Chapter 19: Problem 5
Asin Problem \(9.2\) every Lorentz transformation \(L=\left[L^{1}\right]\) has det \(L=\pm 1\) and either \(L_{4}^{4} \geq 1\) or \(L_{4}^{4} \leq-1\). Hence show that the Lorentz group \(G=O(3,1)\) has four connected components, $$ \begin{aligned} G_{0}=& G^{++}: \operatorname{det} L=1, L_{4}^{4} \geq 1 & G^{+-}: \operatorname{det} L=1, L_{4}^{4} \leq-1 \\ G^{+}+: \operatorname{det} L=-1, L_{4}^{6} \geq 1 & G^{--}: \operatorname{det} L=-1, L_{4}^{4} \leq-1 \end{aligned} $$ Show that the group of components \(G / G_{0}\) is isomorphic with the discrete abelian group \(Z_{2} \times Z_{2}\).
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