Next we shall discuss the possible relation of gravitation to other forces. There is no explanation of gravitation in terms of other forces at the present time. It is not an aspect of electricity or anything like that, so we have no explanation. However, gravitation and other forces are very similar, and it is interesting to note analogies. For example, the force of electricity between two charged objects looks just like the law of gravitation: the force of electricity is a constant, with a minus sign, times the product of the charges, and varies inversely as the square of the distance. It is in the opposite direction — likes repel. But is it still not very remarkable that the two laws involve the same function of distance? Perhaps gravitation and electricity are much more closely related than we think. Many attempts have been made to unify them; the so-called unified field theory is only a very elegant attempt to combine electricity and gravitation; but, in comparing gravitation and electricity, the most interesting thing is the relative strengths of the forces. Any theory that contains them both must also deduce how strong the gravity is.
If we take, in some natural units, the repulsion of two electrons (nature’s universal charge) due to electricity, and the attraction of two electrons due to their masses, we can measure the ratio of electrical repulsion to the gravitational attraction. The ratio is independent of the distance and is a fundamental constant of nature. The ratio is shown in Fig. 5-14. The gravitational attraction relative to the electrical repulsion between two electrons is 1 divided by 4.17 × 1042! The question is, where does such a large number come from? It is not accidental, like the ratio of the volume of the earth to the volume of a flea. We have considered two natural aspects of the same thing, an electron. This fantastic number is a natural constant, so it involves something deep in nature. Where could such a tremendous number come from? Some say that we shall one day find the “universal equation,” and in it, one of the roots will be this number. It is very difficult to find an equation for which such a fantastic number is a natural root. Other possibilities have been thought of; one is to relate it to the age of the universe. Clearly, we have to find another large number somewhere. But do we mean the age of the universe in years? No, because years are not “natural”; they were devised by men. As an example of something natural, let us consider the time it takes light to go across a proton, 10—24 second. If we compare this time with the age of the universe, 2 × 1010 years, the answer is 10—42. It has about the same number of zeros going off it, so it has been proposed that the gravitational constant is related to the age of the universe. If that were the case, the gravitational constant would change with time, because as the universe got older the ratio of the age of the universe to the time which it takes for light to go across a proton would be gradually increasing. Is it possible that the gravitational constant is changing with time? Of course the changes would be so small that it is quite difficult to be sure.
One test which we can think of is to determine what would have been the effect of the change during the past 109 years, which is approximately the age from the earliest life on the earth to now, and one-tenth of the age of the universe. In this time, the gravity constant would have increased by about 10 percent. It turns out that if we consider the structure of the sun — the balance between the weight of its material and the rate at which radiant energy is generated inside it — we can deduce that if the gravity were 10 percent stronger, the sun would be much more than 10 percent brighter — by the sixth power of the gravity constant! If we calculate what happens to the orbit of the earth when the gravity is changing, we find that the earth was then closer in. Altogether, the earth would be about 100 degrees centigrade hotter, and all of the water would not have been in the sea, but vapor in the air, so life would not have started in the sea. So we do not now believe that the gravity constant is changing with the age of the universe. But such arguments as the one we have just given are not very convincing, and the subject is not completely closed.
It is a fact that the force of gravitation is proportional to the mass, the quantity which is fundamentally a measure of inertia—of how hard it is to hold something which is going around in a circle. Therefore two objects, one heavy and one light, going around a larger object in the same circle at the same speed because of gravity, will stay together because to go in a circle requires a force which is stronger for a bigger mass. That is, the gravity is stronger for a given mass in just the right proportion so that the two objects will go around together. If one object were inside the other it would stay inside; it is a perfect balance. Therefore, Gagarin or Titov would find things “weightless” inside a spaceship; if they happened to let go of a piece of chalk, for example, it would go around the earth in exactly the same way as the whole spaceship, and so it would appear to remain suspended before them in space. It is very interesting that this force is exactly proportional to the mass with great precision, because if it were not exactly proportional there would be some effect by which inertia and weight would differ. The absence of such an effect has been checked with great accuracy by an experiment done first by Eötvös in 1909 and more recently by Dicke. For all substances tried, the masses and weights are exactly proportional within 1 part in 1,000,000,000, or less. This is a remarkable experiment.
Gravity and relativity
Another topic deserving discussion is Einstein’s modification of Newton’s law of gravitation. In spite of all the excitement it created, Newton’s law of gravitation is not correct! It was modified by Einstein to take into account the theory of relativity. According to Newton, the gravitational effect is instantaneous, that is, if we were to move a mass, we would at once feel a new force because of the new position of that mass; by such means we could send signals at infinite speed. Einstein advanced arguments which suggest that we cannot send signals faster than the speed of light, so the law of gravitation must be wrong. By correcting it to take the delays into account, we have a new law, called Einstein’s law of gravitation. One feature of this new law which is quite easy to understand is this: In the Einstein relativity theory, anything which has energy has mass — mass in the sense that it is attracted gravitationally. Even light, which has an energy, has a “mass.” When a light beam, which has energy in it, comes past the sun there is an attraction on it by the sun. Thus the light does not go straight, but is deflected. During the eclipse of the sun, for example, the stars which are around the sun should appear displaced from where they would be if the sun were not there, and this has been observed.
Finally, let us compare gravitation with other theories. In recent years we have discovered that all mass is made of tiny particles and that there are several kinds of interactions, such as nuclear forces, etc. None of these nuclear or electrical forces has yet been found to explain gravitation. The quantum-mechanical aspects of nature have not yet been carried over to gravitation. When the scale is so small that we need the quantum effects, the gravitational effects are so weak that the need for a quantum theory of gravitation has not yet developed. On the other hand, for consistency in our physical theories it would be important to see whether Newton’s law modified to Einstein’s law can be further modified to be consistent with the uncertainty principle. This last modification has not yet been completed.