Why do the order of Lorentz transformations matter?

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Homework Help Overview

The discussion revolves around the order of Lorentz transformations, specifically focusing on Lorentz boosts in different spatial directions. Participants explore why the sequence of applying these transformations affects the resultant transformation matrix, contrasting it with the behavior of transformations in the Galilean system.

Discussion Character

  • Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants raise questions about the independence of Lorentz boosts and the implications of their non-commutativity. There is a discussion on whether the order of transformations should matter and how this relates to the properties of matrix multiplication.

Discussion Status

The discussion is ongoing, with some participants providing insights into the mathematical structure of Lorentz boosts and their relationship to groups. There is an exploration of examples related to space rotations and their commutative properties, but no consensus has been reached.

Contextual Notes

Participants reference the differences between Lorentz transformations and Galilean transformations, noting that the former do not form a commutative group, which is a key point of inquiry. The discussion also touches on the nature of matrix multiplication and its implications for the transformations being considered.

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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?
 
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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.
 
dextercioby said:
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?
 
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.
 
Last edited:
I see. Thanks.
 

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