I have read tons of questions about this topic but I think my particular issue is not solved. If so, please let me know.

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?

  • $\begingroup$ No, I mean $\Gamma$. I suppose $\mathcal{T}$ would transform like that in any situation for it is a tensor. $\endgroup$
    – Pablo
    Mar 14, 2021 at 14:06
  • $\begingroup$ 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? $\endgroup$
    – Pablo
    Mar 14, 2021 at 14:30


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