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The chart before and after a Lorentz transform are both valid description of the same history of flat spacetime right? If so, is this just the same as going from one chart to another chart via the intermediate map as we always do in GR? That is, going from X to Y via $Y\circ X^{-1}$, where X, Y are maps from the manifold M $\to$ $R^{d}$. If all the above assumptions are good, then the time direction time level set are different in different chart.

There is no Lorentz transform on the whole manifold for curve spacetime that gives the fake notion of simultaneity, but when we perform a Lorentz transform locally, the local Lorentz transform induces a change of chart on the whole curved spacetime(or manifold) that change our way of modeling spacetime, and all ways are valid? Different observer simply gives us a different chart?

If what I said is terribly wrong, please, point out what I am wrong about. Thanks.

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The transition from local invariance of General Relativity to the global Lorentz (Poincare) symmetry is a subtle issue. Deser puts it well (this paper):

General relativity must be broken down to Special Relativity in a special, but physically natural, way in order for the Poincare or other global groups such as (A)dS to “re-”emerge.

The key word here is "broken". The flat space-time is a symmetry broken phase and global Lorentz (Poincare) symmetry is an emergent phenomenon, akin to the Goldstone mode.

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