188 11 Isotropic Tensors
11.3.2 Properties ofh-Tensors.
The-tensors are antisymmetric against the exchange of the fore and hind subscripts,
viz.
()μ
1 μ 2 ···μ,λ,μ′ 1 μ′ 2 ···μ′
=−()μ′
1 μ′ 2 ···μ′,λ,μ^1 μ^2 ···μ
, (11.23)
furthermore, they vanish whenever the middle index is equal to one of the fore or
hind indices, e.g.
()μ 1 μ 2 ···μ
,μ,μ′ 1 μ′ 2 ···μ′
= 0. (11.24)
For=2, in particular, one has
μν,μ,σ τ= 0.Due to the product properties of epsilon-tensor, cf. Sect. 4 , the multiplication of a
-tensor with an epsilon-tensor or the product of two-tensors, yieldsΔ-tensors
which, in turn, are products ofδ-tensors. For example, one has
εμλμ′()μ 1 μ 2 ···μ
,λ,μ′ 1 μ′ 2 ···μ′=
+ 1
2 + 1
2 − 1
Δ(μ 1 −μ^12 )···μ
− 1 ,μ′ 1 μ′ 2 ···μ′− 1, (11.25)
and
()μ 1 μ 2 ···μ,λ,ν 1 ν 2 ···ν()ν
1 ν 2 ···ν,λ,μ′ 1 μ′ 2 ···μ′=−
+ 1
Δ()μ
1 μ 2 ···μ,μ′ 1 μ′ 2 ···μ′. (11.26)
The special case=1 of this relation corresponds toεμλνενλμ′=− 2 δμμ′,cf.
Sect.4.1.2.
Some additional relations are listed for the case=2, which follow from the
explicit form of the-tensor (11.18):
λκ,ν,σ τσ τ,λ,μκ=5
4
δμν, (11.27)and
μν,λ,σ τμ′ν,λ′,σ τ=1
8
( 9 δμμ′δλλ′− 6 δμλ′δμ′λ−δμλδμ′λ′). (11.28)The relation (11.27) is recovered from (11.28) with the contractionμ=λ′, the use
of (11.23) and the appropriate renaming of indices.