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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 this (making metric and connection independent) physically motivated? Or it just a mathematical curiosity?

\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 this (making metric and connection independent) physically motivated? Or it just a mathematical curiosity?

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