Generators for Lorentz transformations

1. Feb 28, 2012

Wox

Consider Minkowski spacetime with signature (-+++) and coordinates (ct,x,y,z) with respect to the standard orthogonal basis. I'm looking for the smallest set of matrices that can generate any Lorentz transformation with respect to this basis. I came up with 8 matrices (see below). Am I missing something?

Three boosts (in x, y and z direction)
$$\text{Boost}_{x}(\tilde{\alpha})=\left[\begin{matrix} \cosh\tilde{\alpha}&\sinh\tilde{\alpha}&0&0\\ \sinh\tilde{\alpha}&\cosh\tilde{\alpha}&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{matrix}\right]\quad \text{Boost}_{y}(\tilde{\beta})=\left[\begin{matrix} \cosh\tilde{\beta}&0&\sinh\tilde{\beta}&0\\ 0&1&0&0\\ \sinh\tilde{\beta}&0&\cosh\tilde{\beta}&0\\ 0&0&0&1 \end{matrix}\right]\quad \text{Boost}_{z}(\tilde{\gamma})=\left[\begin{matrix} \cosh\tilde{\gamma}&0&0&\sinh\tilde{\gamma}\\ 0&1&0&0\\ 0&0&1&0\\ \sinh\tilde{\gamma}&0&0&\cosh\tilde{\gamma}\\ \end{matrix}\right]\quad$$
Three spatial rotations
$$\text{Rot}_{x}(\alpha)=\left[\begin{matrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&\cos\alpha&\sin\alpha\\ 0&0&-\sin\alpha&\cos\alpha\\ \end{matrix}\right]\quad \text{Rot}_{y}(\beta)=\left[\begin{matrix} 1&0&0&0\\ 0&\cos\beta&0&-\sin\beta\\ 0&0&1&0\\ 0&\sin\beta&0&\cos\beta\\ \end{matrix}\right]\quad \text{Rot}_{z}(\gamma)=\left[\begin{matrix} 1&0&0&0\\ 0&\cos\gamma&\sin\gamma&0\\ 0&-\sin\gamma&\cos\gamma&0\\ 0&0&0&1\\ \end{matrix}\right]$$
Two inversions
$$I=\left[\begin{matrix} 1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1 \end{matrix}\right]\quad \tilde{I}=\left[\begin{matrix} -1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{matrix}\right]$$

Last edited: Feb 28, 2012
2. Feb 28, 2012

George Jones

Staff Emeritus
What do you mean by "generator" and "generate"? I suspect that you take your meanings from the math community. For the physics community, "generator" often means "element of the Lorentz Lie algebra".

Do you really need rotations about all three axes? For example, what does a rotation about the x-axis followed by a rotation about the y-axis give?

If you have all the rotations, do you need boosts in three linearly independent directions?

3. Feb 28, 2012

Wox

To describe an arbitrary rotation I need all three, no?

As for generators: I mean in the mathematical sense. Every Lorentz transformation can be written as a combination of these generators. I was thinking about the generators of a group, but this is different because $\text{Rot}_{x}(\alpha)$ is not one element of the Lorentz group but many elements, so I'm not sure how this is mathematically formalized.

4. Feb 28, 2012

George Jones

Staff Emeritus
I should have written

5. Feb 28, 2012

Wox

Hmmm, I see what you mean. So I can skip one rotation and two boosts. But more importantly, are there Lorentz transformations that can't be written as a combinations of the transformations given?

6. Feb 28, 2012

Ben Niehoff

Your set of transformations covers everything. However, they are not the "generators". The generators of a Lie group are the elements of the Lie algebra. The boosts and rotations you gave are exponentials of Lie algebra elements, so you're close.

7. Feb 28, 2012

George Jones

Staff Emeritus
From Wikipedia (which I think is common usage in the math community)