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 Quote by TrickyDicky So can the affine connection of GR be measured? It is obvious that in the stricter, invariant sense referred to above, it can't. Does this mean it is not "physical"? No. We are certainly feeling their consequences and therefore "observing" it as a force. But what we measure is not so much the connection but the EM resistance of the ground against our natural tendence to follow our geodesic.
Asking if the affine connection can be measured is analogous to asking if the vector potential can be measured in EM. The answer is not exactly "no"; it is more of a "yes, but...". After all, the connection, like the vector potential, does carry real information; but that information is described in a redundant manner.

The caveat is that we can only measure gauge-invariant quantities constructed from these objects. In EM, this means we can measure the E and B fields. In GR, this means we can measure the Riemann tensor. (Where "measure" is defined as a process like I've described before, where we choose a frame and measure contractions against that frame.)

So the answer really depends on the meaning of the question. If the question is "Can we measure the connection independently of the Riemann tensor?", then the answer is certainly "No." In particular, there is no set of measurements we can do that will let us map out exactly what values to assign to each of the components of $\Gamma^\mu_{\nu\rho}$.