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I have recently learned about the equivalence principle and what it says is that( If I have understood it properly) the trajectory of the particles will be the same irrespective of the properties of the particles if the initial conditions are the same. My book says due to this principle we can change our transformation to an accelerated frame and locally the weak gravitational field would look similar to acceleration.
How the author come to the reasoning that only "locally" the accelerated frame would be equivalent to only a "weak" gravitational field?
I understand that if the gravitational field is strong and we are not seeing locally then gravity would not be equivalent to acceleration as it can happen that both bodies are coming closer to each other i.e the effect of curvature.
Also, I know that that Rindler metric in the limit of constant acceleration << c and the metric $g_{ab}$ in the limit of weak gravitational potential is same.
But my doubt is how just from equivalence principle we can deduct that only locally weak gravitational field would be equivalent to the accelerated frame?

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Constant acceleration is exactly equivalent to a perfectly uniform gravitational field. Of course, no real gravitational field is perfectly uniform, but it can be approximately uniform in a small region.

I understand that if the gravitational field is strong and we are not seeing locally then gravity would not be equivalent to acceleration as it can happen that both bodies are coming closer to each other, i.e., the effect of curvature.

That's correct. Over a larger region in a non-uniform gravitational field, eg on the surface of the Earth, we can see that lines pointing downwards are not parallel. We can also see that the strength of the field gets weaker as we go up.

If we were on a planet that's ten times denser than Earth and one tenth Earth's radius, it would have the same surface gravity as Earth. But the reduction in gravity with altitude would happen ten times faster, and the deviation of angle in lines pointing down would also happen ten times faster.

In other words, when gravity is very strong, the gravitational field is only a good approximation to uniform in a small region.

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