Einstein Notation

An arrow in space can be stored as three numbers: how far it reaches along each of the three axes. Call them $a_1$, $a_2$ and $a_3$. The small number is an index; it says which component you mean. To measure how much two arrows point the same way (the dot product), multiply matching components and add the results:

In the sum on the right, the letter being summed over, $i$, appears exactly twice in the term. Einstein noticed that this was true of nearly every sum in his equations, so the Σ sign told the reader nothing that the repeated letter didn’t already say. He stopped writing it:

A repeated index is called a dummy. It is used up by the sum, so its name does not matter: $a_i{b}_i$ and $a_k{b}_k$ are the same number. An index that appears only once in a term is free. It survives, so the expression has one value for each choice of it, and one line of notation stands for several equations.

Two checks catch most mistakes before any algebra is done:

1. No index appears more than twice in one term. Three copies leave it unclear which two are paired.
2. Every term, on both sides, has the same free indices. Otherwise you are claiming that a list labelled by one letter equals a list labelled by another.

The simplest object with two indices is a yes/no test for “are these the same index?”

Its everyday job is renaming. In ${\delta }_{ij}{a}_j$ the sum over $j$ runs through three terms, and the delta switches off every one except the term where $j$ equals $i$:

So a delta next to a matching dummy index can be deleted, as long as the other index takes over its name. Summing the delta with itself counts the matching pairs, which is the number of dimensions:

${\epsilon }_{ijk}$ takes three indices and keeps track of the order they come in. It is defined by three facts:

• In the reference order it equals one: ${\epsilon }_{123}=1$.
• Swapping any two indices flips the sign: ${\epsilon }_{213}=-1$.
• If two indices are equal it is zero: ${\epsilon }_{112}=0$. Swapping the two equal indices would have to flip the sign and also change nothing, and only 0 does both.

Of its 27 entries only 6 are not zero: +1 for the orders 123, 231 and 312, where each one is the last rotated to the front, and −1 for 132, 213 and 321.

The cross product of two arrows, which points at right angles to both, is a single epsilon:

Take $i=1$. Of the nine pairs $(j,k)$, only (2, 3) and (3, 2) give a non-zero epsilon, so the sum leaves two terms:

Choose “cross product” in the expander above to see all nine terms for each $i$, with the zeros dropped. The same symbol also gives the determinant of a 3 × 3 matrix: $\det A={\epsilon }_{ijk}{A}_{1i}{A}_{2j}{A}_{3k}$.

When two epsilons share one summed index, the product can always be rewritten with deltas alone. This single line replaces most vector-identity tables:

Why it works: if $i$ and $j$ differ, exactly one value of $k$ (the third number) makes ${\epsilon }_{kij}$ non-zero. The second epsilon is then non-zero only if $m,n$ are the same two numbers. In the same order the two signs match and multiply to +1; in reversed order they are opposite and give −1. If $i=j$, both sides are 0.

The payoff is speed. The “BAC−CAB” rule for a triple cross product drops out in four lines, with no components written down:

A matrix is a table with two indices, row first, then column. The usual matrix rules become one-line index statements, and the order of factors stops mattering because each factor is just a number:

Pick “matrix product” or “trace” in the expander to check the index form against the sample matrices.

Einstein introduced the convention in his 1916 paper setting out general relativity. There the indices are Greek letters such as $\mu$ and $\nu$, running over four values: 0 for time and 1–3 for space. Almost every line contained sums over them.

Spacetime adds one refinement. Indices sit either upstairs or downstairs, and a sum always pairs one of each. The metric $g_{\mu \nu }$ converts between the two and carries the geometry. The spacetime distance between two nearby events is

The field equations themselves, $G_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }$, have two free indices: 16 equations in one line, 10 of them independent because both sides are symmetric. In ordinary flat space with square axes, upstairs and downstairs components are equal, which is why the sections above only needed lower indices.