Completely General Lorentz Transformation

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Discussion Overview

The discussion revolves around the matrix form of the completely general Lorentz Transformation, which includes both rotations and boosts. Participants explore the complexity of parametrizing these transformations and the implications of the order in which rotations and boosts are applied.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants assert that the general Lorentz Transformation can be written out but acknowledge that it would be cumbersome and dependent on the chosen parametrization.
  • There is a discussion about parametrizing the boost vector, with suggestions including using its magnitude and two angles or its three components.
  • Participants note that since boosts and rotations do not commute, the order of operations is significant and can complicate the transformation.
  • One participant suggests writing out a boost matrix and a rotation matrix and multiplying them, emphasizing the critical nature of the order in which these operations are performed.
  • Another participant recommends using symbolic math software, like Maxima, to handle the complexity of the transformations and references material that may assist in applying it to relativity.

Areas of Agreement / Disagreement

Participants generally agree that the general Lorentz Transformation can be expressed in matrix form, but there is no consensus on the best way to parametrize it or the implications of the order of operations. The discussion remains unresolved regarding the most effective approach to represent these transformations.

Contextual Notes

Limitations include the dependence on the chosen parametrization and the unresolved complexities arising from the non-commutative nature of boosts and rotations.

OniLink++
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Does anyone have the matrix form of the completely general Lorentz Transformation, with rotations AND boosts, or does it not exist?
 
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It can certainly be written out, but it would be very cumbersome. The exact form would depend on how you decided to parametrize it.
 
bcrowell said:
It can certainly be written out, but it would be very cumbersome. The exact form would depend on how you decided to parametrize it.

Could you specify, exactly, what you mean? When you parametrize it, it should just be 3 rotations and 3 boosts, correct?
 
OniLink++ said:
Could you specify, exactly, what you mean? When you parametrize it, it should just be 3 rotations and 3 boosts, correct?

E.g., you could parametrize the boost vector by its magnitude and two angles giving its direction. Or you could parametrize it by its three components.

Since boosts and rotations don't commute, you could parametrize by doing the operations in either order.
 
bcrowell said:
E.g., you could parametrize the boost vector by its magnitude and two angles giving its direction. Or you could parametrize it by its three components.

Since boosts and rotations don't commute, you could parametrize by doing the operations in either order.
Oh, ok then. It just got a lot more complicated. Hrmm... how about the Lorentz Transformations with the 3 components and the rotations applied before the boosts?
 
Why don't you write out a boost matrix and a rotation matrix and just multiply them together? They do form a group. However the order in which boosts and rotations happen is critical. It sounds like another tedious calculation.
 
If you really want to see it written out in all its ugly glory, I'd suggest using symbolic math software. Maxima is free and open-source, and I have some material in this book http://www.lightandmatter.com/genrel/ on how to apply it to relativity. See section 2.5.3 for some similar examples.
 
bcrowell said:
If you really want to see it written out in all its ugly glory, I'd suggest using symbolic math software. Maxima is free and open-source, and I have some material in this book http://www.lightandmatter.com/genrel/ on how to apply it to relativity. See section 2.5.3 for some similar examples.

Thanks very much for that. You've just saved a good fraction of my life:redface:
 

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