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A while back I asked this question Freely falling frame and the use of special relativity? which I later answered using the Hypothesis of Locality which I used to state that $E=-p_\mu u^\mu$ in any reference frame. My concern now is that the measurement of $p_\mu$ cannot really be done at a point and relies on at least two separate space time events. So does my argument in the answer to the linked question stand, given this non-local measurement - either way please can you explain.

I guess I am asking: Can we use all the laws of special relativity at the point of the 'observer' in all GR reference frames - independent of how many derivatives they contain?

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In GR it is possible at any spacetime point to find a reference frame where the metric is Minkowski, and where the derivatives of the metric are 0. The second point means that the Christoffel symbols, i.e. the connection is also zero at that point. That is one derivative of the metric. If you take second derivatives you are now getting to entities from which at that point, you can determine the invariant curvature scalars. They arise from the Riemann tensor, which includes second metric derivatives.

So, the answer is that you can assume the metric and first derivatives are Minkowski, but not second derivatives and higher.

By the way, not in any GR frame, but in a frame you choose that has those properties at that point in spacetime.

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