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So, two frames (one accelerating with respect to the other and one of them having vanishing Christoffel symbols) differ in a very physical sense that one sets up geodesics for the spacetime whereas the other doesn't. This difference stems from the inherent frame-invariant metrical structure that even the spacetime has. Thus, the fact that in one frame, particles go unaccelerated and in the other, particles experience some acceleration also stems from the fact that one of the frames does have a special status owing to the inherent frame-invariant metrical structure that even the empty spacetime has. So, although the principle of relativity of all the kinds of motion isn't respected in the sense in which the principle of relativity of uniform motion is respected in special relativistic set-ups, we get a reason as to why - the existence of definite geodesics (and their relation to each other) in the empty space.

The way principle ofSo, what General Relativity does (rather than establishing true relativity is then preserved is in the formof all kinds of motion) is establishing a duality between accelerated motion and gravity through the principle of general covariance. Which is a statement more about the effects of gravity than about the relativity of all kinds of motion. It says that (since the geodesics define free-fall) the effect of gravity is uniquely determined by the laws of coordinate transformations from the local inertial frame which is the frame attached to a particle (locally) in free-fall. The key physical insight to be absorbed (above the most fundamental fact that we can always go to a local inertial frame) is that gravity exhibits all its effects only and only through determining geodesics. Because, once we know the geodesic, we know the local inertial frame and once we know local inertial frame, we can transform to whatever generic frame we want and we will have taken care of describing effects of gravity in that generic frame!

So, two frames (one accelerating with respect to the other and one of them having vanishing Christoffel symbols) differ in a very physical sense that one sets up geodesics for the spacetime whereas the other doesn't. This difference stems from the inherent frame-invariant metrical structure that even the spacetime has. Thus, the fact that in one frame, particles go unaccelerated and in the other, particles experience some acceleration also stems from the fact that one of the frames does have a special status owing to the inherent frame-invariant metrical structure that even the spacetime has. So, although the principle of relativity of all the kinds of motion isn't respected in the sense in which the principle of relativity of uniform motion is respected in special relativistic set-ups, we get a reason as to why - the existence of definite geodesics (and their relation to each other) in the empty space.

The way principle of relativity is then preserved is in the form of the principle of general covariance. Which is a statement more about the effects of gravity than about the relativity of all kinds of motion. It says that (since the geodesics define free-fall) the effect of gravity is uniquely determined by the laws of coordinate transformations from the local inertial frame which is the frame attached to a particle (locally) in free-fall. The key physical insight to be absorbed (above the most fundamental fact that we can always go to a local inertial frame) is that gravity exhibits all its effects only and only through determining geodesics. Because, once we know the geodesic, we know the local inertial frame and once we know local inertial frame, we can transform to whatever generic frame we want and we will have taken care of describing effects of gravity in that generic frame!

So, two frames (one accelerating with respect to the other and one of them having vanishing Christoffel symbols) differ in a very physical sense that one sets up geodesics for the spacetime whereas the other doesn't. This difference stems from the inherent frame-invariant metrical structure that even the spacetime has. Thus, the fact that in one frame, particles go unaccelerated and in the other, particles experience some acceleration also stems from the fact that one of the frames does have a special status owing to the inherent frame-invariant metrical structure that even the empty spacetime has. So, although the principle of relativity of all the kinds of motion isn't respected in the sense in which the principle of relativity of uniform motion is respected in special relativistic set-ups, we get a reason as to why - the existence of definite geodesics (and their relation to each other) in the empty space.

So, what General Relativity does (rather than establishing true relativity of all kinds of motion) is establishing a duality between accelerated motion and gravity through the principle of general covariance. Which is a statement more about the effects of gravity than about the relativity of all kinds of motion. It says that (since the geodesics define free-fall) the effect of gravity is uniquely determined by the laws of coordinate transformations from the local inertial frame which is the frame attached to a particle (locally) in free-fall. The key physical insight to be absorbed (above the most fundamental fact that we can always go to a local inertial frame) is that gravity exhibits all its effects only and only through determining geodesics. Because, once we know the geodesic, we know the local inertial frame and once we know local inertial frame, we can transform to whatever generic frame we want and we will have taken care of describing effects of gravity in that generic frame!

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So, two frames (one accelerating with respect to the other and one of them having vanishing Christoffel symbols) differ in a very physical sense that one set upssets up geodesics for the spacetime whereas the other doesn't. This difference stems from the inherent frame-invariant metrical structure that even the spacetime has. Thus, the fact that in one frame, particles go unaccelerated and in the other, particles experience some acceleration also stems from the fact that one of the frames does have a special status owing to the inherent frame-invariant metrical structure that even the spacetime has. So, although the principle of relativity of all the kinds of motion isn't respected in the sense in which the principle of relativity of uniform motion is respected in special relativistic set-ups, we get a reason as to why - the existence of definite geodesics (and their relation to each other) in the empty space.

