I wish to study curved spacetime with torsion, however, the trouble is how do I go about with the variational principle? Should I assume the connection $\Gamma^{\alpha}_{\beta\gamma}$ and the metric $g_{\mu\nu}$ as independent variables? Maybe this is fine when metricity is relaxed but what if metricity is imposed i.e. $\nabla_{\mu}g_{\alpha\beta} = 0$ then $$\partial_{\mu}g_{\alpha\beta}-\Gamma^{\lambda}_{~~~\alpha\mu}g_{\lambda\beta}-\Gamma^{\lambda}_{~~~\beta\mu}g_{\alpha\lambda} = 0$$ then clearly the connection is not independent of the metric. Please advice.
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Related: https://physics.stackexchange.com/q/141550/2451 – Qmechanic Jun 19 '20 at 20:50
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@Qmechanic No, it is not related. – Dr. user44690 Jun 20 '20 at 04:02
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Well, the Palatini formalism is sort of the dual situation, where the connection is not metric but the torsion is non-dynamical. – Qmechanic Jun 20 '20 at 06:14
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@Qmechanic But my question is what if the connection is metric? Then the connection and the metric are related via the metricity conditions above, in such a case, what is the procedure for the variation of the action. Clearly, it's a different question. 'Sort of' doesn't answer my question. – Dr. user44690 Jun 20 '20 at 06:20