say, an affine transformation)? And if there is such a constraint, why should it exist? I've seen some places simply define Galilean transformations as compositions of bossts, translations, and rotations, but to me that seems a little less motivated.

It's my first time learning these concepts, so I'm a little confused by what the actual first principles are - the "axioms" that I should take for granted.

• Your transformation does not make sense on dimensional grounds since $vt^2$ is not a distance. Moreover two consecutive transformation do not result in a Galilean transformation because of the non-linear $t^2$ term. Commented Feb 27, 2021 at 17:19
• @ZeroTheHero The dimensions idea make sense - but could you elaborate on the latter point? I think I would end up with $(t,\mathbf{x}+\mathbf{v}(t^2+t^4))$, which seems fine in terms of preserving distances of simultaneous events. Commented Feb 27, 2021 at 17:35
• right... the second part of my comment is in error as I was thinking of something else (basically I was thinking of $t$ as a parameter in the transformation rather than as an element to be transformed). I guess it comes down to dimensional analysis. Alternatively, by definition the transformation of coordinates must be so there is no acceleration, i.e. $d^2x/dt^2 =0$ in both frames since the transformation must connect inertial frames. Note also that your $x$ could be rotated without affecting the result but making the transformation (in particular the composition) more complicated. Commented Feb 27, 2021 at 18:14
• @ZeroTheHero That makes sense. So should I be taking as a definition that $d^2x/dt^2=0$ since we are dealing with inertial frames? Commented Feb 27, 2021 at 20:33
• yes that would be correct. Commented Feb 27, 2021 at 20:50