- #1
nigelscott
- 135
- 4
I am trying to figure how to get 1. from 2. and vice versa where the e's are bases for the vector space and θ's are bases for the dual vector space.
1. T = Tμνσρ(eμ ⊗ eν ⊗ θσ ⊗ θρ)
2. Tμνσρ = T(θμ,θν,eσ,eρ)
My attempt is as follows:
2. into 1. gives T = T(θμ,θν,eσ,eρ)(eμ ⊗ eν ⊗ θσ ⊗ θρ)
Now if I assume that (θμ,θν,eσ,eρ) Ξ (θμ ⊗ θν ⊗ eσ ⊗ eρ) this becomes:
T = T(θμ ⊗ θν ⊗ eσ ⊗ eρ)(eμ ⊗ eν ⊗ eσ ⊗ θρ)
= θμeμ ⊗ θνeν ⊗ eσθσ ⊗ eρθρ
Now using θνeμ = δνμ this becomes:
T = T(I ⊗ I ⊗ I ⊗ I)
So T = T
This seems to work but I'm not sure if this is the correct way to do it. I'm shaky on the tensor product stuff and my interpretation of T(_,_,_,_). Does this look right?
1. T = Tμνσρ(eμ ⊗ eν ⊗ θσ ⊗ θρ)
2. Tμνσρ = T(θμ,θν,eσ,eρ)
My attempt is as follows:
2. into 1. gives T = T(θμ,θν,eσ,eρ)(eμ ⊗ eν ⊗ θσ ⊗ θρ)
Now if I assume that (θμ,θν,eσ,eρ) Ξ (θμ ⊗ θν ⊗ eσ ⊗ eρ) this becomes:
T = T(θμ ⊗ θν ⊗ eσ ⊗ eρ)(eμ ⊗ eν ⊗ eσ ⊗ θρ)
= θμeμ ⊗ θνeν ⊗ eσθσ ⊗ eρθρ
Now using θνeμ = δνμ this becomes:
T = T(I ⊗ I ⊗ I ⊗ I)
So T = T
This seems to work but I'm not sure if this is the correct way to do it. I'm shaky on the tensor product stuff and my interpretation of T(_,_,_,_). Does this look right?