Antisymmetric connection (Torsion Tensor)

[psimeson]

How to show:

T^a_bc = \Gamma^a_bc - \Gamma^a_cb

is a Tensor of rank (1,2)

Attempted solution:
1. Using definition of Covariant Derivative:
D_bT^a= ∂_aT^a+\Gamma^a_bcT^c (1)

D_cT^a= ∂_cT^a+\Gamma^a_cbT^b (2)

I subtracted (2) from (1) but I couldn't really get a Tensor out of it. I just got lost in the mess.

Is this is the right way to start it?

 

[arkajad]

Do you have to use covariant derivatives in your problem? Is it a hint in your problem? There are several ways to show your property.

And why do you say "antisymmetric connection?"

 

[meldraft]

I am also in the process of learning tensor calculus, so I may not be right, but wouldn't it work if you raised the indices and made every tensor ab-contravariant?

 

[arkajad]

Which text are you using? There are different ways of showing your property, but the method should be adapted to what you already know.

 

[psimeson]

@arkajad: Covariant derivative is not a hint in the problem. I am just trying to solve that way. I am following various kind of textbooks. So, any way would work for me.

@meldraft: I am sure if that will work. Since the purpose of this exercise is to show how the difference between two Christoffel symbols that are asymmetric gives rise to torsion tensor.

 

[arkajad]

@arkajad: Covariant derivative is not a hint in the problem. I am just trying to solve that way. I am following various kind of textbooks. So, any way would work for me.

Probably the simples way, for you, is to look at the transformation equations for the connection coefficients, and from that find out how the torsion will transform.

Check Eq. (3.6) in

http://preposterousuniverse.com/grnotes/grnotes-three.pdf

But do not read further than that!

 

[psimeson]

Solved.