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In Classical GR the metric tensor $g_{\mu\nu}$ determines the length of rods and ticking of clocks while the connection $\Gamma^{\alpha}_{\mu\nu}$ determine the equation of geodesic (the free fall motion of particle). Furthermore, in GR the Levi-Civita connection is uniquely determined by the metric tensor as, $$ \Gamma_{\mu\nu}^{\alpha}=\frac{1}{2}g^{\alpha\delta}(g_{\mu\delta,\nu}+g_{\nu\delta,\mu}-g_{\mu\nu,\delta}) $$ In certain theories of gravity and Quantum gravity, the connection and the metric tensor are taken as independent quantities. It appears to me that when this happens the free fall geodesic equation, describing the motion of particle in a manifold, becomes independent of the structure of space-time, the metric tensor (rods and clocks). This makes, to me at least, very little intuitive sense (may be because the picture painted by classical GR is too strongly imprinted on my mind).

So, how is such an action (making metric and connection independent) physically motivated?

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So far my review of the literature available on the matter suggests that the consideration of indepdendent metric and connection stems as a generalization of Einstein-Cartan theory of gravity. The Einstein-Cartan model breaks the symmetry of the Christoffel symbol between 2 and 3 index. This leads to the following definition of Christoffel symbol,

$$\tilde\Gamma^\alpha_{\beta\lambda} = \Gamma^\alpha_{\beta\lambda} + K^\alpha_{\beta\lambda}$$

Where, $\Gamma^\alpha_{\beta\lambda}$ is the standard Christoffel symbol and $K^\alpha_{\beta\lambda}$ is the tortion tensor.

Hence, as far as I understand, there is no physical motivation in considering the independence. So far it appears to be a mathematical curiosity to generalize Einstein-Cartan Theory. For further readingn, see the following reference and the references there in.

Wikipedia: Metric-affine gravitation theory

Metric-Affine Gauge Theory of Gravity: Field Equations, Noether Identities, World Spinors, and Breaking of Dilation Invariance

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