# Torsion tensor and affine connection symbols

So I want to prove that the torsion tensor $$\mathcal{T}$$ actually transforms like a tensor. It is defined as $$\mathcal{T}_{\beta \gamma}^{\alpha}=\Gamma_{\beta \gamma}^{\alpha}-\Gamma_{\gamma \beta}^{\alpha},$$ where the $$\Gamma$$ are the affine connection symbols.

In Relativity class we studied that the transformation of $$\Gamma$$ between two reference systems $$\bar{\alpha}$$ and $$\alpha$$ is given by

$$\begin{equation} \Gamma_{\bar{\beta} \bar{\gamma}}^{\bar{\alpha}}=\Gamma_{\beta \gamma}^{\alpha} \frac{\partial x^{\bar{\alpha}}}{\partial x^{\alpha}} \frac{\partial x^{\beta}}{\partial x^{\bar{\beta}}} \frac{\partial x^{\gamma}}{\partial x^{\bar{\gamma}}}+\frac{\partial^{2} x^{\alpha}}{\partial x^{\bar{\beta}} \partial x^{\bar{\gamma}}} \frac{\partial x^{\bar{\alpha}}}{\partial x^{\alpha}}, \end{equation}$$

and it certainly leads me to $$\mathcal{T}_{\bar{\beta}\bar{\gamma}}^{\bar{\alpha}} = \frac{\partial x^{\bar{\alpha}}}{\partial x^{\alpha}} \frac{\partial x \beta}{\partial x^{\bar{\beta}}} \frac{\partial x^{\gamma}}{\partial x^{\bar{\gamma}}} \mathcal{T}_{\beta \gamma}^{\alpha},$$

which proves that the torsion tensor transforms as a tensor.

My question is: what would happen if the connection represented by $$\Gamma$$ wasn't an affine one? Would the transformation of $$\Gamma$$ be different?

• No, I mean $\Gamma$. I suppose $\mathcal{T}$ would transform like that in any situation for it is a tensor. – Patrick Mar 14 at 14:06
• I think a tensor would still be a tensor. More specifically my question is: what would change in my derivation if the connection weren't affine? – Patrick Mar 14 at 14:30