According to the equivalence principle, it is not possible to distinguish between an inertial system (far from any gravitational interaction) and a system in free fall within a gravitational field. However, if I were inside the system in free fall, wouldn't I notice that I am falling faster and faster? Wouldn't there be any physical sensation indicating that I am accelerating, similar to what I would feel in the free fall of an amusement park ride?
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3$\begingroup$ The sensation of a sudden drop on a ride at an amusement park is identical to abruptly becoming an inertial observer. Your internal organs are usually resisting the acceleration from gravity, and are familiar being in a non-inertial frame, i.e. having to support their own weight. $\endgroup$– AidenCommented Dec 21, 2023 at 20:43
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if I were inside the system in free fall, wouldn't I notice that I am falling faster and faster?
Not if your view were limited to things that are inside "the system." If you are inside a closed box, and you are unable to get any information from outside of the box, then you would have no ability to measure your speed with respect to anything outside of the box.
Wouldn't there be any physical sensation indicating that I am accelerating, similar to what I would feel in the free fall of an amusement park ride?
That sensation you feel when you ride a drop tower at an amusement park is the sensation of free fall. It would feel the same (except, you could feel it for a longer duration) if you were riding a Vomit Comet, or if you were on board the International Space Station, or if you were on a star ship lost in the void, far from any galaxy.
...it is not possible to distinguish between an inertial system at rest...
Actually, that phrase doesn't really make any sense. At rest with respect to what? If you are at rest with respect to one thing, then you almost certainly are in motion with respect to something else.
Usually when people talk about the equivalence principle, they are not talking about an inertial system. The usual formula is to say that, if you are closed inside a box, and you can't get any information from outside the box, then there is no way for you to tell the difference between the case where your box is standing on the surface of a planet, and the case where the box is on-board an accelerating star ship.
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1$\begingroup$ All inertial systems are at rest; they are neither accelerating nor under the influence of a gravitating field. $\endgroup$ Commented Dec 21, 2023 at 20:57
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1$\begingroup$ @AlbertusMagnus A frame moving at constant velocity with respect to an inertial frame is also inertial. Could you mean it is always possible to find an inertial frame in which a non-accelerating observer is at rest? $\endgroup$– AidenCommented Dec 21, 2023 at 22:42