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?(adsbygoogle = window.adsbygoogle || []).push({});

Three boosts (in x, y and z direction)

[tex]

\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

[/tex]

Three spatial rotations

[tex]

\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]

[/tex]

Two inversions

[tex]

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]

[/tex]

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# Generators for Lorentz transformations

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