# Completely General Lorentz Transformation

1. Aug 14, 2011

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

2. Aug 14, 2011

### bcrowell

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

3. Aug 14, 2011

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

4. Aug 14, 2011

### bcrowell

Staff Emeritus
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.

5. Aug 14, 2011

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?

6. Aug 15, 2011

### 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.

7. Aug 15, 2011

### bcrowell

Staff Emeritus
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.

8. Aug 16, 2011

### inottoe

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