Commutativity of Lorentz Boosts & Rotations

In summary, the group property allows for the creation of products of group members, such as the Lorentz boost (B) and rotation (R), but commutativity is not valid for these products (R.B or B.R). The order of these transformations should be considered and it depends on what is being considered. If a particle moves in one direction and an observer performs a finite rotation, the transformation in their frame would be a boost followed by a rotation. The order of these transformations is important and can be seen by performing infinitesimal transformations on the coordinates. The proof that changing the order of R and B leads to an equal transformation can be found using the baker-campbell-hausdorff relation and the explicit algebra.
  • #1
parsikoo
12
0
As per group property, one could make a product of gr members e.g. Lorentz boost (B) and rotation R, as they Commutativity is not valid, R.B or B.R, what should be considered and which order should be preferred? Generally it is known R.B1= B2.R .
 
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  • #2
Nothing is preferred, it depends on what you want to consider. Your question is a bit vague.
 
  • #3
haushofer said:
Nothing is preferred, it depends on what you want to consider. Your question is a bit vague.
Let's be more clear, if a particle moves in a single direction in one coordinate e.g. boost, then an observer has a finite rotation, then the transformation in his frame is simply the product of a boost and rotation, then what are the considerations as regards the order?
 
  • #4
I still don't get it. If you first perform a boost, and then a rotation, then the order is boost-rotation. Are you perhaps puzzled by active vs passive (i.e. boost the particle or boost the observer)?
 
  • #5
I meant, boost in a frame, brought to a new frame which makes a finite rotation, hope it is clear.
 
  • #6
A boost is not a spatial rotation. Maybe it is even easier to visualize it for the Galilei group. A Galilei boost is a spatial translation linear in time. A rotation is a different kind of transformation. In your example it would be boost, and then rotation. You can also first rotate, and then boost. But that will give a different answer, which you can simply see by performing the infinit. transformations on the coordinates.
 
  • #7
Mathematically if you would use iterate many infinitesimal transformations, then the transf. by exp. form would be exp(e1.B+ e2.R) (e=infinitesimal amount), but it would be interesting to see the proof that the outcome of changing the order of R and B would lead to equal transformation
 
  • #8
The order is going to be specified. No one is going to say "boost this way and rotate that way" without explicitly saying which comes first, just for that reason. If they do, they don't have an understanding of the math.

That said, as long as the boost plane and the rotation plane share no common direction, the boost and the rotation commute. But that's rather trivial and obvious. Of course a tx-plane boost won't affect a yz-plane rotation, and vice versa.
 
  • #9
Can we please focus on our specific case.
 
  • #10
parsikoo said:
Mathematically if you would use iterate many infinitesimal transformations, then the transf. by exp. form would be exp(e1.B+ e2.R) (e=infinitesimal amount), but it would be interesting to see the proof that the outcome of changing the order of R and B would lead to equal transformation

You can use the baker- campbell- hausdorff relation for that, and the explicit algebra, to see if that's true.

The algebra ( and also the application of the infinit. transfo. on the coordinates) tells you that rotations and boosts don't commute; the boost parameter is a vector, transforming as a vector under the adjoint action of a rotation. Schematically,

[rot, boost] = boost
 
  • #11
parsikoo, maybe you are referring to the fact that proper Lorentz transformations in 3D also contains a rotation of the three axes, because the composition of two boosts is not just a boost but is a boost followed by a rotation (look up also Thomas precesion).
 
Last edited:

1. What is the commutativity of Lorentz boosts and rotations?

The commutativity of Lorentz boosts and rotations refers to the ability to perform these operations in any order without affecting the final result.

2. Why is the commutativity of Lorentz boosts and rotations important?

This property is important in the study of special relativity, as it allows for the analysis of different reference frames and the transformation of coordinates without changing the physical laws.

3. How is the commutativity of Lorentz boosts and rotations mathematically represented?

The commutativity of Lorentz boosts and rotations is represented by the fact that the matrix multiplication of two Lorentz transformation matrices is commutative.

4. Are there any exceptions to the commutativity of Lorentz boosts and rotations?

Yes, the commutativity property does not hold for non-collinear boosts or rotations, which means that the order of these operations matters when they are not performed along the same axis.

5. What are some practical applications of the commutativity of Lorentz boosts and rotations?

The commutativity of Lorentz boosts and rotations is used in various fields such as particle physics, astrophysics, and engineering to analyze and predict the behavior of objects in different reference frames and to make accurate calculations in special relativity.

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