## 7The Theory of Gravitation

### 7–1Planetary motions

In this chapter we shall discuss one of the most
far-reaching generalizations of the human mind. While we are admiring
the human mind, we should take some time off to stand in awe of a
*nature* that could follow with such completeness and generality
such an elegantly simple principle as the law of gravitation. What is
this law of gravitation? It is that every object in the universe
attracts every other object with a force which for any two bodies is
proportional to the mass of each and varies inversely as the square of
the distance between them. This statement can be expressed
mathematically by the equation
\begin{equation*}
F=G\,\frac{mm'}{r^2}.
\end{equation*}
If to this we add the fact that an object responds to a force by
accelerating in the direction of the force by an amount that is
inversely proportional to the mass of the object, we shall have said
everything required, for a sufficiently talented mathematician could
then deduce all the consequences of these two principles. However, since
you are not assumed to be sufficiently talented yet, we shall discuss
the consequences in more detail, and not just leave you with only these
two bare principles. We shall briefly relate the story of the discovery
of the law of gravitation and discuss some of its consequences, its
effects on history, the mysteries that such a law entails, and some
refinements of the law made by Einstein; we shall also discuss the
relationships of the law to the other laws of physics. All this cannot
be done in one chapter, but these subjects will be treated in due time
in subsequent chapters.

The story begins with the ancients observing the motions of planets
among the stars, and finally deducing that they went around the sun, a
fact that was rediscovered later by Copernicus. Exactly *how* the planets went around the
sun, with exactly *what motion*, took a little more work to
discover. Beginning in the sixteenth century there were great
debates as to whether they really went around the sun or not. Tycho
Brahe had an idea that was different
from anything proposed by the ancients: his idea was that these debates
about the nature of the motions of the planets would best be resolved if
the actual positions of the planets in the sky were measured
sufficiently accurately. If measurement showed exactly how the planets
moved, then perhaps it would be possible to establish one or another
viewpoint. This was a tremendous idea—that to find something out, it
is better to perform some careful experiments than to carry on deep
philosophical arguments. Pursuing this idea, Tycho Brahe studied the positions of the planets for many years
in his observatory on the island of Hven, near Copenhagen. He made
voluminous tables, which were then studied by the mathematician
Kepler, after
Tycho’s death.
Kepler discovered from the data
some very beautiful and remarkable, but simple, laws regarding planetary
motion.

### 7–2Kepler’s laws

First of all, Kepler found that
each planet goes around the sun in a curve called an
*ellipse*, with the sun at a focus of the
ellipse. An ellipse is not just an oval,
but is a very specific and precise curve that can be obtained by using
two tacks, one at each focus, a loop of string, and a pencil; more
mathematically, it is the locus of all points the sum of whose distances
from two fixed points (the foci) is a constant. Or, if you will, it is a
foreshortened circle (Fig. 7–1).

Kepler’s second observation was
that the planets do not go around the sun at a uniform speed, but move
faster when they are nearer the sun and more slowly when they are
farther from the sun, in precisely this way: Suppose a planet is
observed at any two successive times, let us say a week apart, and that
the radius vector^{1} is drawn to the planet for each
observed position. The orbital arc traversed by the planet during the
week, and the two radius vectors, bound a certain plane area, the shaded
area shown in Fig. 7–2. If two similar observations are made
a week apart, at a part of the orbit farther from the sun (where the
planet moves more slowly), the similarly bounded area is exactly the
same as in the first case. So, in accordance with the second law, the
orbital speed of each planet is such that the radius “sweeps out”
equal areas in equal times.

Finally, a third law was discovered by Kepler much later; this law is of a different category from the other two, because it deals not with only a single planet, but relates one planet to another. This law says that when the orbital period and orbit size of any two planets are compared, the periods are proportional to the $3/2$ power of the orbit size. In this statement the period is the time interval it takes a planet to go completely around its orbit, and the size is measured by the length of the greatest diameter of the elliptical orbit, technically known as the major axis. More simply, if the planets went in circles, as they nearly do, the time required to go around the circle would be proportional to the $3/2$ power of the diameter (or radius). Thus Kepler’s three laws are:

- Each planet moves around the sun in an ellipse, with the sun at one focus.
- The radius vector from the sun to the planet sweeps out equal areas in equal intervals of time.
- The squares of the periods of any two planets are proportional to the cubes of the semimajor axes of their respective orbits: $T\propto a^{3/2}$.

