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In the quote cited you could imagine that point particles move in a straight line at a steady velocity and don't rotate when far from massive objects. So to a small accelerating frame, they look like a non rotating point particle moving at a constant velocity would look to an accelerating frame. But maybe point particles near a massive object have similar ...

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Take a future-directed timelike curve $\gamma= \gamma(\tau)$, $\tau$ being the proper time along $\gamma$ in the spacetime $M$. Assume that $p = \gamma(0)$ is the initial point of $\gamma$. Fermi coordinates adapted to $\gamma$ are constructed this way. Consider an orthonormal basis of $T_pM$ with $e_0$ parallel to $\dot{\gamma}$. Transport the basis ...

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The equivalence principle is local (as the comments indicate), only applicable for sufficiently small regions. Another example that highlights the necessity of the word "local" is the following picture, in which the person is able to distinguish between linear acceleration and gravity.

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This is actually a famous theorem known as the Einstein Equivalence Postulate (sometimes Equivalence Principle). It's true that since Earth is spinning, acceleration in a spacecraft isn't quite the same situation we experience daily, but in general, yes, gravity is indistinguishable from uniform acceleration. Specifically, if you are in a box with no windows ...

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Gravitational fields aren't homogeneous. Here on the Earth, a clock on the floor runs more slowly than a clock on the table, and we have clocks precise enough to measure such small differences due to the gravitational gradient. But doesn't a clock in an accelerating spaceship run at the same rate no matter where in the ship you put it? See page ...

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