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No, we should not say that Christoffel symbols are gravity. The big reason, which really should be enough, is that they are coordinate dependent. One of the main tenets of General Relativity is that coordinates don't matter. Everything physical must be expressible in a coordinate independent and/or tensorial manner.

As I said in the comments, personally I think it's a bit ridiculous to suggest that using polar coordinates somehow brings gravity into the mix, while using Cartesian coordinates does not. The equation for a straight line changes, but you can verify using any number of methods that it's still a straight line. If polar coordinates show gravity, then where is that gravity coming from? What physical object is generating it? There was none in Cartesian coordinates.

But let me address your three points:

It is not true in absolute generality that the Christoffel symbols correspond to the gravitational field, for the reasons I gave above. A gravitational field manifests itself in the Christoffel symbols, but not the other way around. Also remember that even in Newtonian gravity the equivalence principle holds, and one might argue that only tidal forces are measurable for someone in free fall, so that's one argument in favor of the curvature. Again, even in flat spacetime there are curved coordinates. "Special relativistic" means that the metric is $\eta_{\mu

u}$ when expressed in locally Lorentzian (i.e. Cartesian) coordinates, not in any coordinates. This is basically the same as 2, but see the following paragraph.

I think the deep issue is that you are misunderstanding gravity for coordinate acceleration. You actually make a very good point in your number 3 argument, but you draw the wrong conclusion. The lesson of the equivalence principle is not that acceleration is relative and hence gravity is relative. You could take it as the conclusion, but then the word "gravity" is not very useful anymore because it is coordinate-dependent.

Instead, the lesson you should take away is that "gravity" should refer to something that has a physical existence independently of the observer, and that something is tidal forces, precisely because of the equivalence principle. Since coordinate acceleration is relative, the smart thing to do is to make "gravity" mean something that is not relative.

I insist with a very important point: this is not just a matter of definition; physical reality has my back here. I say this because it turns out that every time there are tidal forces, one can identify some physical object (planet, star, whatever) responsible for it. However, sometimes objects seem to not obey the Cartestian-flat-space geodesic equation with no apparent source of gravity nearby. To me it just makes much more sense to say that the thing that always manifests itself near a heavy object is gravity and that the thing that sometimes happens as a result of weird coordinates is not gravity.