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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?

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

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  • $\begingroup$ it would be very helpful if you can put the entire derivation of the equation. I would be most grateful. $\endgroup$ – user44690 Jun 5 at 12:03
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    $\begingroup$ 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. $\endgroup$ – mike stone Jun 5 at 13:13
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    $\begingroup$ I've added some extra material to my answer. $\endgroup$ – mike stone Jun 5 at 13:30
  • $\begingroup$ I am glad that somebody agrees with me. I thought I was going crazy or something. $\endgroup$ – user44690 Jun 5 at 15:23
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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 $

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

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