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We can raise and lower indices of any tensor with a non-zero rank by applying the metric tensor with indices properly located.

My question is: why is the metric tensor the tool we use for such operations? If we use another tensor we would perform the same raising or lowering operations but obviously with different numerical results.

In different words, if I have a covariant vector $\tilde T=T_\nu\mathbf e^\nu$, why do I get its contravariant version $\vec T=T^\mu\mathbf e_\mu$ just through the metric tensor?

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The metric tensor is defined in such a way that $\mathbf{e}^\mu = g^{\mu\nu} \mathbf{e}_\nu$. Taking any other tensor $T_\mu e^\mu = T_\mu g^{\mu\nu} \mathbf{e}_\nu$ so it becomes natural to take $T^\nu = T_\mu g^{\mu\nu}$.

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