Tensor calculations involve many sums. Writing $\sum$ everywhere is cumbersome. Einstein’s convention: when an index appears once upstairs and once downstairs in the same term, sum over it.

$$a_i b^i \equiv \sum_{i=0}^{n-1} a_i b^i = a_0 b^0 + a_1 b^1 + \cdots + a_{n-1} b^{n-1}$$

The repeated index is called a dummy index (or contraction index) — its name doesn’t matter. $a_i b^i = a_j b^j$. A free index appears once and labels the components of the result.

Rules

1. Free indices must match on both sides of an equation: $T^i{}_j = A^i{}_k B^k{}_j$ ✓ (both sides have free $i, j$)
2. Dummy indices appear in exactly one upper-lower pair per term: $A^i{}_i$ ✓, $A^i{}_j B^j{}_k$ ✓
3. Never repeat an index more than twice in a single term: $A^i{}_i B^i$ ✗

Examples in 4D spacetime

Indices run from 0 to 3. Greek letters ($\mu, \nu, \alpha, \ldots$) for spacetime indices by convention.

Dot product of four-vectors: $$a_\mu b^\mu = a_0 b^0 + a_1 b^1 + a_2 b^2 + a_3 b^3$$

Matrix multiplication: $$C^\mu{}\nu = A^\mu{}\alpha \, B^\alpha{}_\nu$$

This is matrix $C = AB$ — the dummy index $\alpha$ is summed over, leaving free indices $\mu, \nu$.

Lowering an index with the metric: $$v_\mu = g_{\mu\nu} \, v^\nu$$

The metric tensor $g_{\mu\nu}$ converts an upper index to a lower one. Its inverse $g^{\mu\nu}$ goes the other way: $v^\mu = g^{\mu\nu} v_\nu$.

The Kronecker delta: $$\delta^\mu{}_\nu = \begin{cases} 1 & \text{if } \mu = \nu \ 0 & \text{otherwise} \end{cases}$$

Acts as the identity: $\delta^\mu{}_\nu \, v^\nu = v^\mu$.

Reading tensor equations

When you see an equation in GR like:

$$\nabla_\mu T^{\mu\nu} = 0$$

parse it as: “For each value of the free index $\nu$ (0, 1, 2, 3), sum over $\mu$ from 0 to 3.” That’s four equations, each involving a four-term sum — 16 terms from a single line of notation.

Einstein reportedly said that the summation convention was his greatest contribution to mathematics. (He was joking. Probably.)