Tensor coordinate system help

  • #1

Homework Statement


A tensor and vector have components Tαβγ, and vα respectively in a coordinate system xμ. There is another coordinate system x'μ. Show that Tαβγvβ = Tαβγvβ

Homework Equations


umm not sure...

αvβ = ∂vβ/∂xα - Γγαβvγ

The Attempt at a Solution


Tαβγvβ = (∂xα/∂xi*∂xj/∂xβ*∂xγ/∂xk*Tijk)(∂xβ/∂xa*va)

Tαβγvβ = (∂xα/∂xi*∂xβ/∂xj*∂xγ/∂xk*Tijk)(∂xa/∂xβ*va)

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

Answers and Replies

  • #2
Orodruin
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How do you relate Tijk with Tijk and va with va?
 
  • #3
How do you relate Tijk with Tijk and va with va?
A previous part asked to express Tαβγ in terms of Tαβγ and I put my answer as ∂xα/∂xi*∂xβ/∂xj*∂xγ/∂xk*Tijj or by using the metric tensor

however does that also apply to vectors? if not, I don't know how vα relates to vα...

edit: wait is vα related to vα by vα = ∂xβ/∂xα*v'β?
 
Last edited:
  • #4
Orodruin
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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.
 
  • #5
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 Tαβγ in terms of Tαβγ without taking into account the vector, I use the metric tensor. gαμ(Tμβγ) would then equal gγμ(Tαβμ) = Tαβγ. Is that right?
 
  • #6
Orodruin
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In order to raise a covariant index, you need to contract it with one of the contravariant indices in gij. In the same way, in order to lower a contravariant index, you need to contract it with one of the indices in gij. Remember that you cannot contract covariant indices with each other but must always contract a covariant index with a contravariant one and vice versa.
 

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