general relativity - I think I am misunderstanding Einstein's equivalence principle and his elevator

I'm having difficulties understanding why a gravitational acceleration can be guaranteed to be locally equivalent to an accelerating frame. Doesn't it matter on how the force is being applied? If the floor of the elevator is exerting a force on me (due to some external force accelerating it) then this would be very different from a gravitational acceleration that would accelerate each part of my body equally. If I held a string up during my acceleration in the elevator and I let go, the string should fall, while in the gravitational case, it would remain the same, or am I missing something?

Answer

I'm having difficulties understanding why a gravitational acceleration can be guaranteed to be locally equivalent to an accelerating frame.

Actually, it can't. See section 20 of Relativity: the Special and General Theory where Einstein said this:

“We might also think that, regardless of the kind of gravitational field which may be present, we could always choose another reference-body such that no gravitational field exists with reference to it. This is by no means true for all gravitational fields, but only for those of quite special form. It is, for instance, impossible to choose a body of reference such that, as judged from it, the gravitational field of the earth (in its entirety) vanishes”.

You can't transform away a real gravitational field. So the room you’re in is not exactly equivalent to the room in the rocket. See this article and how it refers to an infinitesimal region? That's a "local" region of zero size. That's no region at all. The principle of equivalence was "Einstein's happiest thought", but it's just a principle, it doesn't mean being in a real gravitational field is exactly the same as being in an accelerating rocket.

Doesn't it matter on how the force is being applied? If the floor of the elevator is exerting a force on me (due to some external force accelerating it) then this would be very different from a gravitational acceleration that would accelerate each part of my body equally.

It's different, but generally speaking, you can't tell the difference between the floor pressing up on you or you pressing down on the floor. Like David Hammen said, in practice you can't distinguish between the two using local experiments. Gennaro Tedesco said much the same re if you just look at the dynamics But note what CuriousOne said: you wouldn't be able to tell, to first order. If you had super-precise measuring equipment, such as NIST optical clocks, you could tell. Especially if your room had a high ceiling.

If I held a string up during my acceleration in the elevator and I let go, the string should fall, while in the gravitational case, it would remain the same, or am I missing something?

I think you're missing something I'm afraid. In both cases the string falls down. Or maybe I'm missing something!