Wheeler, Mechanics

epeated in a single term on the left, while i, j and k are each repeated on the right. The summed indices
are called dummy indices, and single indices are called free indices. In every term of every equation the free
indices must balance. We have a different equation for each value of each free index. The dummy indices
can be changed for our convenience as long as we don’t change their positions or repeat the same dummy
index more than the required two times in any one term. Thus, we can write
or
but not
T ijk v j w k + ω i = S ij u j
T imn v m w n + ω i = S ij u j
T ijj v j w j + ω i = S im u m
because using the same dummy index twice in the term T ijj v j w j means we don’t know whether to sum v j
with the second or the third index of T ijk . We will employ the Einstein summation convention throughout
the remainder of the book, with one further modification occurring in Section 5.
Defining the transpose of R ij by
R t ij = R ji
we may write the defining equation as
R t jix j R ik x k = x i x i
Index notation has the advantage that we can rearrange terms as long as we don’t change the index relations.
We may therefore write
R t jiR ik x j x k = x i x i
Finally, we introduce the identity matrix,
so that we can write the right hand side as
⎛
δ ij = ⎝ 1 1
1
⎞
⎠
x i x i = x j δ jk x k = δ jk x j x k
Since there are no unpaired, or “free” indices, it doesn’t matter that we have i on the left and j, k on the
right. These “dummy” indices can be anything we like, as long as we always have exactly two of each kind
in each term. In fact, we can change the names of dummy indices any time it is convenient. For example, it
is correct to write
x i x i = x j x j
or
We now have
x ′ i = R ij x j = R ik x k
R t jiR ik x j x k = δ jk x j x k
⎞
This expression must hold for all x j and x k . But notice that the product x j x k is really a symmetric matrix,
the outer product of the vector with itself,
⎛
x 2 xy xz
zx zy z 2
x j x k = ⎝ yx y 2 yz ⎠
Beyond symmetry, this matrix is arbitrary.
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