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The following given tensor expression is:$$A^i=B_i+C_i\tag{1}$$ which is invalid expression since the free index on L.H.S is in upper part and on R.H.S it is in lower part so to make valid tensor expression we perform metric multiplication on L.H.S as $$g_{in}A^n=B_i+C_i\tag{2}$$ My question is now we make the free index $i$ in lower part of both sides by metric multiplication so is that above second expression is valid tensor expression of first one?

Are we allowed to do metric mutiplication for the sake of validity of tensor expression?

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  • $\begingroup$ yes, (2) is a valid expression. $\endgroup$
    – Prahar
    Aug 6 at 9:11
  • $\begingroup$ (2) is valid but (1) should never arise in the first place. So the answer to the question about performing metric contractions to make an expression valid is that you can't do this. $\endgroup$
    – Eletie
    Aug 6 at 10:00

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