In Einstein-Cartan theory, the torsion tensor is usually defined as the antisymetric part of the connection: \begin{gather} \nabla_{\mu} \, A^{\lambda} = \partial_{\mu} \, A^{\lambda} + \Gamma_{\mu \nu}^{\lambda} \, A^{\nu}, \tag{1} \\[12pt] T_{\mu \nu}^{\lambda} = \Gamma_{\mu \nu}^{\lambda} - \Gamma_{\nu \mu}^{\lambda}. \tag{2} \end{gather} The contorsion is usually introduced by a simple decomposition of the non-symmetric connection, where $\tilde{\Gamma}_{\mu \nu}^{\lambda}$ is the Levi-Civita (symmetric) connection (or Christoffel symbols): \begin{equation}\tag{3} \Gamma_{\mu \nu}^{\lambda} = \tilde{\Gamma}_{\mu \nu}^{\lambda} + K_{\mu \; \; \, \nu}^{\;\,\lambda}. \end{equation} Now, in many papers I've read on this subject, the upper index may be pushed to the left or to the right, like this: $T_{\mu \nu}^{\;\;\lambda}$ or $T_{\;\;\mu \nu}^{\lambda}$ and $K_{\;\;\mu\nu}^{\lambda}$ or $K_{\mu\nu}^{\;\;\lambda}$. This is a very frustrating source of confusion. Authors are using lots of different index conventions and the relatively old litterature on torsion in General Relativity is a real mess! (I'm not even counting the metric signature convention and index position for the Riemann tensor!)

About the upper index in (2) and (3) above, is there a common consensus or a more frequent convention in use today? Or is it still a total freak show of random conventions?


When it comes to torsion there are lot of different conventions because there has never been much interest in it from GR practitioners. It's only when many people work on a topic that notation becomes standardized. This is true even for definitions of the curvature tensor. This is why MTW devote the inside of their front cover to list their and other people's conventions. For example, if I remember correctly, MTW define $$ \nabla_\mu V^\nu= \partial_\mu V^\nu+ V^\alpha {\Gamma^\nu}_{\alpha\mu}, $$ which puts the suffixes on the Christoffel tensor in the opposite order to what you have. Then the torsion tensor $T(X,Y)=\nabla_X Y-\nabla_Y X$ has components
$$ {T^\alpha}_{\mu\nu} = {\Gamma^\alpha}_{\nu\mu}-{\Gamma^\alpha}_{\mu\nu} $$ that are "backwards". We just have to live with the mess.

  • $\begingroup$ I'm sorry if I overwrote your edit. Didn't notice we were editing nearly the same time. $\endgroup$ – Thomas Fritsch Aug 30 at 12:50
  • $\begingroup$ While I agree with your answer, the MTW covariant derivative (with its connection on the right of the vector) is much less in use today (AFAIK). I believe that (1) and (2) in my question are closer to a consensus. However, the position of the upper index is "problematic". $\endgroup$ – Cham Aug 30 at 14:13

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