the rule of raising and lowering indices

wdlang

i am new to relativistic quantum mechanics

i am always puzzled by the magic trick of raising and lowering indices

for a lorentz transform

we write [tex]\tilde{x}^{\mu}=a^{\mu}_{\;\;\nu}x^{\nu}[/tex]

is it okay to write it as [tex]\tilde{x}^{\mu}=a^{\mu}_{\nu}x^{\nu}[/tex]?

and [tex]g^{\mu}_{\:\:\nu}g_{\mu}^{\:\:\tau}=g^{\tau}_{\:\:\nu}[/tex] or [tex]g^{\:\:\tau}_{\nu}[/tex]?