Tensors for Physics

(Marcin) #1

11.6 Coupling of Second Rank Tensors with Irreducible Tensors 195


bνbλ bμ 1 bμ 2 ···bμ =bνbλbμ 1 bμ 2 ···bμ

+

2 

2 + 3

b^2 Δ(,μ 12 μ,) 2 ···μ,νλ,ν 1 ν 2 ···νbν 1 bν 2 ···bν (11.57)

+

(− 1 )

( 2 + 1 )( 2 − 1 )

b^4 Δ()μ 1 μ 2 ···μ,νλν 1 ν 2 ···ν− 2 bν 1 bν 2 ···bν− 2.

For=1, the (11.54) is recovered. Of special interest is the case=2. Here (11.57)
is equivalent to


bμbν bλbκ =bμbνbλbκ+

4

7

b^2 Δμν,λκ,σ τbσbτ +

2

15

b^4 Δμν,λκ. (11.58)

Multiplication of this equation with symmetric tensorsaμνandcλκyields


aμνbμbνcλκbλbκ =aμνcλκbμbνbλbκ +

4

7

b^2 aμνbνbλcλμ+

2

15

b^4 aμνcμν.
(11.59)
Applications of these relations involving the products of irreducible tensors are found
in the following sections.


11.7 Scalar Product of Three Irreducible Tensors


11.7.1 Scalar Invariants


Consider three symmetric irreducible tensorsa,b,c,viz.aμ 1 μ 2 ·μ 1 ,bν 1 ν 2 ·ν 2 and
cκ 1 κ 2 ·κ. Provided that= 1 + 2 , their product and total contraction yields the
scalar


aμ 1 μ 2 ·μ 1 bν 1 ν 2 ·ν 2 cμ 1 μ 2 ···μ 1 ν 1 ν 2 ···ν 2.

However, using in the product of the tensorsa,btwo subscripts which are equal
and summed over, one also obtains a scalar, when= 1 + 2 −2 applies and the
appropriate contractions are performed. In this case one has


aμ 1 ·μ 1 − 1 λbν 1 ·ν 2 − 1 λcμ 1 μ 2 ···μ 1 − 1 ν 1 ν 2 ···ν 2 − 1.

Obviously, this can be generalized to form scalars from the product of three tensors
of ranks 1 , 2 andwith


= 1 + 2 , 1 + 2 − 2 ,...,| 1 − 2 |.
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