### 7–3Development of dynamics

While Kepler was discovering
these laws, Galileo was studying
the laws of motion. The problem was, what makes the planets go around?
(In those days, one of the theories proposed was that the planets went
around because behind them were invisible angels, beating their wings
and driving the planets forward. You will see that this theory is now
modified! It turns out that in order to keep the planets going around,
the invisible angels must fly in a different direction and they have no
wings. Otherwise, it is a somewhat similar theory!)
Galileo discovered a very
remarkable fact about motion, which was essential for understanding
these laws. That is the principle of *inertia*—if something is
moving, with nothing touching it and completely undisturbed, it will go
on forever, coasting at a uniform speed in a straight line. (*Why*
does it keep on coasting? We do not know, but that is the way it is.)

Newton modified this idea, saying
that the only way to change the motion of a body is to use *force*.
If the body speeds up, a force has been applied *in the direction
of motion*. On the other hand, if its motion is changed to a new
*direction*, a force has been applied *sideways*.
Newton thus added the idea that a
force is needed to change the speed *or the direction* of motion of
a body. For example, if a stone is attached to a string and is whirling
around in a circle, it takes a force to keep it in the circle. We have
to *pull* on the string. In fact, the law is that the acceleration
produced by the force is inversely proportional to the mass, or the
force is proportional to the mass times the acceleration. The more
massive a thing is, the stronger the force required to produce a given
acceleration. (The mass can be measured by putting other stones on the
end of the same string and making them go around the same circle at the
same speed. In this way it is found that more or less force is required,
the more massive object requiring more force.) The brilliant idea
resulting from these considerations is that no *tangential* force
is needed to keep a planet in its orbit (the angels do not have to fly
tangentially) because the planet would coast in that direction anyway.
If there were nothing at all to disturb it, the planet would go off in a
*straight line*. But the actual motion deviates from the line on
which the body would have gone if there were no force, the deviation
being essentially *at right angles* to the motion, not in the
direction of the motion. In other words, because of the principle of
inertia, the force needed to control the motion of a planet
*around* the sun is not a force around the sun but *toward*
the sun. (If there is a force toward the sun, the sun might be the
angel, of course!)

### 7–4Newton’s law of gravitation

From his better understanding of the theory of motion,
Newton appreciated that *the
sun* could be the seat or organization of forces that govern the motion
of the planets. Newton proved to
himself (and perhaps we shall be able to prove it soon) that the very
fact that equal areas are swept out in equal times is a precise sign
post of the proposition that all deviations are precisely
*radial*—that the law of areas is a direct consequence of the
idea that all of the forces are directed exactly *toward the sun*.

Next, by analyzing Kepler’s third law it is possible to show that the farther away the planet, the weaker the forces. If two planets at different distances from the sun are compared, the analysis shows that the forces are inversely proportional to the squares of the respective distances. With the combination of the two laws, Newton concluded that there must be a force, inversely as the square of the distance, directed in a line between the two objects.

Being a man of considerable feeling for generalities,
Newton supposed, of course, that
this relationship applied more generally than just to the sun holding
the planets. It was already known, for example, that the planet Jupiter
had moons going around it as the moon of the earth goes around the
earth, and Newton felt certain
that each planet held its moons with a force. He already knew of the
force holding *us* on the earth, so he proposed that this was a
*universal force*—*that everything pulls everything else*.

The next problem was whether the pull of the earth on its people was the
“same” as its pull on the moon, i.e., inversely as the square of the
distance. If an object on the surface of the earth falls $16$ feet in
the first second after it is released from rest, how far does the moon
fall in the same time? We might say that the moon does not fall at all.
But if there were no force on the moon, it would go off in a straight
line, whereas it goes in a circle instead, so it really *falls in*
from where it would have been if there were no force at all. We can
calculate from the radius of the moon’s orbit (which is about
$240{,}000$ miles) and how long it takes to go around the earth
(approximately $29$ days), how far the moon moves in its orbit in
$1$ second, and can then calculate how far it falls in one
second.^{2} This distance turns out to be roughly $1/20$ of an
inch in a second. That fits very well with the inverse square law,
because the earth’s radius is $4000$ miles, and if something which is
$4000$ miles from the center of the earth falls $16$ feet in a second,
something $240{,}000$ miles, or $60$ times as far away, should fall
only $1/3600$ of $16$ feet, which also is roughly $1/20$ of an inch.
Wishing to put this theory of gravitation to a test by similar
calculations, Newton made his
calculations very carefully and found a discrepancy so large that he
regarded the theory as contradicted by facts, and did not publish his
results. Six years later a new measurement of the size of the earth
showed that the astronomers had been using an incorrect distance to the
moon. When Newton heard of this,
he made the calculation again, with the corrected figures, and obtained
beautiful agreement.

