Why do the order of Lorentz transformations matter?

[ehrenfest]

lets say you apply a Lorentz boost in the x direction with velocity v and a Lorentz boost in the y direction with velocity v'. Why does it makes that the order in which you apply the transformations affects the resultant transformation matrix? These are two independent directions, so shouldn't you be free to apply the transformations in whatever order you want. Interestingly, I get the transpose matrix when I reverse the order of application. Why does that make sense? In the Galilean system, the order does not matter, right?

 

[dextercioby]

Well, the set of all Lorentz boosts don't even make up a group, not to mention a commutative group. Or think about it this way: do matrices generally commute under multiplication ?

Spacetime translations form a commutative group. Space rotations don't form a commutative group, but they form a group.

 

[ehrenfest]

Well, the set of all Lorentz boosts don't even make up a group, not to mention a commutative group. Or think about it this way: do matrices generally commute under multiplication ?

Spacetime translations form a commutative group. Space rotations don't form a commutative group, but they form a group.

What is an example of two space rotations that are not commutative? They do form a group in two dimensions, correct?

 

[dextercioby]

By space rotations i meant just that, "space" rotations, i.e. the SO(3) group. The plane rotations, or SO(2), form an abelian group, since one can show that $SO(2)\simeq U(1)$, with the latter group being abelian.

 

[ehrenfest]

I see. Thanks.