Construction of an affine tensor of rank 4

[Whitehole]

Homework Statement

In E_n the quantities B_ij are the components of an affine tensor of rank 2. Construct two affine tensors each of rank 4, with components C_ijkl and D_ijkl for which

∑_k ∑_l C_ijkl B_kl = B_ij + B_ji

∑_k ∑_l D_ijkl B_kl = B_ij - B_ji

are identities.

Homework Equations

The Attempt at a Solution

Can anyone give me a hint on how to start? I just don't know how to start.

 

[Samy_A]

Homework Statement

In E_n the quantities B_ij are the components of an affine tensor of rank 2. Construct two affine tensors each of rank 4, with components C_ijkl and D_ijkl for which

∑_k ∑_l C_ijkl B_kl = B_ij + B_ji

∑_k ∑_l D_ijkl B_kl = B_ij - B_ji

are identities.

Homework Equations

The Attempt at a Solution

Can anyone give me a hint on how to start? I just don't know how to start.

Maybe the first post of this old thread can help you on the way: https://www.physicsforums.com/threads/isotropic-tensors.106292/

 

[Whitehole]

Maybe the first post of this old thread can help you on the way: https://www.physicsforums.com/threads/isotropic-tensors.106292/

The link in the first post in that thread is broken. Can you help me by just giving a hint?

 

[Samy_A]

The link in the first post in that thread is broken. Can you help me by just giving a hint?

There is no link in the first post of that thread.
The tensor mentioned in the first post of that thread was the hint, more specifically $\delta_{ab}\delta_{cd}$.

 

[Whitehole]

There is no link in the first post of that thread.
The tensor mentioned in the first post of that thread was the hint, more specifically $\delta_{ab}\delta_{cd}$.

This is what I did,

Let C_ijkl = δ_ik δ_jl + δ_il δ_jk

Multiply both sides by B_kl then take the sum

∑_k∑_l C_ijkl B_kl = ∑_k∑_l (δ_ik δ_jl B_kl + δ_il δ_jk B_kl)

Then from the dirac delta in the first term k=i and l=j, as for the second term l=i and k=j

Thus ∑_k∑_l C_ijkl B_kl = B_ij + B_ji

The same process goes for the second question. Is this correct?

 

[Samy_A]

This is what I did,

Let C_ijkl = δ_ik δ_jl + δ_il δ_jk

Multiply both sides by B_kl then take the sum

∑_k∑_l C_ijkl B_kl = ∑_k∑_l (δ_ik δ_jl B_kl + δ_il δ_jk B_kl)

Then from the dirac delta in the first term k=i and l=j, as for the second term l=i and k=j

Thus ∑_k∑_l C_ijkl B_kl = B_ij + B_ji

The same process goes for the second question. Is this correct?

Yes, that's what I meant.

 

[Whitehole]

Yes, that's what I meant.

Oh, thanks! But what does that relation mean?

 

[Samy_A]

Oh, thanks! But what does that relation mean?

I don't know. Contraction of a rank 2 tensor with C gives a symmetric rank 2 tensor. Contraction of a rank 2 tensor with D gives an antisymmetric rank 2 tensor.
Whether is means something, and if so, what it means, I don't know.

Was this an exercise, or is this used somewhere in a theoretical context?

 

[Whitehole]

I don't know. Contraction of a rank 2 tensor with C gives a symmetric rank 2 tensor. Contraction of a rank 2 tensor with D gives an antisymmetric rank 2 tensor.
Whether is means something, and if so, what it means, I don't know.

Was this an exercise, or is this used somewhere in a theoretical context?

Yes this is an exercise in Tensors, Differential Forms, and Variational Principles by Lovelock and Rund. Problem 2.7. Thank you very much!