Understanding Contortion Tensor in Teleparallel Gravity

[yenchin]

I have been reading papers on teleparallel gravity as well as the recent modified gravity called f(T) (see e.g. http://arxiv.org/abs/1005.3039v2" ).

I am confused about the contortion tensor, which is defined as the difference between the teleparallel connection and the usual GR connection. In the paper above, as well as say, equation 7 of http://arxiv.org/abs/1001.3407" , it is given by:

However, page 254 of Nakahara "Geometry, Topology and Physics" as well as some papers such as http://iopscience.iop.org/0264-9381/18/12/307" (which quoted Nakahara), give

I am confused why there's discrepancies in the extra minus signs. Granted that the torsion tensor in the teleparallel papers seem to differ from Nakahara's text and differential geometry texts by an extra minus sign, i.e.

instead of

shouldn't the contortion then differ by an *overall* minus sign instead of having only one of the terms differ in sign?

 

[arkajad]

Did you take into account the fact that $T^\lambda_{\mu\nu}$ is antisymmetric in $\mu,\nu$ whichever way it is defined?

 

[yenchin]

Did you take into account the fact that $T^\lambda_{\mu\nu}$ is antisymmetric in $\mu,\nu$ whichever way it is defined?

Yes, as far as I can tell, well, maybe there are mistakes in my calculations somewhere...

 

[arkajad]

One possible reason for discrepancies may be due to the fact that in gr-qc/0607138v1 they have index $\theta$ which refers to an orthonormal tetrad. They can mean something else by the connection coefficients. It is not clear. This needs to be checked. Nakhara based formulas for the coordinate frame are certainly correct.

 

[yenchin]

One possible reason for discrepancies may be due to the fact that in gr-qc/0607138v1 they have index $\theta$ which refers to an orthonormal tetrad. They can mean something else by the connection coefficients. It is not clear. This needs to be checked. Nakhara based formulas for the coordinate frame are certainly correct.

OK. I will look into that. Thanks :-)

 

[dextercioby]

I have checked Nakahara's calculations and they are correct. He uses the convention for covariant differentiation different than here: http://en.wikipedia.org/wiki/Covariant_derivative. Specifically, Nakahara uses

$$\nabla_{\mu}g_{\nu\lambda}:=\partial_{\mu}g_{\nu\lambda}-\Gamma^{\kappa}{}_{\mu\nu}g_{\kappa\lambda}-\Gamma^{\kappa}{}_{\mu\lambda}g_{\nu\kappa} =0$$

by the metricity condition imposed to the linear connection. If you compare this formula with the Wikipedia definition (semicolon notation), you can see the difference. The Wiki formula for the (0,2) tensor is wrong, it is not a consequence of the (0,1) tensor written there, but from a formula for the (0,1) tensor written with the lower indices of the connection backwards.

Checking the Wiki comment page, I see that somenone noticed this error, too:

light incoherent Christoffel symbols index position
The article does not assume a symetric (torsion-free) connection, so it is not necessarily true that

Γkij = Γkji.
For example, in the section "coordinate description" the definition of the covariant derivative uses the "derivative index" of the Christoffel symbol to be the first lower index. The expression for the covariant derivative of a vector field is coherent with the definition used. On the other hand, the expressions for the covariant derivative of a dual field and for a general tensor field use the second lower index of the Christoffel symbol as the "derivative index".

My suggestion is to choose one convention and use it coherently. Personally, I'd rather use the second lower index of the Christoffel symbol as the "derivative index" because it maintains some coherence with the position of the index added by the semi-colon notation, but I have seen different texts using both of these conventions.

Marcelo Roberto Jimenez 15:19, 15 July 2010 (UTC)

 

[dextercioby]

I have been reading papers on teleparallel gravity as well as the recent modified gravity called f(T) (see e.g. http://arxiv.org/abs/1005.3039v2" ).

I am confused about the contortion tensor, which is defined as the difference between the teleparallel connection and the usual GR connection. In the paper above, as well as say, equation 7 of http://arxiv.org/abs/1001.3407" , it is given by:

However, page 254 of Nakahara "Geometry, Topology and Physics" as well as some papers such as http://iopscience.iop.org/0264-9381/18/12/307" (which quoted Nakahara), give

I am confused why there's discrepancies in the extra minus signs. Granted that the torsion tensor in the teleparallel papers seem to differ from Nakahara's text and differential geometry texts by an extra minus sign, i.e.

instead of

shouldn't the contortion then differ by an *overall* minus sign instead of having only one of the terms differ in sign?

Adding to my post above, I'm telling you that there are basically 2 different conventions and definitions used throughout the literature. Nakahara used the above-mentioned convention/definition for the covariant derivative of a (0,2) tensor AND the following definition of the torsion tensor:

$$T^{\lambda}{}_{\mu\nu}:= \Gamma^{\lambda}{}_{\mu\nu}-\Gamma^{\lambda}{}_{\nu\mu}$$

,while Aldrovandi & Pereira (quoted by 2 of the article you mention oposing Nakahara) use the following conventions/definitions:

$$\nabla_{\mu}g_{\nu\lambda}:=\partial_{\mu}g_{\nu\lambda}-\Gamma^{\kappa}{}_{\nu\mu}g_{\kappa\lambda}-\Gamma^{\kappa}{}_{\lambda\mu}g_{\nu\kappa}$$

AND

$$T^{\lambda}{}_{\mu\nu}:= \Gamma^{\lambda}{}_{\nu\mu}-\Gamma^{\lambda}{}_{\mu\nu}$$.

Both approaches are correct. The authors follow the conventions they choose with precision, so that a confusion as the one pointed on the Wiki page does not exist.

The conventions of Aldrovandi and Pereira are probably more common in physics articles/books than the ones used by Nakahara (I remember the literature on the PGT), but, nonetheless, both are equally correct.

 

[arkajad]

I suggest you check http://www.phys.sinica.edu.tw/~heptheory/files/pereira.pdf" paper by Pereira, where the explicit relation between Weitzenböck connection coefficients (used by Aldrovandi et al) and Christoffel symbols of the Levi-Civita connection is given. The rest should be an easy calculation.

 

[yenchin]

Thanks to all the help guys. Really appreciated!