This idea that the moon “falls” is somewhat confusing, because, as you
see, it does not come any *closer*. The idea is sufficiently
interesting to merit further explanation: the moon falls in the sense
that *it falls away from the straight line that it would pursue if
there were no forces*. Let us take an example on the surface of the
earth. An object released near the earth’s surface will fall $16$ feet
in the first second. An object shot out *horizontally* will also
fall $16$ feet; even though it is moving horizontally, it still falls
the same $16$ feet in the same time. Figure 7–3 shows an
apparatus which demonstrates this. On the horizontal track is a ball
which is going to be driven forward a little distance away. At the same
height is a ball which is going to fall vertically, and there is an
electrical switch arranged so that at the moment the first ball leaves
the track, the second ball is released. That they come to the same depth
at the same time is witnessed by the fact that they collide in midair.
An object like a bullet, shot horizontally, might go a long way in one
second—perhaps $2000$ feet—but it will still fall $16$ feet if it is
aimed horizontally. What happens if we shoot a bullet faster and faster?
Do not forget that the earth’s surface is curved. If we shoot it fast
enough, then when it falls $16$ feet it may be at just the same height
above the ground as it was before. How can that be? It still falls, but
the earth curves away, so it falls “around” the earth. The question
is, how far does it have to go in one second so that the earth is
$16$ feet below the horizon? In Fig. 7–4 we see the earth
with its $4000$-mile radius, and the tangential, straight line path that
the bullet would take if there were no force. Now, if we use one of
those wonderful theorems in geometry, which says that our tangent is the
mean proportional between the two parts of the diameter cut by an equal
chord, we see that the horizontal distance travelled is the mean
proportional between the $16$ feet fallen and the $8000$-mile diameter
of the earth. The square root of $(16/5280)\times8000$ comes out very
close to $5$ miles. Thus we see that if the bullet moves at $5$ miles a
second, it then will continue to fall toward the earth at the same rate
of $16$ feet each second, but will never get any closer because the
earth keeps curving away from it. Thus it was that Mr. Gagarin
maintained himself in space while going $25{,}000$ miles around the
earth at approximately $5$ miles per second. (He took a little longer
because he was a little higher.)

Any great discovery of a new law is useful only if we can take more out
than we put in. Now, Newton
*used* the second and third of Kepler’s laws to deduce his law of gravitation. What
did he *predict?* First, his analysis of the moon’s motion was a
prediction because it connected the falling of objects on the earth’s
surface with that of the moon. Second, the question is, *is the
orbit an ellipse?* We shall see in a later chapter how it is possible to
calculate the motion exactly, and indeed one can prove that it should be
an ellipse,^{3} so no
extra fact is needed to explain Kepler’s *first* law. Thus
Newton made his first powerful
prediction.

The law of gravitation explains many phenomena not previously
understood. For example, the pull of the moon on the earth causes the
tides, hitherto mysterious. The moon pulls the water up under it and
makes the tides—people had thought of that before, but they were not
as clever as Newton, and so they
thought there ought to be only one tide during the day. The reasoning
was that the moon pulls the water up under it, making a high tide and a
low tide, and since the earth spins underneath, that makes the tide at
one station go up and down every $24$ hours. Actually the tide goes up
and down in $12$ hours. Another school of thought claimed that the high
tide should be on the other side of the earth because, so they argued,
the moon pulls the earth away from the water! Both of these theories are
wrong. It actually works like this: the pull of the moon for the earth
and for the water is “balanced” at the center. But the water which is
closer to the moon is pulled *more* than the average and the water
which is farther away from it is pulled *less* than the average.
Furthermore, the water can flow while the more rigid earth cannot. The
true picture is a combination of these two things.

