References: Anthony Zee, Einstein Gravity in a Nutshell, (Princeton University Press, 2013) – Chapter I.4, Problem 2.
In Zee’s book, he defines a tensor as “something that transforms like a tensor”. For a tensor with indices, under a rotation specified by the matrix , the transformation of the tensor is given by multiplying the original tensor by one copy of for each index. For a 2-index tensor, for example
Another way of looking at this transformation is to think of each component of the tensor as a separate object in its own right. We can then arrange these objects in a column matrix (I’m avoiding calling this column matrix a ‘vector’ since, as Zee points out, vectors have a specific transformation property that this column matrix doesn’t have, namely that it must transform under a rotation by a multiplication by a single instance of a rotation matrix ). For 3-d, for example, we have the 9-component matrix
Under a rotation, we see from 1 that the transformed tensor component is a linear combination of the original components , where the coefficients of this linear transformation are found from the elements of the rotation matrix . This means that we could define a matrix which, in 3-d, is of size and whose elements are composed of combinations of the elements of . That is
where the index is summed from to . We can read off the first row of from 1, as this is the row of which provides the coefficients for producing the transformed component .
For a general rank 2 tensor (a tensor having 2 indices), there aren’t any pre-defined symmetries, so all the elements are independent of each other. As such, a transformed component could have a contribution from all 9 of the original components . However, it’s possible to create linear combinations of the original s such that a subset of these linear combinations transform into each other.
One such subset contains the antisymmetric combinations
Zee shows that an antisymmetric component transforms as
That is, antisymmetric components transform as linear combinations of only other antisymmetric components. In 3-d the index in can have 3 values, while can have only 2 (since is always zero by definition, we don’t count it). Also, since we’re after only components that are linearly independent of each other, we don’t count once we’ve counted , so there are a total of independent . In dimensions, there are independent antisymmetric combinations. These components transform entirely within their own private subset.
We can also define a set of symmetric components as
These components transform as follows:
In the third line, we swapped the dummy summed indices and . Thus the symmetric combinations also transform within their own subset. There are plus the diagonal components (no sum) which are, in general, non-zero, for a total of symmetric components. Together the antisymmetric and symmetric components contain all independent linear combinations in the original tensor . This means that any of the original tensor components can be written as a combination of the and as
This decomposition also works for diagonal elements since and (no sums).
If we write the original tensor in terms of the and , then (in 3-d) the matrix decomposes into a block diagonal matrix with a block for the and a block for the . That is, the transformation equation becomes
For example, if we want we have
The sums over and can now be worked out using the symmetry properties of these elements. For we have
Thus the third row of the block in the matrix (which is used to calculate ) is
We could do a similar calculation for except this time we’d get 6 terms in the transformation.
In fact the symmetric part of can be decomposed further by observing that the trace of the symmetric submatrix is invariant under rotation, as Zee shows in his equation 6 (sum implied over ):
Therefore the matrix breaks into a matrix and a matrix. Zee shows that the components of the block (or in the -dimensional case) are given by
Zee gives an example in 3-d showing that the components of do indeed transform into themselves.