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Tetrads in General Relativity

I. Metric and Connection

Torsion Tensor

   The Levi-Civita connection coefficients $\Gamma $ depend on the chosen coordinates and are not tensors. However, the difference of two connections can be show to be tensorial. Hence, the torsion defined as the difference

1.23

\[  T_{\mu \nu }^\kappa : = \Gamma_{\mu \nu }^\kappa - \Gamma_{\nu \mu }^\kappa = 2\Gamma_{[\mu \nu]}^\kappa  \]

transforms as a tensor. Equivalently, the torsion differential 2-form

1.24

\[  {{\mathbf{T}}^\kappa } : = \frac{1}{2}T_{\mu \nu }^\kappa {{\mathbf{g}}^\mu } \wedge {{\mathbf{g}}^\nu } = \Gamma _{[\mu \nu ]}^\kappa {{\mathbf{g}}^\mu } \wedge {{\mathbf{g}}^\nu }  \]

measuring the anti-symmetry of the Levi-Civita connection, transforms as a vector. Both representations show that torsion vanishes if and only if the connection coefficients are symmetric with respect to the lower indices.

   Since the embedding space is smooth, partial spacetime derivatives commute: $ \, 0 = {\partial _\mu }{\partial _\nu }{\mathbf{x}} - {\partial _\nu }{\partial _\mu }{\mathbf{x}} = {\partial _\mu }{{\mathbf{g}}_\nu } - {\partial _\nu }{{\mathbf{g}}_\mu } $. Then, this only holds for the coderivative

1.25

\[  {D_\mu }{{\mathbf{g}}_\nu } - {D_\nu } {{\mathbf{g}}_\mu } = \left( {\Gamma _{\mu \nu }^\kappa - \Gamma _{\nu \mu }^\kappa } \right){{\mathbf{g}}_\kappa } = 0  \]

if $\Gamma _{\mu \nu }^\kappa = \Gamma _{\nu \mu }^\kappa$ is symmetric. This imposes that the Levi-Civita connection is torsion-free, which implies

1.26

\[  D \wedge {{\mathbf{g}}^\mu } = {{\mathbf{g}}^\nu } \wedge {D_\nu }{{\mathbf{g}}^\mu } = - \Gamma _{[\nu \kappa ]}^\mu {{\mathbf{g}}^\nu } \wedge {{\mathbf{g}}^\kappa } = 0  \]

   Torsion vanishes by assumption in GRT. Other gravity theories such as, for example, the Einstein-Cartan theory (1921), allow for torsion to incorporate possible new effects beyond Einstein gravity.