What do we mean by “balanced”? What balances? If the moon pulls the
whole earth toward it, why doesn’t the earth fall right “up” to the
moon? Because the earth does the same trick as the moon, it goes in a
circle around a point which is inside the earth but not at its center.
The moon does not just go around the earth, the earth and the moon both
go around a central position, each falling toward this common position,
as shown in Fig. 7–5. This motion around the common center
is what balances the fall of each. So the earth is not going in a
straight line either; it travels in a circle. The water on the far side
is “unbalanced” because the moon’s attraction there is weaker than it
is at the center of the earth, where it just balances the “centrifugal
force.” The
result of this imbalance is that the water rises up, away from the
center of the earth. On the near side, the attraction from the moon is
stronger, and the imbalance is in the opposite direction in space, but
again *away* from the center of the earth. The net result is that
we get *two* tidal bulges.

### 7–5Universal gravitation

What else can we understand when we understand gravity? Everyone knows
the earth is round. Why is the earth round? That is easy; it is due to
gravitation. The earth can be understood to be round merely because
everything attracts everything else and so it has attracted itself
together as far as it can! If we go even further, the earth is not
*exactly* a sphere because it is rotating, and this brings in
centrifugal effects which tend to oppose gravity near the equator. It turns out that
the earth should be elliptical, and we even get the right shape for the
ellipse. We can thus deduce that the sun, the moon, and the earth should
be (nearly) spheres, just from the law of gravitation.

What else can you do with the law of gravitation? If we look at the
moons of Jupiter we can understand everything about the way they move
around that planet. Incidentally, there was once a certain difficulty
with the moons of Jupiter that is worth remarking on. These satellites
were studied very carefully by Rømer, who noticed that the moons sometimes seemed to be ahead
of schedule, and sometimes behind. (One can find their schedules by
waiting a very long time and finding out how long it takes on the
average for the moons to go around.) Now they were *ahead* when
Jupiter was particularly *close* to the earth and they were
*behind* when Jupiter was *farther* from the earth. This would
have been a very difficult thing to explain according to the law of
gravitation—it would have been, in fact, the death of this wonderful
theory if there were no other explanation. If a law does not work even
in *one place* where it ought to, it is just wrong. But the reason
for this discrepancy was very simple and beautiful: it takes a little
while to *see* the moons of Jupiter because of the time it takes
light to travel from Jupiter to the earth. When Jupiter is closer to the
earth the time is a little less, and when it is farther from the earth,
the time is more. This is why moons appear to be, on the average, a
little ahead or a little behind, depending on whether they are closer to
or farther from the earth. This phenomenon showed that light does not
travel instantaneously, and furnished the first estimate of the speed of
light. This was done in 1676.

If all of the planets pull on each other, the force which
controls, let us say, Jupiter in going around the sun is not just the
force from the sun; there is also a pull from, say, Saturn. This force
is not really strong, since the sun is much more massive than Saturn,
but there is *some* pull, so the orbit of Jupiter should not be a
perfect ellipse, and it is not; it is slightly off, and “wobbles”
around the correct elliptical orbit. Such a motion is a little more
complicated. Attempts were made to analyze the motions of Jupiter,
Saturn, and Uranus on the basis of the law of gravitation. The effects
of each of these planets on each other were calculated to see whether or
not the tiny deviations and irregularities in these motions could be
completely understood from this one law. Lo and behold, for Jupiter and
Saturn, all was well, but Uranus was “weird.” It behaved in a very
peculiar manner. It was not travelling in an exact ellipse, but that was
understandable, because of the attractions of Jupiter and Saturn. But
even if allowance were made for these attractions, Uranus *still*
was not going right, so the laws of gravitation were in danger of being
overturned, a possibility that could not be ruled out. Two men,
Adams and Le Verrier, in England and
France, independently, arrived at another possibility: perhaps there is
*another* planet, dark and invisible, which men had not seen. This
planet, $N$, could pull on Uranus. They calculated where such a planet
would have to be in order to cause the observed perturbations. They sent
messages to the respective observatories, saying, “Gentlemen, point
your telescope to such and such a place, and you will see a new
planet.” It often depends on with whom you are working as to whether
they pay any attention to you or not. They did pay attention to
Le Verrier; they looked, and
there planet $N$ was! The other observatory then also looked very
quickly in the next few days and saw it too.

