If we write the Minkowski metric with “down” indices

$$ \eta_{\mu\upsilon}=\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix} $$

then convention is to write the inverse matrix with “up” indices

$$ \eta^{\mu\upsilon}=\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix} $$

so that $\eta_{\mu\upsilon}\eta^{\mu\upsilon}=\delta^\upsilon_\mu$ and $\delta^\upsilon_\mu$ is the Kronecker-$\delta$ with Lorentz indices. It has special properties that it is $1$ if $\mu=\upsilon$ and is zero otherwise. It plays the role of the identity matrix

🗒️ Note: In this specific case $\eta^{-1}=\eta$ and $\eta^2=I$, but in more genera cases which will not be touched upon it can be different

Vocabulary

• “up” index $A^\mu$ is known as contravariant
• “down” index $A_\mu$ is known as covariant

💰 Example: if we define $A_\mu=\eta_{\mu\upsilon} A^\upsilon$ then

$$ \eta^{\mu\alpha}A_\alpha=\eta^{\mu\alpha}\eta_{\alpha \beta}A^{\beta}=\delta^\mu_\beta A^\beta=A^\mu $$

If $A^\mu=(A^0,\vec A)=(A^0,A^1,A^2,A^3)$ and $A_\mu=(A_0,A_1,A_2,A_3)$ then $A^i=-A_i$ for $i=1,2,3$ with this definition we see that

$$ \overset{↝}{\bold A}\cdot \overset{↝}{\bold B}=\eta_{\mu\upsilon}A^{\mu}B^\upsilon=A_\mu B^\mu=A^\mu B_\mu $$

The summation convention is in operation in this new context.

Standard rules:

• Summed indices must be “up” and the other “down”
• There must be the same number of “up” and “down” indices on each side of an equation and in all terms added together in an equation

💰 Example: ✅ Obey the rules ❌ Don’t obey the rules

$$ \begin{aligned} ✅\;\;\; A^\mu&=B^{\mu\upsilon} C_\upsilon+D^\mu \\ ✅\;\;\;A^\mu&=X^\mu Y_\alpha Z^\alpha Y^\Beta Z_\beta=X^\mu(Y_\alpha Z^\alpha)^2 \\ ✅\;\;\;A^\mu_\upsilon&=B^\mu_{\upsilon\rho} C^\rho \\ (1)\;❌ \;\;\; A^\mu&=B^{\mu\upsilon\rho}C_\rho \\ (2)\;❌ \;\, A_{\mu\upsilon}&=B_{\mu\upsilon}^\rho C^\rho \\ (3)\;❌ \;\;\; A^\upsilon&=B^\upsilon+C_\rho \end{aligned} $$

• $(1)$ is wrong because on the left there is 1 free index $\mu$ and on the right there are 2 $\mu\upsilon$
• $(2)$ has both of the summed indices $\rho$ in the “up” position
• $(3)$ makes no sense as you are summing since its adding different components $\upsilon\ne\rho$

🗒️ Note: in Euclidean space we did not have to bother with this writing as the covariant and the contravariant components are equal in $\delta_{ij}$

💰 Example: Consider a 4-vector $X^\mu=(ct,\vec x)=(ct,x,y,z)$