42.1.1 Kinematics of an anti-symmetric tensor

Anti-symmetry introduces constraints on the form of mixing described by anti-symmetric tensors. All these constraints originate from the property that the tensor contraction of a symmetric tensor with an anti-symmetric tensor vanishes. Explicitly, consider a symmetric tensor ( S^{m n} = S^{n m}) and its contraction (under a flat Euclidean metric) with an anti-symmetric tensor ( A^{m n} = - A^{n m}).

S_{m n}A^{m n} = S^{1 2} A^{1 2} + S^{2 1} A^{2 1} + S^{1 3} A^{1 3} + S^{3 1} A^{3 1} + S^{2 3} A^{2 3} + S^{3 2} A^{3 2} (42.4)

= S^{1 2} A^{1 2} - S^{1 2} A^{1 2} + S^{1 3} A^{1 3} - S^{1 3} A^{1 3} + S^{2 3} A^{2 3} - S^{2 3} A^{2 3} (42.5)

= 0, (42.6)

where the symmetry of S^{m n} and the anti-symmetry of A^{m n}were used. This property implies, for example, that

   

$$A^{m n} \partial_{m}T \partial_{n} T$$ = 0 (42.7)

$$A^{m n} \partial_{m}\partial_{n}T$$ = 0 (42.8)

$$\partial_{m} \partial_{n} A^{m n}$$ = 0. (42.9)