Torsion tensor: definition

Question

The definition of torsion tensor is the following:

$$ \mathbf{T}(\mathbf{X},\mathbf{Y})=\nabla_{\mathbf{X}}\mathbf{Y}-\nabla_{\mathbf{Y}}\mathbf{X} -\left[\mathbf{X},\mathbf{Y}\right]. $$

In an holonomic base $\left[\mathbf{e}_a,\mathbf{e}_b\right]=0$ the coordinates are

$$ T{^a_{\ \ bc}}e_a=\nabla_b e_c-\nabla_c e_b=(\Gamma^a_{\ cb}-\Gamma^a_{\ bc})e_a $$

due to $ \nabla_b\mathbf{e}_c=\Gamma^a_{\ \ cb}\mathbf{e}_a$. This result is different from the general result

$$ T^a_{\ \ bc}= \Gamma^a_{bc}-\Gamma^a_{cb}. $$

Where am I wrong in this calculations?

Answer 1

It is just a matter of convention. MTW uses the convention that you have followed. Others, like Wald, use different one. Just a nuisance.

Answer 2

The connection (in this case the Levi-Civita conn.) $\Gamma^\alpha{}_\beta$ is a one-form. We can express it as (in a coordinate (holonomic) basis) $$\Gamma^\alpha{}_\beta \equiv \Gamma_\mu{}^\alpha{}_\beta\, dx^\mu \tag 1$$ on another hand in the case of the spin-connection we have $$\omega^I{}_J \equiv \omega_\mu{}^I{}_J\, dx^\mu \tag 2$$ There are two types of the indices on the above equations. I would call the indices that appearLHS of the equations only: $\{\alpha,\beta\}$ for eqn(1) and $\{ I,J \}$ for eqn (2) the vector space indices or transformation indices (not the standard words), and I would call the index $\mu$ on both eqn. the form index.

Consider an equation write with only the vector space indices $$ \nabla {\bf e}^a = \Gamma^a{}_b {\bf e}_a \;.$$ There are two conventions to write in term with the form index

CONVENTION 1 $$ \nabla_c {\bf e}^a = \Gamma_c{}^a{}_b {\bf e}_a =: \Gamma^a_{\underline{c}b}{\bf e}_a\;,$$ where the underline indicate that it is a form index.

CONVENTION 2 $$ \nabla_c {\bf e}^a = \Gamma^a{}_{bc} {\bf e}_a =: \Gamma^a_{b \underline{c}}{\bf e}_a\;,$$ where the underline indicate that it is a form index. In the both case $c$ is a form index.

If you notice that which index is a form index you will never confuse.

Your second equation $$T{^a_{\ \ bc}}e_a=\nabla_b e_c-\nabla_c e_b=(\Gamma^a_{\ cb}-\Gamma^a_{\ bc})e_a \tag 3$$ is obviously use the CONVENTION 2 , we can write $$T{^a_{\ \ bc}}e_a=\nabla_b e_c-\nabla_c e_b=(\Gamma^a_{\ c\underline b}-\Gamma^a_{\ b\underline c})e_a \tag 4$$ but your third equation us the CONVENTION 1 we can write it as $$ T^a_{\ \ bc}= \Gamma^a_{\underline b c}-\Gamma^a_{\underline c b}. \tag 5 $$

$T^a_{\ \ bc}$ in (4) and (5) are the same.