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Unformatted text preview: The Theory of Gravitation 7—1 Planetary 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 state
ment can be expressed mathematically by the equation mm’
r2 F=G 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 dis
cuss 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 mys
teries 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 redis
covered later by Copernicus. Exactly how the planets went around the sun,
with exactly what motion, took a little more work to discover. In the beginning of
the ﬁfteenth 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 pro
posed 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—2 Kepler’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 71 7—1 Planetary motions 7—2 Kepler’s laws 7—3 Development of dynamics
74 Newton’s law of gravitation
7—5 Universal gravitation 7—6 Cavendish’s experiment
7—7 What is gravity? 7—8 Gravity and relativity Fig. 7—l. An ellipse. Fig. 7—2. Kepler’s law of areas. 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* 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: I. Each planet moves around the sun in an ellipse, with the sun at one focus. II. The radius vector from the sun to the planet sweeps out equal areas in
equal intervals of time. 111. The squares of the periods of any two planets are proportional to the
cubes of the semimajor axes of their respective orbits: T ~ (13/ 2. 7—3 Development 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 dis
covered a very remarkable fact about motion, which was essential for under
standing 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 direc
tion, 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.) * A radius vector is a line drawn from the sun to any point in a planet’s orbit.
7—2 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 oil“ 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—4 Newton’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 propor
tional 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.* 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 * That is, how far the circle of the moon’s orbit falls below the straight line tangent
to it at the point where the moon was one second before. 73 '\ fLHETROMAGNET
, ,
/ /
/ l
I
I /
"COLLISION'
h l hh
"2 I 2 Fig. 7—3. Apparatus for showing the
independence of vertical and horizontal
motions. From
Plane Geomtry
X _ ZFS  2_R_ a x x 'R‘  Mina of
earth l£00) miles 'x';d1!tlnee
"invalled hori
zontally" in one
second '5' dunno:
"fallen" in one
second (16 fee‘) Fig. 7—4. Acceleration toward the
center of a circular path. From plone
geometry, x/s = (2R — Sl/x a: 2R/x,
where R is the radius of the earth, 4000
miles; x is the distance ”travelled hori
zontally" in one second; and S is the
distance "fallen" in one second (16 feet). 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
movmg 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 4000mile radius,
and the tangential, straightline 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 8000mile diameter of the earth.
The square root of (16/5280) X 8000 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 approxi—
mately 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,* 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 * The proof is not given in this course. 7—4 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 [me either; 1t 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 itJust 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 attrac
tion from the moon is stronger, and the imbalance is in the opposite d1rection1n space, but again away from the center of the earth. The net result 15 that we get
two tidal bulges. 7—5 Universal 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 etTects 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 Roemer,
who noticed that the moons sometimes seemed to be ahead of schedule, and some
times 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 dis
crepancy 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 1656. If all of the planets push and 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. 1.0 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 under
standable, 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 Leverrier, in England and France, independently, 7—5 H20 Fig. 7—5.
with tides. The POINT AROUND WHICH
EARTH 8 MOON ROTATE EARTH earthmoon system, Fig. 7—6. A doublestar system. 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 perturba
tions. 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 atten
tion to you or not. They did pay attention to Leverrier; 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. 10" SCALE Fig. 7—7. Orbit of Sirius B with respect to Sirius A.
7—6 Fig. 7—8. A globular star cluster. 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 Fig. 7—9. A galaxy. 7—7 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% 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! Fig. 7—10. A cluster of galaxies. Fig. 7—] l. An interstellar dust cloud. Fig. 7—12. The formation of new stars?
7—8 7—6 Cavendish’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 mm’ F=Gr2 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.” Cavendish claimed he was weighing the earth, but what he was meas
uring was 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 6.670 X 10’11 newton  mZ/kgz. 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 confu
sion, the lack of conﬁdence, the incomplete knowledge that prevailed in the earlier
ages, when there were endless debates and paradoxes, with the clarity and simplic
ity of this 1aw——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—7 What 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 con
sider 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 7—9 91 19 Fig. 7—13. A simpliﬁed diagram of
the apparatus used by Cavendish to
verify the law of universal gravitation for
small obiects and to measure the gravita
tional constant G. Gr: wratuy A tt nthn = / 4/ , 42
flcztmu/ Reﬁll/[Ann / 7 /0 : //4§ / 70, 000, am; 009 00
O \ Fig. 7—14. The relative strengths of
electrical and gravitational interactions
between two electrons. 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 im
pulse 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 in
volves 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
headl). 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 electricityis 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 remark
able that the two laws involve the same function of distance? Perhaps grav1tation
and electricity are much more closely related than we think. Many attempts have
been made to unify them; the socalled 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 X 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 pos
sibilities 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 X 1010 years, the answer is 10‘“. 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 onetenth of the age of the universe.
In this time, the gravity constant would have increased by about 10 percent. It 7—10 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 argu
ments 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,
w111 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 wrll 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
Eotvos in 1909 and more recently by Dicke. For all substances tried, the masses
and weights are exactly proportlonal within 1 part in 1,000,000,000, or less. This
IS a remarkable experiment. 7—8 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 gravi
tation is not correct! It was modiﬁed by Einstein to take into account the theory
of relativity. According to Newton, the gravrtational 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 gravi
tation. One feature of this new law which 18 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 de
ﬂected. 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 quantummechanical 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. 7—11 ...
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 Spring '09
 LeeKinohara
 Physics

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