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