As is well established, Special Relativity ensures that no observer can ever tell from the experiments he has been doing in his car whether the car is moving or not as long as the car constitutes an inertial frame of reference. This I find in complete accord with the Special Principle of Relativity.
Now, the most fundamental issue Einstein himself had with the Special Relativity was that it still picks out some mysterious inertial frames which we find no reason to exist if we completely embrace the general principle of relativity. There should not be any standard of absolute acceleration. Thus, I thought that General Relativity must be of such a nature that it abolishes the absolute acceleration standards just as Special Relativity abolished the notion of absolute velocity.
Now, imagine this scenario. There is a lift and there is the vacuum in the lift. Thus, $R_{\mu \nu}=0$ in every possible coordinate system. Here, I can choose a coordinate system in which $\Gamma^{\alpha}_{\beta\gamma}=0$ but I can also choose a coordinate system in which $\Gamma^{\alpha}_{\beta\gamma}\neq0$. Now, can't I assert that the frames in which, the Ricci tensor is trivial and yet the connections are not, are accelerated and the ones in wich Ricci tensor is trivial and the connections are also trivial are inertial? This would not be an artifact as there is a definite way of making a distinction. (Whereas in SR, there was not such way of making a distinction between a rest frame and a moving frame.)
Edit: So the problem is that if I can identify a frame as inertial and the others as non-inertial then based on the same I can establish some local standards of non-acceleration or those of acceleration. This is completely against the spirit of the general principle of relativity. According to the general principle of relativity, there should be absolutely no way to tell which object is moving and which is at rest, i.e., I should be able to call $A$ to be at rest and $B$ to be accelerating as well as $B$ to be at rest and $A$ to be accelerating- and do the Physics in any of them the same way. But this essence is spoilt here.
P.S.:
I know that there is a gauge freedom in choosing the components of the metric even if I have been provided with the curvature tensor and this results in multiple possible connections for a single curvature tensor. But this is the mechanism of how this different possibility of connections arises- not a valid way of denying that I can make distinction between coordinate systems which I can, in turn, use to define absolute acceleration.
I also know that the observer which is using the coordinate system with non-trivial connections can attribute these effects to a gravitational field. But this seems like an excuse to me when I imagine a spacetime which is entirely flat and has the stress-energy tensor identically zero everywhere. In such a universe, the introduction of a homogeneous gravitational field to explain the non-triviality of connection in certain coordinate systems would be an utter artifact – in the sense that whom should I ascribe the origin of such a field?
And even more strikingly, in a completely empty universe, in one particular set of frames, my symbols will be trivial and in all the rest they won't be. This is a measurable and clear distinction. Put it another way, only in a particular class of coordinates, I will be able to synchronize a non-local array of clocks whereas, in the rest, I won't be able to do so.
Best Answer
I think I understand what you're asking so I'll answer accordingly. Ignore this answer if I've got the wrong end of the stick.
General relativity tells us that the four acceleration is given by:
$$ A^\alpha = \frac{\mathrm d^2x^\alpha}{\mathrm d\tau^2} + \Gamma^\alpha_{\,\,\mu\nu}U^\mu U^\nu \tag{1} $$
So there are two contributions, the time dependence of the coordinates and the term in the Christoffel symbols. Since the four-acceleration is a four-vector the norm of the four-acceleration, the proper acceleration, is an invariant so it will be the same in all coordinate systems.
If we consider a freely falling observer in Minkowski spacetime (i.e. your lift) then the norm of the four-acceleration is zero. As you say, we can choose coordinates where $\mathrm d^2x^\alpha/\mathrm d\tau^2=0$ and $\Gamma^\alpha_{\,\,\mu\nu}=0$ and this is what we'd call an inertial frame. Alternatively we could choose accelerating coordinates, like the Rindler coordinates, where neither $\mathrm d^2x^\alpha/\mathrm d\tau^2=0$ nor $\Gamma^\alpha_{\,\,\mu\nu}=0$ but of course the proper acceleration of our freely falling observer would still come out as zero.
I'd guess we agree so far, but where we disagree is that I don't see that there's anything different between GR and SR or indeed classical mechanics. The invariant is the proper acceleration of the observer and that is always unambiguously measurable because the observer just has to weight themselves. The same equation (1) applies to curved spacetime, flat spacetime and indeed to non-relativistic motion where the manifold is Riemannian.