Completely General Lorentz Transformation

[OniLink++]

Does anyone have the matrix form of the completely general Lorentz Transformation, with rotations AND boosts, or does it not exist?

 

[bcrowell]

It can certainly be written out, but it would be very cumbersome. The exact form would depend on how you decided to parametrize it.

 

[OniLink++]

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?

 

[bcrowell]

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.

 

[OniLink++]

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?

 

[keithdow]

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.

 

[bcrowell]

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.

 

[inottoe]

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