I've been having a bit of trouble with the idea of coordinate independence in general relativity. Let me start with a simple example that I think illustrates my question conceptually:
Say you have two objects, A and B. A is at rest and B accelerates directly away from A. From a mathematical point of view, there is absolutely nothing which distinguishes the two objects – A is accelerating away from B as much as B is from A. To distinguish the two, we need something extra – something empirical which tells us that B feels a force acting on it and A does not.
In this sense, there are infinitely many coordinate motions which might be the truly unaccelerated ones, and they can only be distinguished by the physical fact that they feel no forces. So, at some point, doesn't any physical law, even if it is written in such a way that it is satisfied in all coordinate systems, have to make reference to truly unaccelerated frames, or something similar? In other words, in order to connect an abstract mathematical description of motion, which in some sense can be transformed in any way, to physical observations, doesn't one have to anchor the mathematical reference somehow to physically meaningful concepts like unaccelerated frames?
Now, as I've been studying the Einstein field equations, I've wondered about the following: if you compare the equations for empty, flat space, in inertial coordinates versus some weird accelerating coordinates they'd be exactly the same – the space must be Ricci flat, or $R^{\alpha \beta} = 0$. Of course, there are many solutions of $R^{\alpha \beta} = 0$ (including non-flat ones, but ignore those for a second). Without some physical knowledge, wouldn't it be impossible to say which coordinates lead to $g_{\alpha \beta} = \eta_{\alpha \beta}$, and which coordinates lead to $g_{\alpha \beta}$ being some complicated (still Ricci flat) function of the coordinates? I imagine that this means that while the field equations must be satisfied, they alone are not enough to say what someone will observe – one must also know how his coordinates relate to locally inertial ones.
So I now ask: by the "coordinate independence of General Relativity", does one really just mean that the expression for curvature makes no reference to a coordinate system – that the expression for curvature is coordinate independent, and so the law relating curvature and mass-energy, the Einstein field equation, is valid in all coordinate systems? And yet, even though the law must be satisfied, in order to know what one will observe, it's necessary to also know which coordinate systems are (locally) inertial?
Finally, if you're feeling up to it, is this a real point I'm making in general? Don't the mathematical laws of physics all have to "give up" and some point and make reference to the solely empirically-defined concept of unaccelerated motion?
EDIT: I don't think I made myself totally clear – let's just say this. If I asked you to solve Einstein's field equations in flat space in the coordinates $x^\mu$, wouldn't you, in general, have to say that there's not enough information? How could you possibly know whether $x^\mu$ were inertial coordinates and $g_{\mu \nu} = \eta_{\mu \nu}$, or if $x^\mu$ were some weird accelerated coordinates, and $g_{\mu \nu}$ were the same flat metric but written in these accelerated coordinates? Both are solutions of the Einstein field equations. In this way, aren't the field equations underdetermined? Don't they have to be supplemented with information about how the coordinates relate to locally inertial coordinates?
Best Answer
Take something simpler that General Relativity.
Consider cartesian coordinates $ds^2= dx^2+dy^2$, it is an inertial frame (relatively to coordinates $x,y$).
However polar coordinates $ds^2=dr^2 + r^2 d\theta^2$ corresponds to a non-inertial frame (relatively to coordinates $r,\theta$), with $g_{\theta\theta} = r^2$. There are non-zero Christoffel symbols , even if the Riemann/curvature tensor is zero here.
However, the laws of physics expressed in polar coordinates, are no more better or worse that the laws of physics expressed in cartesian coordinates. You have only a diffeomorphism between coordinates, which translate the laws of physics, from one reference frame to the other.