Of the 16 components Aμν, the four components Aμμ vanish, the rest are equal and opposite in pairs; so that there are only 6 numerically different components present (Six-vector).
Thus we also see that the antisymmetrical tensor Aμνσ (3rd rank) has only 4 components numerically different, and the antisymmetrical tensor Aμνστ only one. Symmetrical tensors of ranks higher than the fourth, do not exist in a continuum of 4 dimensions.
§ 7. Multiplication of Tensors.
Outer multiplication of Tensors:—We get from the components of a tensor of rank z, and another of a rank z′, the components of a tensor of rank (z + z′) for which we multiply all the components of the first with all the components of the second in pairs. For example, we obtain the tensor Τ from the tensors A and B of different kinds:—
Τμνσ = AμνBσ,
Ταβγδ = AαβBγδ,
Ταβγδ = AαβBγδ.
The proof of the tensor character of Τ, follows immediately from the expressions (8), (10) or (12), or the transformation equations (9), (11), (13); equations (8), (10) and (12) are themselves examples of the outer multiplication of tensors of the first rank.
Reduction in rank of a mixed Tensor.
From every mixed tensor we can get a tensor which is two ranks lower, when we put an index of co-variant character equal to an index of the contravariant character and sum according to these indices (Reduction). We get for example, out of the mixed tensor of the fourth rank Aαβγδ, the mixed tensor of the second rank