In Ta-Pei Cheng's Relativity, Gravitation and Cosmology book, he stated the equivalence principle as
Physics in a frame freely falling in a gravity field is equivalent to physics in an inertial frame without gravity.
But from wikipedia, an inertial frame is defined as a frame of reference that is not undergoing acceleration. So why is the freely falling frame of reference in a gravity field, which is accelerating, an inertial frame?
We have the that the standard form of stating the Principle of Equivalence is dependent on providing a definition of inertial frame.
My preferred way of dealing with the definition of inertial frame is to go back to the thought demonstration that Galileo offered. Except, rather than using a ship moving along over smooth water we let the protagonists of the thought demonstration be the crew of a spaceship.
Inside the spaceship a device shoots darts to a dartboard, the device can be aimed with arbitrarily high precision. The frame of the dartboard represents a coordinate system. Operate the device in multiple orientations, covering all angles. If you find that the same aim produces the same result for every orientation then the spaceship is in free motion (either in orbital motion or at an arbitrarily large distance away from any gravitating body) Conversely, if the ship is using any form of propulsion then the motion of the darts that are aimed by the device will not be isotropic with respect to a coordinate system that is co-moving with the spaceship.
The whole point of the concept of inertial frame is that in order to be meaningful the assessment is local. This applies in all three contexts that use the concept of inertial frame: Galilean relativity, Special relativity, General relativity.
What you quoted from wikipedia was the opening sentence of that article, which is not at all intended as an exhaustive definition. If that opening sentence would instead be an immediate attempt at exhaustive definition it would be unreadable.
In general the only way to define the concept of inertial frame is to provide an operational definition. The only way is to describe the kind of operation a device must perform in order to make the assessment.
So why is the freely falling frame of reference in a gravity field, which is accelerating, an inertial frame?
Note that the book is saying,
Physics in a frame freely falling in a gravity field is equivalent to physics in an inertial frame without gravity.
When we talking of motion under the influence of gravity so that $$\mathbf{F}=m\mathbf{a}=-mg\hat{k}$$ that means we are using Newton's second law which is valid only in inertial frames. We are observing the motion of particle from an inertial frame. That's the reason we are able to compute the force with the help of Newton's second law.
If we suppose an accelerating frame so that there is a pseudo force which is given by $$\mathbf{F}=-m\mathbf{a}$$ Now if acceleration of frame is equal to gravity then $$\mathbf{F}=-mg\hat{k}$$ So that the situation of equivalent. The person in a elevator will not able to tell whether he is feeling force due to influence of gravity or due to the acceleration of elevator ( in space where gravity is not present).
