Lorentz transformation

Proposition

Given that the interval ${\displaystyle x^{\mu }x_{\mu }}$  is invariant under a Lorentz transformation, prove that the Lorentz transformation is orthogonal.

 1 ${\displaystyle x'^{\mu }x'_{\mu }=x^{\alpha }x_{\alpha }}$ . Given. 2 ${\displaystyle x'^{\mu }x'_{\mu }=g_{\mu \nu }x'^{\mu }x'^{\nu }}$ metric tensor 3 ${\displaystyle x'^{\mu }=\Lambda ^{\mu }{}_{\alpha }x^{\alpha }}$ Lorentz transformation 4 ${\displaystyle x'^{\nu }=\Lambda ^{\nu }{}_{\beta }x^{\beta }}$ Lorentz transformation 5 ${\displaystyle x'^{\mu }x'_{\mu }=g_{\mu \nu }\Lambda ^{\mu }{}_{\alpha }x^{\alpha }\Lambda ^{\nu }{}_{\beta }x^{\beta }}$ Substitute 3 and 4 into 2. 6 ${\displaystyle x^{\alpha }x_{\alpha }=g_{\alpha \beta }x^{\alpha }x^{\beta }}$ metric tensor 7 ${\displaystyle g_{\mu \nu }\Lambda ^{\mu }{}_{\alpha }x^{\alpha }\Lambda ^{\nu }{}_{\beta }x^{\beta }=g_{\alpha \beta }x^{\alpha }x^{\beta }}$ From 5, 1, and 6. 8 ${\displaystyle g_{\mu \nu }\Lambda ^{\mu }{}_{\alpha }\Lambda ^{\nu }{}_{\beta }x^{\alpha }x^{\beta }=g_{\alpha \beta }x^{\alpha }x^{\beta }}$ Rearrange 7. 9 ∴ ${\displaystyle g_{\mu \nu }\Lambda ^{\mu }{}_{\alpha }\Lambda ^{\nu }{}_{\beta }=g_{\alpha \beta }}$ From 8, since ${\displaystyle x^{\alpha }}$  and ${\displaystyle x^{\beta }}$  may be arbitrary. 10 ${\displaystyle g_{\alpha \beta }g^{\alpha \gamma }=\delta ^{\gamma }{}_{\beta }}$ Kronecker delta 11 ${\displaystyle g_{\mu \nu }\Lambda ^{\mu }{}_{\alpha }\Lambda ^{\nu }{}_{\beta }g^{\alpha \gamma }=\delta ^{\gamma }{}_{\beta }}$ Multiply both sides of 9 by ${\displaystyle g^{\alpha \gamma }}$ , then apply 10. 12 ${\displaystyle g_{\mu \nu }\Lambda ^{\mu \gamma }\Lambda ^{\nu }{}_{\beta }=\delta ^{\gamma }{}_{\beta }}$ Contracting the indices ${\displaystyle \alpha }$  in 11. 13 ${\displaystyle \Lambda _{\nu }{}^{\gamma }\Lambda ^{\nu }{}_{\beta }=\delta ^{\gamma }{}_{\beta }}$ Contracting the indices ${\displaystyle \mu }$  in 12. 14 ${\displaystyle (\Lambda ^{T})^{\gamma }{}_{\nu }\Lambda ^{\nu }{}_{\beta }=\delta ^{\gamma }{}_{\beta }}$ Swap the order of indices in order to transpose the first ${\displaystyle \Lambda }$  of 13. 15 ∴ ${\displaystyle \Lambda ^{T}=\Lambda ^{-1}}$  ${\displaystyle \Box }$ 14 may be paraphrased as ${\displaystyle \Lambda ^{T}\Lambda =I}$ .

Reference: http://www.physics.gla.ac.uk/~dmiller/lectures/RQM_2008.pdf, page 9.