So, two frames (one accelerating with respect to the other and one of them having vanishing Christoffel symbols) differ in a very physical sense that one set ups geodesics for the spacetime whereas the other doesn't. This difference stems from the inherent frame-invariant metrical structure that even the spacetime has. Thus, the fact that in one frame, particles go unaccelerated and in the other, particles experience some acceleration also stems from the fact that one of the frames does have a special status owing to the inherent frame-invariant metrical structure that even the spacetime has. So, although the principle of relativity of all the kinds of motion isn't respected in the sense in which the principle of relativity of uniform motion is respected in special relativistic set-ups, we get a reason as to why - the existence of definite geodesics (and their relation to each other) in the empty space.

So, two frames (one accelerating with respect to the other and one of them having vanishing Christoffel symbols) differ in a very physical sense that one sets up geodesics for the spacetime whereas the other doesn't. This difference stems from the inherent frame-invariant metrical structure that even the spacetime has. Thus, the fact that in one frame, particles go unaccelerated and in the other, particles experience some acceleration also stems from the fact that one of the frames does have a special status owing to the inherent frame-invariant metrical structure that even the spacetime has. So, although the principle of relativity of all the kinds of motion isn't respected in the sense in which the principle of relativity of uniform motion is respected in special relativistic set-ups, we get a reason as to why - the existence of definite geodesics (and their relation to each other) in the empty space.

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Let's consider a completely empty spacetime with $T_{\mu\nu}=0$ everywhere (in every coordinate set-up). Now, consider two events $A$ and $B$. Now, consider several clocks who all simultaneously tick their zeros at $A$ and then do some motion and again meet at $B$ and stop ticking (assume that the events $A$ and $B$ are of such a nature as to allow such a motion - this is the only qualification on their otherwise generic nature). The numbers on their dials at $B$ are certainly frame-invariant - though these numbers can differ from one another, of course. Now, it's an experimental fact that there exists a limit as to how big this number can be. And only a unique clock out of all the possible clocks that connect $A$ and $B$ displays this number. Now, consider all such possible pairs of events that can be connected by such clocks. And for each pair of events, there exists a unique clock which displays a maximum number. Now, form a set (call $\Lambda$) of all these unique clocks determined by all the possible pairs of events $A$ and $B$. It is an experimental fact that every member of this set is in uniform motion with respect to every other member (i.e. if you attach indefinitely extended rulers of Euclidean kind with any of these clocks (with that clock being at the center) and put extra clocks at every point of the coordinate set-up synchronized with the central clock via a symmetric procedure then the coordinate velocity of all the clocks of the set $\Lambda$ will be constant.) This means that a completely empty space also has a very definite intrinsic structure to it which determines the extremum distance (and corresponding geodesic) between each pair of events and, in addition, has a (global) structure of such a kind that particles on these geodesics see each other going with constant relative velocity. (This second property of its structure seems to be a derivative of the fact that a global synchronization of clocks is possible in empty space. But I am not sure.) Thus, this set $\Lambda$ (determined by the intrinsic frame-invariant metrical structure of the spacetime) creates a local (and global) standard of non-acceleration.

So, two frames (one accelerating with respect to the other and one of them having vanishing Christoffel symbols) differ in a very physical sense that one set ups geodesics for the spacetime whereas the other doesn't. This difference stems from the inherent frame-invariant metrical structure that even the spacetime has. Thus, the fact that in one frame, particles go unaccelerated and in the other, particles experience some acceleration also stems from the fact that one of the frames does have a special status owing to the inherent frame-invariant metrical structure that even the spacetime has. So, although the principle of relativity of all the kinds of motion isn't respected in the sense in which the principle of relativity of uniform motion is respected in special relativistic set-ups, we get a reason as to why - the existence of definite geodesics (and their relation to each other) in the empty space.

The fact that timelike geodesics, in full GR with the energy-matter, maximizes the proper time can also be thought of in the perspective described above. The existence of mass-energy-momentum determines specific paths in spacetime called geodesics which maximizes the proper time. And the local standards of acceleration (or non-acceleration) are determined by the particles following these geodesics. There isn't a full symmetry between frames because of this very fact that all the particles following the geodesic (as determined by mass-energy-momentum) are in uniform motion with respect to each other. This makes these particles' frames the local standard of non-acceleration. Similarly, in empty space, one should possibly expect the absence of the standards of acceleration only if there is the absence of geodesics. But that not being the case, there certainly exists the standards of (non) acceleration in empty space as well.

The way principle of relativity is then preserved is in the form of the principle of general covariance. Which is a statement more about the effects of gravity than about the relativity of all kinds of motion. It says that (since the geodesics define free-fall) the effect of gravity is uniquely determined by the laws of coordinate transformations from the local inertial frame which is the frame attached to a particle (locally) in free-fall. The key physical insight to be absorbed (above the most fundamental fact that we can always go to a local inertial frame) is that gravity exhibits all its effects only and only through determining geodesics. Because, once we know the geodesic, we know the local inertial frame and once we know local inertial frame, we can transform to whatever generic frame we want and we will have taken care of describing effects of gravity in that generic frame!