## the rule of raising and lowering indices

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 $$\tilde{x}^{\mu}=a^{\mu}_{\;\;\nu}x^{\nu}$$

is it okay to write it as $$\tilde{x}^{\mu}=a^{\mu}_{\nu}x^{\nu}$$?

and $$g^{\mu}_{\:\:\nu}g_{\mu}^{\:\:\tau}=g^{\tau}_{\:\:\nu}$$ or $$g^{\:\:\tau}_{\nu}$$?
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