This discovery shows that Newton’s laws are absolutely right in the solar system; but do they extend
beyond the relatively small distances of the nearest planets? The first
test lies in the question, do *stars* attract *each other* as
well as planets? We have definite evidence that they do in the
*double stars*. Figure 7–6 shows a
double star—two stars very close together (there is also a third star
in the picture so that we will know that the photograph was not turned).
The stars are also shown as they appeared several years later. We see
that, relative to the “fixed” star, the axis of the pair has rotated,
i.e., the two stars are going around each other. Do they rotate
according to Newton’s laws? Careful
measurements of the relative positions of one such double star system
are shown in Fig. 7–7. There we see a beautiful ellipse, the
measures starting in 1862 and going all the way around to 1904 (by now
it must have gone around once more). Everything coincides with
Newton’s laws, except that the star
Sirius A is *not at the focus*. Why should that be? Because the
plane of the ellipse is not in the “plane of the sky.” We are not
looking at right angles to the orbit plane, and when an ellipse is
viewed at a tilt, it remains an ellipse but the focus is no longer at
the same place. Thus we can analyze double stars, moving about each
other, according to the requirements of the gravitational law.

That the law of gravitation is true at even bigger distances is
indicated in Fig. 7–8. If one cannot see gravitation acting
here, he has no soul. This figure shows one of the most beautiful things
in the sky—a globular star cluster. All of the dots are stars.
Although they look as if they are packed solid toward the center, that
is due to the fallibility of our instruments. Actually, the distances
between even the centermost stars are very great and they very rarely
collide. There are more stars in the interior than farther out, and as
we move outward there are fewer and fewer. It is obvious that there is
an attraction among these stars. It is clear that gravitation exists at
these enormous dimensions, perhaps $100{,}000$ times the size of the
solar system. Let us now go further, and look at an *entire
galaxy*, shown in Fig. 7–9. The shape of this galaxy
indicates an obvious tendency for its matter to agglomerate. Of course
we cannot prove that the law here is precisely inverse square, only that
there is still an attraction, at this enormous dimension, that holds the
whole thing together. One may say, “Well, that is all very clever but
why is it not just a ball?” Because it is *spinning* and has
*angular momentum* which it cannot give up
as it contracts; it must contract mostly in a plane. (Incidentally, if
you are looking for a good problem, the exact details of how the arms
are formed and what determines the shapes of these galaxies has not been
worked out.) It is, however, clear that the shape of the galaxy is due
to gravitation even though the complexities of its structure have not
yet allowed us to analyze it completely. In a galaxy we have a scale of
perhaps $50{,}000$ to $100{,}000$ light years. The earth’s distance from
the sun is $8\tfrac{1}{3}$ light *minutes*, so you can see how
large these dimensions are.

Gravity appears to exist at even bigger dimensions, as indicated by
Fig. 7–10, which shows many “little” things clustered
together. This is a *cluster of galaxies*, just like a star
cluster. Thus galaxies attract each other at such distances that they
too are agglomerated into clusters. Perhaps gravitation exists even over
distances of *tens of millions* of light years; so far as we now
know, gravity seems to go out forever inversely as the square of the
distance.

Not only can we understand the nebulae, but from the law of gravitation
we can even get some ideas about the origin of the stars. If we have a
big cloud of dust and gas, as indicated in Fig. 7–11, the
gravitational attractions of the pieces of dust for one another might
make them form little lumps. Barely visible in the figure are “little”
black spots which may be the beginning of the accumulations of dust and
gases which, due to their gravitation, begin to form stars. Whether we
have ever seen a star form or not is still debatable.
Figure 7–12 shows the one piece of evidence which suggests
that we have. At the left is a picture of a region of gas with some
stars in it taken in 1947, and at the right is another picture, taken
only $7$ years later, which shows two new bright spots. Has gas
accumulated, has gravity acted hard enough and collected it into a ball
big enough that the stellar nuclear reaction starts in the interior and
turns it into a star? Perhaps, and perhaps not. It is unreasonable that
in only seven years we should be so lucky as to see a star change itself
into visible form; it is much less probable that we should see
*two!*

