# Understanding calculation in Feynman's Lecture on Theory of Gravitation about age of universe and gravitational constant

I am reading Feynman's Lectures. In the chapter entitled "The Theory of Gravitation", there is the following part

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. 7–14. The gravitational attraction relative to the electrical repulsion between two electrons is 1 divided by $$4.17\times 10^{42}$$! 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, $$2×10^{10}$$ 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 $$10^9$$ 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.

My question is about the calculations in the second paragraph above.

Here are the calculations I did

The universe is $$2\cdot 10^{10}$$ years old, which is $$6.311385044\cdot 10^{17}$$ seconds.

If we divide the time it takes for light to cross a proton, $$10^{-24}$$ by the age of the universe in seconds, we get

$$\frac{10^{-24}}{6.311385044\cdot 10^{17}}=1.584438270\cdot 10^{-42}\tag{1}$$

As the universe ages, this ratio goes down (because the denominator increases).

Now, the ratio between the gravitational force and the electric force between two electrons is about $$\frac{1}{4.17}\cdot 10^{-42}\tag{2}$$

Suppose this ratio is related to the age of the universe through a relationship between the gravitational constant and the age of the universe. i.e., as the universe ages, the gravitational constant decreases, thus decreasing the ratio in (2).

Consider the situation $$10^9$$ years ago.

The universe was $$5.995815792\cdot 10^{17}$$ seconds old at that point, and the ratio in (1) was

$$\frac{10^{-24}}{5.995815792 \cdot 10^{17}}=1.667829758\cdot 10^{-42}\tag{3}$$

That is, in $$10^9$$ years (3) decreased by about 5%.

Are these calculations correct, and how do they relate to the claim that the gravitational constant "would have increased by about 10%" in this $$10^9$$ year time frame?

Your main point is correct. There is no problem with your calculations. Good catch. The root of the problem is that Feynman was inconsistent in his language. He said that "the age of the universe" was $$20\times10^9$$ years. He wanted to consider "the effect of the change during the past" $$10^9$$ years, which he indicated was "one-tenth of the age of the universe", which one can easily calculate to be $$2\times10^9$$ years, using his information. But $$10^9$$ is not $$2\times10^9$$. If the time change had been "one-tenth of the age of the universe" or $$2\times10^9$$ years instead of $$10^9$$ years, then he would have correctly found that "the gravity constant would have increased by about 10 percent.", if one is looking back in time. $$10^9$$ out of $$20\times10^9$$ is only five percent. He might have gotten confused, according to increases in the estimates of "the age of the universe". He might have remembered some old estimates. It is more likely that it was just a typo in his book with one of his two numbers ($$20\times10^9$$ years or $$10^9$$ years) being wrong by a factor of two to then yield his more emphasized 10%. Based on the location of the minimal potential typo, the typo was most likely in the $$10^9$$ years (which was only used once), since he also used the $$20\times10^9$$ years in another calculation and also there is much evidence that "the age from the earliest life on the earth to now" was much larger than $$10^9$$ years (currently by at least a factor of 3.5, which is certainly greater than a potential missing factor of two). Anyway, it appears that you were correct in that there was a decrease in gravity by 5% using the likely typo number of $$10^9$$.
If you used his other number of $$2\times10^9$$ years or "one-tenth of the age of the universe" (that is ignoring his $$10^9$$ years), then you would have verified his 10% estimate. In that case he would have been correct that "the gravity constant would have increased by about" 10%, if we are looking back in time (but you would have needed to ignore his typo/error). By Feynman using the word "increased", we should give him the benefit of the doubt meaning he was looking back to the earlier time. For your own edification, you might redo your calculations using the $$2\times10^9$$ years so that you would get the 10% as Feynman did (that is ignoring his $$10^9$$ years typo/error). If you don't redo your comments, that might better serve to point out the typo/error in Feynman's words so that others need not repeat your correct calculations, using the likely typo/error number of $$10^9$$ years.