# Commutator of derivatives with torsion

I am currently looking at "Physical aspect of space-time torsion" by IL Shapiro. There in eq. 2.10, it mentions that in space-time with torsion the commutator of covariant derivatives acting on the scalar $$\phi$$ gives $$[\nabla_\mu, \nabla_\nu]\phi = K^\lambda_{~~~\mu\nu}\partial_\lambda \phi$$ Where $$K$$ is the contorsion tensor. Clearly this is incorrect. Shouldnt we instead have $$[\nabla_\mu, \nabla_\nu]\phi = T^\lambda_{~~~\mu\nu}\partial_\lambda \phi$$ Where $$T$$ is torsion? Or am I missing something?

• You should specify the equation no. or page no. etc. while citing a specific equation from a reference. – Dvij D.C. Jun 4 at 15:18

Torsion is always defined by $$T(X,Y)= \nabla_X Y-\nabla_y X -[X,Y]$$ and the curvature is defined by the commutator on a vector field as $$[\nabla_X,\nabla_Y] Z - \nabla_{[X,Y]}Z=R(X,Y)Z.$$ This last equation holds both with and without torsion in the connection.

On a scalar the commutator is given by $$[\nabla_X,\nabla_Y]\phi - \nabla_{[X,Y]}\phi= 0$$ as the scalar does not see curvature. So, using the coordinate basis vectors $$X=\partial_\mu$$, $$Y=\partial_\nu$$, we have $$[\nabla_{\partial_\mu},\nabla_{\partial_\nu}]\phi=0.$$

It's not safe to write $$\nabla_{\partial_\mu}$$ as $$\nabla_\mu$$ as this gives the impression that you need an extra connection term because of the $$\nu$$ index when $$\nabla_\mu$$ acts on $$\nabla_\nu$$ etc. If you make this interpretation then then the $$\nabla_\mu$$ changes the tensor character of the object it acts on, unlike the usual covariant derivative $$\nabla_X$$ which does not change the tensor character. If you insist of changing the character then $$\nabla_\mu$$ and $$\nabla_\nu$$ are acting on different spaces depending on their order, and the "commutator" is not really a commutator and its properties are rather ill defined. This is harmless in torsion-free GR but becomes a probelm when you have torsion as it leads to notational ambguities that I suspect you have in your paper.

The torsion definition applied to $$X=\partial_\mu$$, $$Y=\partial_\nu$$ gives $$\nabla_{\partial_\mu} \partial_\nu - \nabla_{\partial_\nu}\partial_\mu = T^{\lambda}(\partial_\mu,\partial_\nu) \partial_\lambda.$$ With $$T^\lambda(\partial_\mu,\partial_\nu)= {T^\lambda}_{\mu\nu}$$ as the coordinate componets of the torsion tensor, and applying the vector field $${T^\lambda}_{\mu\nu}\partial_\lambda$$ to a scalar $$\phi$$ we get
$$(\nabla_{\partial_\mu} \partial_\nu - \nabla_{\partial_\nu}\partial_\mu) \phi = {T^{\lambda}}_{\mu\nu} \partial_\lambda \phi.$$ which suggests that you opinion that $$K$$ should be replaced by $$T$$ is correct.

Also, by the definition of the Christoffel symbols $$\nabla_{\partial_\mu} \partial_\nu= {\Gamma^\lambda}_{\nu\mu} \partial_\lambda$$ we see that the above dequation gives the usual $${T^{\lambda}}_{\mu\nu}= {\Gamma^\lambda}_{\nu\mu}-{\Gamma^\lambda}_{\mu\nu}$$ where I am using MTW's placement of the indices on the $$\Gamma$$'s, so the indices might seem to be backwards compared to some other defs.

• it would be very helpful if you can put the entire derivation of the equation. I would be most grateful. – user44690 Jun 5 at 12:03
• The formulae for the torsion and the curvature are the usual definitions in differential geometry, so there is no real "derivation." There is a good discussion in Misner, Thorn and Wheeler, although I suspect that they do not consider torsion. I learnt them from books on differential geometry. Lots of people use $\nabla_\mu$ and have extra torsion terms in their "commutators" and you can be consistent if you are careful. It's just always safer to use $\nabla_X = X^\mu\nabla_\mu$ fo these things. – mike stone Jun 5 at 13:13
• I've added some extra material to my answer. – mike stone Jun 5 at 13:30
• I am glad that somebody agrees with me. I thought I was going crazy or something. – user44690 Jun 5 at 15:23

You are right, this must be a typo, as the following shows: $$[\nabla_\mu, \nabla_\nu] \phi = \nabla_\mu \partial_\nu \phi - \nabla_\nu \partial_\mu \phi = (\partial_\mu \partial_\nu \phi - \Gamma^\alpha_{\mu\nu} \partial_\alpha \phi) - (\partial_\nu \partial_\mu \phi - \Gamma^\alpha_{\nu\mu} \partial_\alpha \phi) = - (\Gamma^\alpha_{\mu\nu} - \Gamma^\alpha_{\nu\mu}) \partial_\alpha \phi = - T^\alpha_{\mu\nu} \partial_\alpha \phi$$

• Yeah. But unfortunately, he is using this wrong equation in various parts of this paper and making some conclusions. It's really bizzare. – user44690 Jul 23 at 5:18
• I guess, it is a typo, for in 2.11 he writes $T^\tau_{\cdot\alpha\beta}$ at the same place. – Nikodem Jul 23 at 11:00