Problems understanding index notation

Question

I'm reading Spacetime and Geometry and I have problems understanding index notation of matrix multiplication. Why is $$ \eta=\Lambda^T \eta \Lambda $$ in index notation $$ \eta_{\rho \sigma}=\Lambda^{\mu'}_{\rho}\eta_{\mu'\nu'}\Lambda^{\nu'}_\sigma?$$

Maybe someone can explain how the index notation works in this case (I know it's a very stupid question).

Answer 1

Use $(AB)_{ij} = \sum_k A_{ik} B_{kj}$ (defn of matrix multiplication).

Repeated use of that gives -

$\begin{array}{rcl}(\Lambda \eta \Lambda)_{ij}&=&\sum_k(\Lambda\eta)_{ik}\Lambda_{kj}\\ &=&\sum_k\sum_l\Lambda_{il}\eta_{lk}\Lambda_{kj} \end{array}$

In the Einstein notation, it is customary to use Greek symbols for indices.

Answer 2

The index convention being used, sometimes called the "Einstein summation convention," is that any index that appears twice, once as a contravariant index (a superscript) and once as a covariant index (a subscript) 9is to be summed over from $1$ to for (in the convention that $x = x_1, y=x_2, z=x_3, t = x_4$).

That notation convention in general $d$-dimensional spacetime refers to summation over however many spacetime indices are present.

Caution - if you ever do renormilzation using dimensional regularizatoin (working in $4+\epsilon$ dimensions to control certain infinities that come up in quantum field theory expansions), you have to be careful about which summations of $\delta_a^b$ give $4$ and which give $4+\epsilon$.