**Einstein Summation**, introduced by Einstein in 1916 in his generalized theory of relativity paper, is a shorthand way to avoid the tedium of writing summations. It is stated as:

“If a suffix occurs twice in a term, once in the lower position and once in the upper position, that suffix implies sum over a defined range.”

This convention tells us that expressions with a repeated index (one raised and one lowered) can have the sign of summation omitted; Instead, you can implicitly sum over that index from 1 to n (Weinstein, 2016). If the range isn’t specified, you can assume it is from 1 to n (Islam, 2006).

The Einstein summation convention is used frequently in physics. For example, index notation is useful for keeping track of vector components **v** = (v_{1}, v_{2}, v_{3}). It is common in physics for Greek indices (used for space and time components), but not necessarily Latin indices (used for spatial components) (Helrich, 2016).

## Examples of Einstein Summation

A very simple example of Einstein summation is the following definition of a dot product:

**Aâ€¢B** = A_{i} B_{i}

A slightly more complex example involves a series:

A repeated index (usually

*i*) means to sum over that index.

## Einstein Summation Rules

Einstein summation only applies to very specific summations which follow **four basic rules** (Evans, 2020):

- The summation sign is omitted.
- A summation is implied if the index appears twice. For example, A
_{i}B_{i}= A_{1}sub>B_{1}+ A_{2}B_{2}+ A_{3}B_{3}, where*i*is a dummy index. It’s called a “dummy index” as you can use anything here (i, j, k,…) and it doesn’t change the summation. - A suffix appearing just once can take any value. For example, A
_{i}= B_{i}holds for i = 1, 2, 3. (there can be multiple free indices*i*).

Note that there may be more than one free index and they must be the same on both sides of an equation. For example, G_{i}= H_{j}is wrong. - A suffix can’t appear more than two times.

## References

Evans, M. (2020). Lecture 5: More About Suffix Notation. Retrieved November 18, 2020 from: https://www2.ph.ed.ac.uk/~mevans/mp2h/VTF/lecture05.pdf

Helrich, C. (2016). Analytical Mechanics. Springer.

Weinstein, G. (2016). General Relativity Conflict and Rivalries: Einstein’s Polemics with Physicists. Cambridge Scholar Publishing.

Islam, N. (2006). Tensor Algebra. New Age International.