Source: cirosantilli/implicit-metric-signature-in-einstein-notation
= Implicit metric signature in Einstein notation
Then a specific <metric> is involved, sometimes we want to automatically add it to products.
E.g., in a context considering the common <Minkowski inner product matrix> where the $\eta$ 4x4 matrix and $\mu$ is a vector in <\R^4>
$$
x^{\mu} x_{\mu} = x^{\mu} \eta_{\mu \nu} x^{\nu} = -x_0^2 + x_1^2 + x_2^2 + x_3^2;
$$
which leads to the change of sign of some terms.