Matthew Litwin

The Einstein Summation Convention

"I have made a great discovery in mathematics; I have suppressed the summation sign every time that the summation must be made over an index which occurs twice..." Albert Einstein (Kollros 1956; Pais 1982, p. 216).

The Einstein summation notation simplifies expressions involving sums over multiple indices. Typically, such expressions are sums of products, with some indices summed over ("dummy" indices) and others left free ("free" indices).

The convention is: if an index appears exactly twice in a term, summation over that index is implied.

Dummy indices can be renamed without changing the meaning; free indices remain unchanged. In tensor notation, summed indices often pair as one superscript (upper) and one subscript (lower).

Cheat-sheet

Type Description Summation Notation
Vector expansion xieix^i e_i
Co-Vector expansion yieiy_i e^i
Metric Tensor basis change ei=gijeje^i = g^{ij}e_j
Metric Tensor dot product ⟨x,y⟩=x⋅y=gijxiyj\langle x, y \rangle = x \cdot y = g_{ij} x^i y^j
Matrix multiplication ui=Ajivju^i = A^i_j v^j
Matrix multiplication Cji=AkiBjkC^i_j = A^i_k B^k_j
Matrix inverse Aki(A−1)jk=δjiA^i_k (A^{-1})^k_j = \delta^i_j