Gravitation vs acceleration

Consider the following experiment: a person is put in a room-size box high above the moon (chosen because there is no air and hence no air friction) with a bunch of measuring devices. This box is then taken high above the lunar surface and then let go: the box is then freely falling. The question is now, can the observer determine whether he/she is falling or whether he/she is in empty space unaffected by external forces (of course the answer is supposed to come before the box hits the surface). The answer to that is a definite NO! The observer can do experiments by looking at how objects move when initially at rest and when given a kick, he/she will find that they appear to move as is there were no gravitational forces at all! Similarly any experiment in physics, biology, etc. done solely inside the box will be unable to determine whether the box is freely falling or in empty space.

Why is that? Because of the equality of the gravitational and inertial masses. All objects are falling together and are assumed to be rather close to each other (the box is not immense) hence the paths they will follow will be essentially the same for each of them. So if the observer lets go of an apple, the apple and the observer follow essentially the same trajectory, and this implies that the observer will not see the apple move with respect to him. In fact, if we accept the priniciple of equivalence, nothing can be done to determine the fact that the observer is falling towards the Moon, for this can be done only if we could find some object which behaved differently from all the rest, and this can happen only if its gravitational and inertial masses are different. The principle of equivalence then implies that the observer will believe that he/she is an inertial frame of reference...until disabused of the notion by the crash with the surface.

The principle of equivalence is of interest neither because of its simplicity, nor because it leads to philosophically satisfying conclsions. It's importance is based on the enormous experimental evidence which confirms it; as with the Special Theory, the General Theory of Relativity is falsifiable.

 

Figure 7.3: An observer cannot distinguish between acceleration produced by a rocket and the acceleration produced by gravity.  

The lesson is that for any gravitational force we can always choose a frame of reference in which an observer will not experience any gravitational effects in his/her immediate vicinity (the reason for this last qualification will become clear below). Such a frame of reference is, as stated above, freely falling.

Conversely one can take the box an attach it to a machine that accelerates it (Fig. 7.3). If an observer drops an apple in such an accelerated box he/she will see the apple drop to the floor, the observer will also feel hi/her-self pressed against the bottom of the box, etc. The observer cannot distinguish between this situation and the one he/she would experience in the presence of gravitational forces! As long as we do experiments in a small region, the effects produced by a gravitational force are indistinguishable from those present in an accelerated reference frame.

Does this mean that the gravitational forces are a chimera, an illusion? Of course no. Consider for example Fig. 7.4, two apples fall to the Moon inside a box which is also falling. If they are separated by a sufficiently large distance an observer falling with the apples and box will find that the distance between the apples shortens as time goes on: this cannot be an inertial frame he argues (or else it is, but there is some force acting on the apples).

 

Figure 7.4: Experiment that differentiates between a gravitational effect and the effects of uniform acceleration: for an observer in the box the apples will draw closer.  

This same set-up can be used to distinguish between a box under the influence of a gravitational force and one being pulled by a machine; again we need a very big box (planet-sized). An observer places an two apples at the top of the box and releases them, he/she carefully measures its initial separation. The apples fall to the bottom of the box and the observer measures their separation there. If it is the same as above, and is the same irrespective of their initial separation, the observer is being pulled by a machine (box and all). If the separation is different, he/she can conclude that he/she is experiencing the effects of a gravitational force.