# Torsion tensor definition doubt

[Ref. Core Principles of Special and General Relativity by Luscombe, page 246]

Let's say we have any two covariant derivative operators $$\nabla$$ and $$\nabla'$$. Then there exists a tensor $$C^{\alpha}_{\mu\nu}$$ such that for all covariant vectors $$\omega_{\nu}$$, $$\nabla_{\mu}\omega_{\nu}=\nabla'_{\mu}\omega_{\nu}-C^{\alpha}_{\mu\nu}\omega_{\alpha}$$

Now I'm quoting the relevant section on torsion tensor definition:

What if the no-torsion requirement is dropped? Set $$\omega_{\nu}=\nabla_{\nu}\phi=\nabla'_{\nu}\phi$$: (which gives) $$\nabla_{\mu}\nabla_{\nu}\phi=\nabla'_{\mu}\nabla'_{\nu}\phi-C^{\alpha}_{\mu\nu}\nabla_{\alpha}\phi$$. Antisymmetrize over $$\mu$$ and $$\nu$$, and assume $$\nabla'$$ is torsion free, but $$\nabla$$ is not. In that case $$\nabla_{[\mu}\nabla_{\nu]}\phi=-C^{\alpha}_{[\mu\nu]}\nabla_{\alpha}\phi$$. The torsion tensor is defined as $$T^{\alpha}_{\mu\nu}\equiv 2C^{\alpha}_{[\mu\nu]}$$, implying that $$(\nabla_{\mu}\nabla_{\nu}-\nabla_{\nu}\nabla_{\mu})\phi=-T^{\alpha}_{\mu\nu}\nabla_{\alpha}\phi$$

I don't understand why this is so. I mean the LHS is can also be notationally represented as $$\nabla_{[\mu}\nabla_{\nu]}\phi$$, so either there should be a factor of $$1/2$$ on the RHS, or the torsion tensor should be defined as $$T^{\alpha}_{\mu\nu}\equiv C^{\alpha}_{[\mu\nu]}$$, or am I missing something?

If your confusion is with the apparently missing factor of $$1/2$$, note that

$$\nabla_{[a}\nabla_{b]} \equiv \frac{1}{2}(\nabla_a\nabla_b-\nabla_b\nabla_a)$$

Symmetrization and antisymmetrization brackets come defined with a factor of $$1/2$$, because they are meant to extract the symmetric and antisymmetric parts of the relevant tensors. With this in mind, your equation becomes

$$\nabla_\mu\nabla_\nu \phi - \nabla_\nu\nabla_\mu\phi = 2\nabla_{[\mu}\nabla_{\nu]}\phi = -2C^\alpha_{[\mu\nu]}\nabla_a \phi \equiv -T^\alpha_{\mu\nu}\nabla_\alpha\phi$$

The if $$X$$ and $$Y$$ are (contravariant) vector fields, the torsion tensor $$T(X,Y)$$ is defined as $$\nabla_X Y-\nabla_Y X-[X,Y]=T(X,Y)$$ where $$T(X,Y)^\lambda= {T^\lambda}_{\mu\nu}X^\mu Y^\nu$$, and $$[X,Y]^\nu= X^\mu \partial_\mu Y^\nu- Y^\mu\partial_\mu X^\nu$$ is the Lie bracket of the vector fields.

If $$(\nabla_X Y)^\lambda = X^\mu (\nabla_\mu Y^\lambda + {\Gamma^\lambda}_{\mu \nu} Y^\nu)$$ then $${T^\lambda}_{\mu\nu}= {\Gamma^\lambda}_{\mu \nu}-{\Gamma^\lambda}_{\nu \mu}$$.

The notation $$[\nabla_\mu,\nabla_\nu]$$ is potentialy unsafe, although common. This is because $$"\nabla_\mu"$$ is a acting on a different tenor space depending whether is acts before or after $$\nabla_\nu$$, so the "commutator" is not really a commutator. That why I prefer to use $$\nabla_X$$. With this notation $$[\nabla_X,\nabla_Y]Z- \nabla_{[X,Y]}Z = R(X,Y)Z$$ whether there is torsion or not.