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Wox
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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)
[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]
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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