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## Main Question or Discussion Point

I have questions about the infinitesimal Lorentz transformation. but specifically about index manipulations.

[tex]\Lambda^{\mu}_{}_{\nu}=\delta^{\mu}_{\nu}+\delta\omega^{\mu}_{}_{\nu}[/tex]

where [tex]\delta\omega^{\mu}_{}_{\nu} << 1[/tex]

as found in many textbooks, we substitute this into

[tex]g_{\mu\nu}\Lambda^{\mu}_{}_{\alpha}\Lambda^{\nu}_{}_{\beta}=g_{\alpha\beta}[/tex]

and do the tedious(?) algebra...

[tex]g_{\mu\nu}(\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}+\delta^{\mu}_{\alpha}\delta\omega^{\nu}_{}_{\beta}+\delta\omega^{\mu}_{}_{\alpha}\delta^{\nu}_{\beta}+\delta\omega^{\mu}_{}_{\alpha}\delta\omega^{\nu}_{}_{\beta})=g_{\alpha\beta}[/tex]

and the last term is negligible because it too small (correct me if wrong!)

my question is the second and third terms. they are supposed to become...

[tex]...\delta\omega_{\mu\nu}+\delta\omega_{\nu\mu}...[/tex]

why is that? because [tex]\delta^{\mu}_{\nu}'s[/tex] are identity matrices, so whichever order we multiply, we get the same result.

and I look up some articles and found something about "abstract index notation". but is this the one?

[tex]\Lambda^{\mu}_{}_{\nu}=\delta^{\mu}_{\nu}+\delta\omega^{\mu}_{}_{\nu}[/tex]

where [tex]\delta\omega^{\mu}_{}_{\nu} << 1[/tex]

as found in many textbooks, we substitute this into

[tex]g_{\mu\nu}\Lambda^{\mu}_{}_{\alpha}\Lambda^{\nu}_{}_{\beta}=g_{\alpha\beta}[/tex]

and do the tedious(?) algebra...

[tex]g_{\mu\nu}(\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}+\delta^{\mu}_{\alpha}\delta\omega^{\nu}_{}_{\beta}+\delta\omega^{\mu}_{}_{\alpha}\delta^{\nu}_{\beta}+\delta\omega^{\mu}_{}_{\alpha}\delta\omega^{\nu}_{}_{\beta})=g_{\alpha\beta}[/tex]

and the last term is negligible because it too small (correct me if wrong!)

my question is the second and third terms. they are supposed to become...

[tex]...\delta\omega_{\mu\nu}+\delta\omega_{\nu\mu}...[/tex]

why is that? because [tex]\delta^{\mu}_{\nu}'s[/tex] are identity matrices, so whichever order we multiply, we get the same result.

and I look up some articles and found something about "abstract index notation". but is this the one?