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Since you mention the following in one of your comments I'm less interested in Einsteins historical struggles and would love a more modern perspective on how to get to this insight. I hereby unashamedly ignore history, and offer instead a quick plausibility argument. Let's start with the equivalence principle which, loosely speaking, says that a (...

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This is not what I, and I would posit most physicists, understand as a physical treatment of what general covariance is in physics. General covariance is that the equations look the same in any coordinate frame - any meaning that the transformations can be any function. The only limitation is that the functions be differentiable, maybe n or infinite times (...

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The development of general relativity has led to a lot of misconceptions about the significance of general covariance. It turns out that general covariance is a manifestation of a choice to represent a theory in terms of an underlying differentiable manifold. Basically, if you define a theory in terms of the geometric structures native to a differentiable ...

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Would you decrease your impact impulse by jumping during the fall? Yes When? Soon enough that it's before impact, late enough that you don't hit the ceiling of the elavator. Beyond that I don't think it matters much. Would it help if you jump inside a free falling elevator? Probably not. Indeed I expect it would make things worse. The ...

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I suppose you are asking about locally inertial frames? Chapter 2.4: Postulate (2) of general relativity implies that at each point of spacetime it is possible to choose locally inertial coordinates: $\xi^m$ Say you have coordinates $x^\mu$ and you want to transform to inertial coordinates $\xi^m$ which are in locally inertial frame $ds^2=\eta_{m ... 1 In the context of GR and the equivalence principle, given a Lorentzian manifold$(M,g)$, the following comments seem relevant: If the (Levi-Civita) Riemann curvature tensor does not vanish in a point$p\in M$, then there does not exist a neighborhood$U \subseteq M$of$p$(and a coordinate system defined on$U$) such that the metric$g_{\mu\nu}\$ becomes ...

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