Torsion tensor: definition

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?

• This looks fine to me. – Lawrence B. Crowell Jul 10 '17 at 19:39
• The definition is actually $\nabla_b e_c = \Gamma^a_{bc}$ – Slereah Jul 10 '17 at 19:51
• @Slereah according to Wikipedia it is not, if i'm not mistaken wheeler too uses my convention – raskolnikov Jul 10 '17 at 20:03
• @Slereah hawking in the structure of space time uses my – raskolnikov Jul 10 '17 at 20:04
• The answer is of course that there are varying sign conventions for torsion/curvature and varying conventions for the definition of $\Gamma^i{}_{jk}$. – Ryan Unger Jul 11 '17 at 0:24

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

• Ok, but if I want to use my convention $\nabla_b e_c = \Gamma^a_{cb}$ and to be self consistent with the definition of $\mathbf{T}$ with covariant derivative what I have to do? Because if I leave the definition as it is written in the question I have two different expression of the torsion tensor. May I have to put a minus in front of the definition of $\mathbf{T}$? (Indeed, I have to define the torsion tensor with the opposite sign convention $\mathbf{T}(\mathbf{X},\mathbf{Y})=-\left(\nabla_{\mathbf{X}}\mathbf{Y}-\nabla_{\mathbf{Y}}\mathbf{X} -\left[\mathbf{X},\mathbf{Y}\right]\right)$) – raskolnikov Jul 11 '17 at 8:46
• Can be a correct solution? O it's wrong? – raskolnikov Jul 11 '17 at 8:47

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.