Similar conclusions would apply were we to consider those other species of inertial forces called centrifugal and Coriolis forces. From the standpoint of an observer who selects a rotating platform as frame of reference (just as prior to Copernicus men were wont to take the earth as frame of reference), centrifugal and Coriolis forces are real. But the moment we refer events to a Galilean frame, these forces become fictitious, and can be interpreted in terms of the law of inertia, just as the force in the train became fictitious when events were viewed from the embankment.
Contrast this situation with that of a stone falling towards the ground. Here, we may refer events to any frame we please, but in any case the law of inertia can never be made to account for the stone’s motion towards the earth. The situation appears to be entirely different from that of the passenger in the train. And so it follows that we are compelled to conceive of a real force, a force of gravitation pulling the stone towards the centre of the earth. Thus it would appear that gravitational force, in contradistinction to inertial force, could not be ascribed to mere conditions of observation. It was assumed, therefore, by classical science that there existed around a body like the earth a real absolute field of force distributed radially through space, whereas in the accelerated enclosure the forces were merely fictitious or relative.
There is yet another way of bringing out the difference between forces of inertia and forces of gravitation. Forces of inertia are generated by motion, whereas forces of gravitation are generated by matter. Thus, if we arrest the rotation of a disk, the observer, fixed to the disk, will note that the forces of inertia have vanished. But to remove the force of gravitation, it would be necessary to annihilate the entire mass of the earth, and this would constitute an operation of an entirely different nature.
Now, in every type of field of force, the magnitude of the pull which is exerted on a body susceptible to the influence of the field is governed by a certain definite characteristic of the body. Thus, in a given electric field, the magnitude of the pull exerted on a body at a given point of the field is proportional to the electric charge of the body. Likewise, in an inertial field, the magnitude of the pull is proportional to what is known as the inertial mass of the body, and in a gravitational field the pull is proportional to the gravitational mass of the body. Every material body was assumed, therefore, to possess two types of masses, the inertial and the gravitational, corresponding to the two types of fields to which the body would react.
A concrete illustration of the difference between inertial and gravitational mass is given by the following example: Consider a billiard ball at rest on a table. We should discover that the more massive the ball, the greater would be the effort necessary to set it in motion, as also to arrest its motion once started. The type of mass with which we should here be concerned would be inertial mass; and we may say that it is inertial mass which opposes a departure from rectilinear uniform motion or rest, hence which opposes acceleration. If, now, we were to lift the ball from the table and hold it at arm’s length, the effort we should have to exert to prevent our arm and ball from falling would again vary with the mass of the ball. This time, however, the mass whose effects we were seeking to resist would be the gravitational mass, i.e., the mass which is responsible for weight. It is this same gravitational mass which enters into the expression of Newton’s law of attraction.
There is no a priori reason why any connection should exist between these two different types of masses. Thus, if a billiard ball is found to weigh twice as much as another, there is no logical necessity to anticipate that its inertial mass will also be twice that of the other. Yet it so happens that the most refined physical experiments have invariably proved the strictest proportionality between these two types of masses; so that by choosing our units suitably it was always possible to represent both the inertial and the gravitational mass of a given body by the same number. Classical science had accepted the equality of the two types of masses as an empirical fact, but had found itself incapable of suggesting any theoretical justification for it. It appeared to be due to some miraculous coincidence.
This equality in the numerical values of the two types of masses will entail important consequences. For suppose, indeed, that the equality did not hold. This would imply that two billiard balls might present exactly the same gravitational mass, yet differ in the value of their respective inertial masses. If, then, the two balls were to be released simultaneously and allowed to fall from the same height towards the earth, their gravitational masses being equal, both balls would be subjected to exactly the same pull by the earth. But their inertial masses being assumed unequal, the ball whose inertial mass was greater would oppose the acceleration of the fall more strenuously than would be the case with the ball of lesser inertial mass. As a result, the balls would not reach the earth’s surface simultaneously. On the other hand, if the two types of masses are identical, we may be assured that any two objects released from the same point will reach the earth’s surface at the same instant (provided we operate in vacuo so that the resistance of the air does not interfere with their fall). It was precisely because experiment had demonstrated the existence of the same rate of fall for all bodies released from the same height, that Galileo was led to recognise the equality of the two masses. It may be mentioned, however, that ultra-precise experiments (notably with Eötvös’ torsion balance) have since verified these results with extreme accuracy.[80]
We see, therefore, that the equality of the two types of masses allows us to anticipate that an observer situated in a falling elevator would have no weight. He might be standing on a weighing machine, but the needle would always register zero. The fact is that both observer and weighing balance would be falling with the same motion towards the earth; hence the observer’s feet would never press against the balance.
These anticipations may appear to be self-evident; but it is important to note that they are in no wise self-evident until we have recognised the perfect equality of the two types of masses. If such an equality did not exist, the observer in the elevator might still press against the scales of the weighing machine or might hit the ceiling of the elevator, and it would no longer be correct to anticipate that elevator, scales and observer would all fall with exactly the same velocity and acceleration.
We may illustrate the example of the falling elevator in a more general way by considering an observer situated in the interior of a hollow projectile describing its parabola under the influence of the earth’s attraction. The observer would detect no manifestation of weight in his enclosure. He might tip a glass full of water upside down: the water would not spill. If he threw a handful of pebbles into the air, each pebble would follow a straight course with rectilinear motion through the projectile, just as would be the case in a Galilean frame. In short, no mechanical experiment could ever reveal the presence of the earth’s field of gravitation in the interior of any frame of reference moving freely under the action of this field of gravitation. Experiments conducted inside such falling bodies are very difficult to perform, but since the results we have mentioned are the immediate consequences of the equality of the two types of masses, which highly refined experiment has established, we are able to anticipate indirectly the results of such experiments.