The fact that all masses accelerate at the same rate when subjected to purely gravitational forces but do not do so for any other type of force indicates the singular nature of gravity. Another peculiarity of gravity is that although we can shield bodies from the effects of other forces, we cannot shield them from gravitational forces. Although the concept of gravity has been around for a long time and we have used it to explain so much, from the motion of planets to terrestrial events, gravity remains a somewhat mysterious force, difficult to incorporate into more general schemes. Attempts to create unified theories of all the forces have been foiled by gravity. Trying to unify quantum mechanics with gravity has also proven to be extremely difficult, resisting the most determined efforts from some the best minds in physics, including Einstein. This is partly because gravity, unlike electric and magnetic forces, is a non-linear force and non-linear forces are notoriously difficult to handle mathematically.
A linear force like electricity can be understood by using the following example. Suppose I measure the force on a test electric charge Q at a location A due to the presence of another electric charge R that is located at some other point B. I then remove R and bring in a a different charge S and place it at yet a third point C and again measure the force on the test charge Q. If I now place both R at B and S at C, the total force on the test charge due to both will be just the sum of the two earlier forces as measured separately. If we have a complicated system of charges, we can use our powerful mathematical techniques, such as calculus, to add up all the forces due to each charge.
But in the case of gravity forces, if we replace the electric charges with masses in the above example, the resultant force due to both will not be the sum of the forces due to each separately. That is what we mean by non-linearity.
Why is gravity non-linear? The reason was hinted at earlier when we said that matter affects the nature of space. Using a crude metaphor, in the traditional view of space, we treat it as being like a stage on which all the events of nature, including the particles and forces, move around like actors. Whatever they do, the stage remains unchanged and so they can perform predictably and without much difficulty. But introduce the idea of matter affecting space and then the introduction of matter as actors makes the stage itself part of the performance as it starts to warp and shift and change in response to their presence, distorting and moving around as the matter moves, affecting even the behavior of the non-gravitational entities. The shifting stage affects the motion of matter which in turn makes the stage move even more. This makes it hard for the ensemble to perform together.
But don’t we routinely treat the gravitational force as if it is linear? Isn’t that is how we calculate the motion of pretty much everything including planets, etc.? For the most part, we get away with ignoring the non-linearity because the force of gravity is so extremely weak that the nonlinear effects do not have an appreciable effect, except when we have to deal with the large scale structure of space time or massive objects like black holes or are trying to use it exactly as part of a unifying scheme with other forces or quantum mechanics. It is only then that we have to invoke the nonlinear field equations of general relativity and it is that that causes difficulties. It is like how we can get away with treating the Earth as an inertial frame for many things because the non-inertial effects introduced by its rotations are small, except when we are doing high-precision work or are dealing with large-scale systems like air currents.
I said earlier that unlike with the other forces, we cannot shield ourselves from the force of gravity. That is not strictly true. Einstein said that the ‘happiest thought’ of his life came in 1907 when he was thinking about how to reconcile Newton’s theories with his own theory of special relativity and he realized that there was a way in which to make the force of gravity disappear. The basic idea is that if I have an object in my hand and let it go, it will fall away from me to the ground and we say that it is because of gravity. But if I happened to be in free fall and let go of the object, it will remain where it is relative to me. As far as I am concerned, the force of gravity has ‘disappeared’ while I am in free fall.
Einstein is a remarkably lucid writer and when he is at the top of his game, it is a waste of time to paraphrase his writings to try and make them clearer because you end up making things more obscure. In his collected works can be found a recounting of how he arrived at his idea that the gravity can be made to disappear.
In an example worth considering, the gravitational field has a relative existence only in a manner similar to the electric field generated by magneto-electric induction. Because for an observer in free-fall from the roof of a house there is during the fall—at least in his immediate vicinity—no gravitational field. Namely, if the observer lets go of any bodies, they remain relative to him, in a state of rest or uniform motion, independent of their special chemical or physical nature. The observer, therefore, is justified in interpreting his state as being “at rest.”
The extremely strange and confirmed experience that all bodies in the same gravitational field fall with the same acceleration immediately attains, through this idea, a deep physical meaning. Because if there were just one single thing to fall in a gravitational field in a manner different from all others, the observer could recognize from it that he is in a gravitational field and that he is falling. But if such a thing does not exist—as experience has shown with high precision—then there is no objective reason for the observer to consider himself as falling in a gravitational field. To the contrary, he has every right to consider himself in a state of rest and his vicinity as free of fields as far as gravitation is concerned. [Italics in original]
This realization led him to what we know as the Principle of Equivalence that is heavily used in addressing the radiation paradoxes.
(To be continued.)