We have now acquired most of the background knowledge needed to start directly addressing the question that this series of posts started with, as to whether an electric charge and a neutral particle dropped from the ceiling will hit the ground at the same time. The first stage of that is what is known as the Principle of Equivalence.
Recall the Postulate #1 we started with that said that all objects falling freely in a gravitational field will fall at the same rate and hit the ground at the same time. We were able to explain this by saying that it followed from the fact that gravitational and inertial masses were equal. But we did not explain why they should be equal. The two masses were arrived at, after all, by distinct methods using different operational definitions. But by considering accelerating frames and the Principle of Equivalence, we have a simple explanation for it.
Consider two objects floating freely in space that are at rest with respect to each other and to an observer S, to signify space. No suppose we enclose the two objects in a closed room and accelerate the room ‘upwards’ (i.e., in the direction from the ‘floor’ to the ‘ceiling’) with a value g. We will call this frame E to signify an elevator or Earth. Then in the frame E, both objects will seem to be accelerated ‘downwards’ (i.e., towards the floor of the room) with a value g, just as if they were falling freely on Earth, and will both hit the floor at the same time. They will behave just as if they were falling down near the surface of the Earth and since they hit the floor at the same time, we can infer that their gravitational and inertial masses are equal, rather than tacitly assuming it to be the case as we did before.
Einstein’s insight was that the two situations are indistinguishable, that we can never tell if we are in a uniform gravitational field that is acting downwards (due to a large mass somewhere below us) or whether there is no mass anywhere near that is producing a gravitational field and instead we are in a frame that is accelerating upwards. There is no experiment that we can do within the enclosed room (i.e., within the frame E) that would enable us to distinguish between the two systems.
This insight was derived from the behavior of objects in a gravitational field and explains Postulate #1 and thus also the equivalence of gravitational and inertial masses. But crucially Einstein extended it into a general rule, that there is no experiment whatsoever, involving any and all forces, that will enable us to make the distinction between the two frames, a stationary frame in a uniform downward gravitational field or an upwardly accelerating frame in the absence of any gravity. That added level of generality has massive implications for all of physics and laid the foundation for the development of general relativity. This is called the Principle of Equivalence (PoE).
It should be noted that this equivalence principle as stated above says that we cannot distinguish between an accelerating frame and a uniform gravitational field, i.e., a field that has the same magnitude and direction everywhere. But real gravitational fields are not uniform because they are due to masses that have limited size. So for example, if we were to drop two objects in an enclosed room, then if we were in an upwardly accelerating frame and there were no gravitational field, the two objects would fall along parallel lines. But if the room were stationary on the surface of the Earth and the objects were falling due to its gravitational field, then the paths of the two objects would ever so slightly converge towards each other because each one is headed towards the same single point, that is the center of the Earth. In that case, we could distinguish the two situations. So the PoE is formulated either in terms of a hypothetical uniform field or by applying it only within a very small region of space, where the field is approximately uniform. To get a truly uniform gravitational field over a large volume one would need to hypothesize the existence of an infinitely large, uniform slab of matter, with the two sides being flat and parallel. This is of course, not physically possible. The closest one might get to it in the real world is close to the surface of the plane of stars in the spiral arms of galaxies.
With all this in hand, we can tackle Postulate #2, that asserts that an accelerating charge will radiate energy, thus reducing its kinetic energy. If so, an electric charge falling in a gravitational field will fall more slowly than a neutral charge that does not radiate and lose any kinetic energy, thus seeming to violate Postulate #1. Doing an actual experiment to test this would be extraordinarily difficult and has not been done as far as I am aware. But Einstein’s PoE gives us a way to address this question theoretically by allowing us to switch between a frame at rest in a gravitational field (which is our familiar world) and an accelerating frame in the absence of gravity, and requiring the observable results to be the same.
Consider an electric charge and a neutral particle both floating freely in space, far from any gravitational and other forces. If observe them to be at rest or moving with uniform velocity, then that means that we are in an inertial frame S. The electric charge will not radiate any energy because according to Maxwell’s equations, which are valid in inertial frames, a charge must have an acceleration in an inertial frame in order to radiate energy. Now consider another frame E that is moving with an acceleration g with respect to S. In the frame E, both the charged and neutral particles will have the same acceleration g but in the opposite direction to E. The situation in E is exactly as if the two particles are accelerating in a uniform gravitational field. But all we have done is change the observer from one in S to one in E. The state of motion of the charge is unchanged. By the PoE, observable results have to be the same in both frames and hence the charge still should not radiate in E. Hence it seems as if we can conclude that an electric charge will not radiate if its acceleration is entirely due to it falling freely in a gravitational field. Hence Postulate #1 is satisfied but Postulate #2 is found to be only valid under certain conditions, when a charge is accelerating in an inertial frame. We arrived at this conclusion purely theoretically, by invoking the PoE.
This appears to resolve the paradox that we started with but at the expense of Postulate #2, which has to be modified to say that an electric charge does not radiate when it is falling in a uniform gravitational field. This resolution requires us to go from a universal statement that an accelerating charge always radiates to having to first determine whether its acceleration is due to falling freely in a uniform gravitational field because then that situation is equivalent to the charge not accelerating in an inertial frame. Thus it seems as if a charge will radiate if it is accelerating in a horizontal direction on the Earth’s surface but not if it is falling in a vertical direction, because in that case it is not accelerating in an inertial frame. If it is falling at an angle, so that the acceleration has a vertical component and a horizontal component, that would presumably require appropriate adjustments.
All that is is a little awkward but some closed the book at this point, reconciled to the situation.
But wait, there’s more! That is by no means the end of the story. While we seem to have resolved this one issue, if we start looking a little more closely, we will find that that this resolution does not hold up and looking deeper leads to another paradox and resolving that leads to another, and so on.
To misquote Bette Davis from the film All About Eve, “Fasten your seatbelts. It is going to be a bumpy ride.”
(To be continued.)