How do I correctly manipulate tensor components in different coordinate systems?

[sweetdreams12]

Homework Statement

A tensor and vector have components T^α_β^γ, and v^α respectively in a coordinate system x^μ. There is another coordinate system x'^μ. Show that T^α_β^γv^β = T^αβγv_β

Homework Equations

umm not sure...

∇_αv_β = ∂v^β/∂x^α - Γ^γ_αβv_γ

The Attempt at a Solution

T^α_β^γv^β = (∂x^α/∂xⁱ*∂x^j/∂x^β*∂x^γ/∂x^k*Tⁱ_j^k)(∂x^β/∂x^a*v^a)

T^αβγv_β = (∂x^α/∂xⁱ*∂x^β/∂x^j*∂x^γ/∂x^k*T^ijk)(∂x^a/∂x^β*v^a)

and the two are not equal which they should be. I really don't know where I've went wrong...

 

[Orodruin]

How do you relate Tⁱ_j^k with T^ijk and v_a with v^a?

 

[sweetdreams12]

How do you relate Tⁱ_j^k with T^ijk and v_a with v^a?

A previous part asked to express T^αβγ in terms of T^α_β^γ and I put my answer as ∂xα/∂xi*∂x^β/∂x^j*∂xγ/∂xk*Ti^jj or by using the metric tensor

however does that also apply to vectors? if not, I don't know how v_α relates to v^α...

edit: wait is v_α related to v^α by v_α = ∂x^β/∂x^α*v'_β?

 

[Orodruin]

You are confusing the concepts of how a vector transforms under coordinate transformations to how a contravariant tensor relates to a covariant one. A contravariant index can be turned into a covariant one by contraction with the metric. Covariant and contravariant indices transform differently under coordinate transformations.

A vector is a tensor of rank one and its indices therefore follow precisely the same rules as any tensor indices - what distinguishes a vector is that it only has one.

 

[sweetdreams12]

You are confusing the concepts of how a vector transforms under coordinate transformations to how a contravariant tensor relates to a covariant one. A contravariant index can be turned into a covariant one by contraction with the metric. Covariant and contravariant indices transform differently under coordinate transformations.

A vector is a tensor of rank one and its indices therefore follow precisely the same rules as any tensor indices - what distinguishes a vector is that it only has one.

Okay so if I take it one step at a time: expressing T^αβγ in terms of T^α_β^γ without taking into account the vector, I use the metric tensor. g^α_μ(T^μ_β^γ) would then equal g^γ_μ(T^αβ_μ) = T^αβγ. Is that right?

 

[Orodruin]

In order to raise a covariant index, you need to contract it with one of the contravariant indices in g^ij. In the same way, in order to lower a contravariant index, you need to contract it with one of the indices in g_ij. Remember that you cannot contract covariant indices with each other but must always contract a covariant index with a contravariant one and vice versa.