Defining the spin tensor using the matter Lagrangian

[Yusuf Emre BARAN]

Hi everyone,

I am using Cyril Pitrou's  ActionVariation_Metric_Fields.nb on xAct-contrib/examples on GitHub to derive the field equations of Einstein-Cartan theory. I have managed to derive the vacuum field equations. To include matter, one must introduce the spin tensor, which is given by the variation of the matter Lagrangian with respect to the contorsion tensor. More precisely, one has (in LaTeX notation)

\tau^{abc}=\frac{2}{\sqrt{-g}}\frac{\delta L_m}^{\delta C_{abc}}

where C_{abc} is the contorsion tensor and L_m is the matter Lagrangian.

In classical mechanics, we always implicitly assume that L=L(q,q'). However,  since I do not know how to specify that the matter Lagrangian depends on the contorsion tensor and its derivatives, the variation with respect to the contorsion tensor gives me zero. Could you help me figure out how I can tell xPert about this dependence? There is a similar thread on how to define a scalar field that depends on R, but that does not seem to help me.

Also, a more general question is how we can define a scalar function (or a scalar field) that depends on some parameters such as the affine connection, contorsion tensor, non-metricity tensor, etc.? For convenience, I have attached my work.

Best regards,
Yusuf