### 7–6Cavendish’s experiment

Gravitation, therefore, extends over enormous distances. But if there is
a force between *any* pair of objects, we ought to be able to
measure the force between our own objects. Instead of having to watch
the stars go around each other, why can we not take a ball of lead and a
marble and watch the marble go toward the ball of lead? The difficulty
of this experiment when done in such a simple manner is the very
weakness or delicacy of the force. It must be done with extreme care,
which means covering the apparatus to keep the air out, making sure it
is not electrically charged, and so on; then the force can be measured.
It was first measured by Cavendish with an apparatus which is schematically indicated in
Fig. 7–13. This first demonstrated the direct force between
two large, fixed balls of lead and two smaller balls of lead on the ends
of an arm supported by a very fine fiber, called a torsion fiber. By
measuring how much the fiber gets twisted, one can measure the strength
of the force, verify that it is inversely proportional to the square of
the distance, and determine how strong it is. Thus, one may accurately
determine the coefficient $G$ in the formula
\begin{equation*}
F=G\,\frac{mm'}{r^2}.
\end{equation*}
All the masses and distances are known. You say, “We knew it already
for the earth.” Yes, but we did not know the *mass* of the earth.
By knowing $G$ from this experiment and by knowing how strongly the
earth attracts, we can indirectly learn how great is the mass of the
earth! This experiment has been called “weighing the earth” by some
people, and it can be used to determine the coefficient $G$ of the
gravity law. This is the only way in which the mass of the earth can be
determined. $G$ turns out to be
\begin{equation*}
6.670\times10^{-11}\text{ newton}\cdot\text{m}^2/\text{kg}^2.
\end{equation*}

It is hard to exaggerate the importance of the effect on the history of
science produced by this great success of the theory of gravitation.
Compare the confusion, the lack of confidence, the incomplete knowledge
that prevailed in the earlier ages, when there were endless debates and
paradoxes, with the clarity and simplicity of this law—this fact that
all the moons and planets and stars have such a *simple rule* to
govern them, and further that man could *understand* it and deduce
how the planets should move! This is the reason for the success of the
sciences in following years, for it gave hope that the other phenomena
of the world might also have such beautifully simple laws.

### 7–7What is gravity?

But is this such a simple law? What about the machinery of it? All we
have done is to describe *how* the earth moves around the sun, but
we have not said *what makes it go*. Newton made no hypotheses about this; he was satisfied to find
*what* it did without getting into the machinery of it. *No
one has since given any machinery*. It is characteristic of the physical
laws that they have this abstract character. The law of conservation of
energy is a theorem concerning quantities that have to be calculated and
added together, with no mention of the machinery, and likewise the great
laws of mechanics are quantitative mathematical laws for which no
machinery is available. Why can we use mathematics to describe nature
without a mechanism behind it? No one knows. We have to keep going
because we find out more that way.

Many mechanisms for gravitation have been suggested. It is interesting
to consider one of these, which many people have thought of from time to
time. At first, one is quite excited and happy when he “discovers” it,
but he soon finds that it is not correct. It was first discovered about
1750. Suppose there were many particles moving in space at a very high
speed in all directions and being only slightly absorbed in going
through matter. When they *are* absorbed, they give an impulse to
the earth. However, since there are as many going one way as another,
the impulses all balance. But when the sun is nearby, the particles
coming toward the earth through the sun are partially absorbed, so fewer
of them are coming from the sun than are coming from the other side.
Therefore, the earth feels a net impulse toward the sun and it does not
take one long to see that it is inversely as the square of the
distance—because of the variation of the solid angle that the sun
subtends as we vary the distance. What is wrong with that machinery? It
involves some new consequences which are *not true*. This
particular idea has the following trouble: the earth, in moving around
the sun, would impinge on more particles which are coming from its
forward side than from its hind side (when you run in the rain, the rain
in your face is stronger than that on the back of your head!). Therefore
there would be more impulse given the earth from the front, and the
earth would feel a *resistance to motion* and would be slowing up
in its orbit. One can calculate how long it would take for the earth to
stop as a result of this resistance, and it would not take long enough
for the earth to still be in its orbit, so this mechanism does not work.
No machinery has ever been invented that “explains” gravity without
also predicting some other phenomenon that does *not* exist.

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. 7–14. The gravitational attraction
relative to the electrical repulsion between two electrons is $1$
divided by $4.17\times10^{42}$! 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*, $2\times10^{10}$ 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.

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 space ship; 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 space ship, 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.

### 7–8Gravity 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.