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yenchin

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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?

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?

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