Einstein summation convention

1. Einstein summation convention The convention was introduced by Einstein in 1916, who later jested to a friend, "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..." According to this convention, when an index variable appears twice in a single term, it implies that we are summing over all of its possible values. In typical applications, the index values are 1,2,3 (representing the three dimensions of physical Euclidean space), or 0,1,2,3 or 1,2,3,4 (representing the four dimensions of space-time, or Minkowski space), but they can have any range, even (in some applications) an infinite set. The starting point for the index notation is the concept of a basis of vectors. A basis is a set of linearly independent vectors that span the vector space. For example, in three-dimensional space, basis vectors Griffiths x1=x, x2=y, x3=z

2. summation convention rules http://ocw.mit.edu/NR/rdonlyres/Physics/8-07Fall-2005/ Summation Convention Rule #1 Repeated, doubled indices in quantities multiplied together are implicitly summed. Doubled indices in quantities multiplied together are sometimes called paired indices. If the writer’s intent is not to have the repeated index summed over, then this must be made explicit. Example: The index i has a particular value When the index appears only once, the index i is called a free index: it is free to take any value, and the equation must hold for all values. Indices that are summed over are called dummy indices. The names of dummy indices are arbitrary. Summation Convention Rule #2 Indices that are not summed over (free indices) are allowed to take all possible values unless stated otherwise.

3. Summation Convention Rule #3 It is illegal to use the same dummy index more than twice in a term unless its meaning is made explicit. free indices paired indices illegal

4. Vector operations Linear superposition: The dot product of two vectors: By Rule #1, there is an implied sum on both i and j where they occur paired. By Rule #3, it is mandatory that different indices be used for the expansion of A and B. Here we used the orthonormality property of basis vectors: The curl of two vectors: Kronecker delta squared length of vector A vector expansion in the vector basis let us write: Levi-Civita symbol (tensor)

5. A permutation of (123) is defined to be a rearrangement of them obtained by exchanging elements of the set. Even permutations have an even number of exchanges; odd permutations have an odd number. For example, (213) is an odd permutation of (123) but (231) is an even permutation. The even permutations (231) and (312) are often called cyclic permutations of (123) because they are obtained by rolling around in a cycle like links on a bicycle chain. There are three odd permutations of (123): (213), (132), and (321). Thus, the Levi-Civita symbol is zero aside from 6 terms. Swapping any two indices gives a sign-change: which explains why the Levi-Civita tensor is sometimes called the completely anti-symmetric tensor. Compare with Griffiths:

6. Important identity: Example 1: calculate Since Example 2: calculate In particular:

7. Example 3: calculate

8. Matrix (tensor) operations Second-rank tensor: trace of matrix M product of two matrices symmetric antisymmetric