I am reading Wheeler and Taylor's Spacetime Physics. In Ch2, Wheeler mentioned:
"for gravity, any free-fall frame is an inertial frame." (roughly)
I am left wondering if that is true for electrical force:
Consider one charge is under a statistic electrical field. The charge is in free-fall. Is the electron's free fall frame an inertial frame?
(if yes, then can we say electrical force is a pseudo-force too?)
The special quality of gravitational fields that is not shared by electric (or magnetic) fields is the Equivalence Principle. The thought experiment you need to do is something like this...
Imagine being in a laboratory which is floating in outer space in the absence of any external fields and closed to the outside world. Do a series of experiments in that laboratory and record a video of what happens.
Now imagine that, while you are sleeping, somebody switches on a uniform gravitational field so that (in the usual way we describe things) your laboratory accelerates along the field lines. The question is, would you be able to perform an experiment to deduce the existence of that field? It turns out the answer (so far as we have been able to tell) is No, and we call this fact the Equivalence Principle. The equivalence principle means that the so-called 'free-fall' frame of the laboratory is just as 'inertial' as the one floating in outer space.
Finally, suppose instead that while you were sleeping somebody turns on a uniform electric field, and let's ask the same question (i.e. would you be able to tell when you wake up?). This time the answer is a definite Yes. As a simple example, a positive and negative charge aligned with the field would now experience an attractive or repulsive force in addition to their previous attractive force and so the trajectory of the particles would be completely different from the one you recorded in the earlier experiment. (Actually, the effects around you would probably be so obvious you probably wouldn't even need to perform an explicit laboratory experiment!)
In short: A uniform gravitational field doesn't make any difference to the internal dynamics of a system, provided the system is allowed to 'free-fall' in that field. The falling frame is just as inertial as one floating in empty space. An electric (or electromagnetic) field absolutely does effect the internal dynamics of a system, however, and so does not allow us to create a 'free-fall' frame which behaves inertially..
The force of gravity can be eliminated by using a free-fall frame because all massive objects are affected in the same way by a uniform gravitational field. This is not true for electrically charged objects in a uniform electric field.
To see why, we need Newton's second law, $F = ma$, as well es the force laws for a uniform gravitational field, $F_g = mg$, and for a uniform electric field, $F_q = qE$.
The effect of the force of gravity on an object with mass $m$ is $$ F_g = ma\\ mg = ma\\ a = g$$
The effect of the electric force on an object with charge $q$ and mass $m$ is $$ F_q = ma\\ qE = ma\\ a = \frac{q}{m} E$$
What does this tell us? If we have only objects with the same ratio of charge to mass (or to be precise, charge and mass density), the situation is analogous to the gravitational case: By performing experiments on these objects only, we cannot tell whether we are in free fall in a uniform electric field or whether there's no field at all.
In a real experiment, all objects having the same ratio of charge to mass is a very unrealistic condition already because most macroscopic things are uncharged. So in practise, we can immediately tell the difference because some objects are affected by the electric field and some are not.
In the electric case, we have one quantity which determines how strong the force is (electric charge) and another quantity which determines how big the acceleration due to this force is (mass). In the case of gravity, oddly both quantities are the same. Noticing this lead Einstein to general relativity.
Perhaps I'm misunderstanding the quote, but I think there's a distinction between the false statement...
Anything falling in a gravitational field is in an inertial frame
... and what I believe the quote means...
In the frame of reference gravity itself, something falling [in a gravitational field] is inertial, because the gravity frame measures no acceleration in the falling object's frame (my frame when I'm falling in an elevator is inertial w.r.t. the frame of anything else in said elevator).
So, to kind-of answer your question, the electron is accelerating towards the positive charge, since the magnitude of force between the two is...
$$\frac{1}{4 \pi \epsilon_0} \frac{q^- q^+}{r^2}$$
The closer they get, the more force attracts them—they are accelerating towards one another. To construct an inertial reference frame w.r.t. the electron, one must construct a frame that is accelerating at the same rate, and in the same direction, as the electron.
An object is in an inertial frame if no forces are acting on this object. If you feel weightless than you are in an inertial frame. So jumping down from a wall you are for a moment in an inertial frame.
Such a definition of the inertial fram implicates two things:
I am left wondering if that is true for electrical force:
Consider one positive charge and one negative charge (say, electron). They attract each other; hence, both charges are in free fall. Is the electron's free fall frame an inertial frame?
You are mixing the electric force between charged particles with their behavior under the influence of gravitation (from the mass of the earth). As long as an electric force is acting between particles or objects they are under acceleration. It does not matter will this be an attractive or repulsive force, it is still an acceleration. This particles are not in an inertial frame.
