Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 6; Problem 6.6.
Tensors, like matrices, can be symmetric or anti-symmetric. Since a tensor can have a rank higher than 2, however, a single tensor can have more than one symmetry. For a rank-2 tensor , it is symmetric if and anti-symmetric if . In matrix terminology, a symmetric rank-2 tensor is equal to its transpose, and an anti-symmetric rank-2 tensor is equal to the negative of its transpose.
A higher rank tensor can be symmetric or anti-symmetric in any pair of its indices, provided both indices are either upper or lower. For example, can be symmetric or anti-symmetric in any pair selected from or in the pair , but not in one upper and one lower index. Thus if and , then is symmetric in and , and anti-symmetric in and .
Returning to rank-2 tensors, we can show that the symmetry property is an invariant:
If a tensor is symmetric in a pair of upper indices, then if both indices are lowered, the resulting tensor is also symmetric in the two lower indices:
Similarly, the anti-symmetric property persists through lowering of indices. If
If then all diagonal elements must be zero, since has only zero as a solution. Also, the trace is
In line 3, we used , since in terms of the basis vectors, , and thus the metric tensor is symmetric. Thus the trace is also zero for an anti-symmetric tensor.
A rank 2 symmetric tensor in dimensions has all the diagonal elements and the upper (or lower) triangular set of elements as independent components, so the total number of independent elements is . An anti-symmetric tensor has zeroes on the diagonal, so it has independent elements.