Fear of Physics (11 page)

Hawking showed that similar phenomena can occur near the black hole. Particles can tunnel through the gravitational barrier at the surface of the black hole and escape. This demonstration was a tour de force because it was the first time the laws of quantum mechanics had been utilized in connection with general relativity to reveal a new phenomenon. Again, however, it was possible only because, like the hydrogen atom, the quantum-mechanical states of particles around a black hole are “separable”—that is, the three-dimensional calculation can be effectively turned into a one-dimensional problem and an independent two-dimensional problem. If it weren’t for this simplification, we might still be in the dark about black holes.

Interesting as these technical tricks might be, they form only the tip of the iceberg. The real reason we keep repeating ourselves as we discover new laws is not so much due to our character, or lack thereof, as it is due to the character of nature. She keeps repeating herself. It is for this reason that we almost universally check to see whether new physics is really a reinvention of old physics. Newton, in discovering his Universal Law of Gravity, benefited tremendously from the observations and analyses of Galileo, as I have described. He also benefited from another set of careful observations by the Danish astronomer Tycho Brahe, as analyzed by his student Johannes Kepler—a contemporary of Galileo.

Both Brahe and Kepler were remarkable characters. Brahe, from a privileged background, became the most eminent astronomer in Europe after his observations of the supernova of 1572. He was given an entire island by the Danish monarch King Frederick II to use as an observatory site, only to be forced to move some years later by Frederick’s successor. Unhindered by, or perhaps because of, his arrogance (and a false nose made of
metal), Brahe managed to improve in one decade the precision in astronomical measurement by a factor of 10 over that which it had maintained for the previous thousand years—and all of this without a telescope! In Prague, where he had gone in exile from Denmark, Brahe hired Kepler a year before his own death to perform the intricate calculational analysis required to turn his detailed observations of planetary motions into a consistent cosmology.

Kepler came from another world. A child of a family of modest means, his life was always at the edge, both financially and emotionally. Besides his scientific pursuits, Kepler found time to defend his mother successfully from prosecution as a witch and to write what was probably the first science fiction novel, about a trip to the moon. In spite of these diversions, Kepler approached the task of analyzing the data in Brahe’s notebooks, which he inherited upon Brahe’s death, with an uncommon zeal. Without so much as a Macintosh, much less a supercomputer, he performed a miracle of complicated data analysis that would occupy the better part of his career. From the endless tables of planetary positions, he arrived at the three celebrated laws of planetary motion that still bear his name, and that provided the key clues Newton would use to unravel the mystery of gravity.

I mentioned one of Kepler’s Laws earlier—namely, that the orbits of the planets sweep out equal areas in equal times—and how Newton was able to use this to infer that there was a force pulling the planets toward the sun. We are so comfortable with this idea nowadays that it is worth pointing out how counterintuitive it really is. For centuries before Newton, it was assumed that the force needed to keep the planets moving around the sun must emanate from something
pushing
them around. Newton quite simply relied on Galileo’s Law of Uniform Motion to see that this
was unnecessary. Indeed, he argued that Galileo’s result that the motion of objects thrown in the air would trace out a parabola, and that their horizontal velocity would remain constant, would imply that an object thrown sufficiently fast could orbit the Earth. Due to the curvature of the Earth an object could continue to “fall” toward the earth, but if it were moving fast enough initially, its constant horizontal motion could carry it far enough that in “falling” it would continue to remain a constant distance from the Earth’s surface. This is demonstrated in the following diagram, copied from Newton’s
Principia:

Having recognized that a force that clearly pulled downward at the Earth could result in a body continually falling toward it for eternity—what we call an orbit—it did not require too large a leap of imagination to suppose that objects orbiting the sun, such as the planets, were being continually pulled toward the sun, and not pushed around it. (Incidentally, it is the fact that objects in orbit are continually “falling” that is responsible for the weightlessness experienced by astronauts. It has nothing to do with an absence of
gravity, which is almost as strong out at the distances normally traversed by satellites and shuttles as it is here on Earth.)

In any case, another of Kepler’s laws of planetary motion provided the icing on the cake. This law yielded a quantitative key that unlocked the nature of the gravitational attraction between objects, for it gave a mathematical relation between the length of each planet’s year—the time it takes for it to go around the sun and its distance from the sun. From this law, one could easily derive that the velocity of the planets around the sun falls in a fixed way with their distance from the sun. Specifically, Kepler’s laws showed that their velocity falls inversely with the square root of their distance from the sun.

Armed with this knowledge, and his own generalization from the results of Galileo that the acceleration of moving bodies must be proportional to the force exerted on them, Newton was able to show that if planets were attracted toward the sun with a force proportional to the product of their mass and the sun’s mass, divided by the square of the distance between them, Kepler’s velocity law would naturally result. Moreover, he was able to show that the constant of proportionality would be precisely equal to the mass of the sun times the strength of the gravitational force. If the strength of the gravitational force between all objects is universal, this could be represented by a constant, which we now label
G.

Even though it was beyond the measuring abilities of his time to determine the constant
G
directly, Newton did not need this to prove that his law was correct. Reasoning that the same force that held the planets around the sun must hold the moon in orbit around the Earth, he compared the predicted motion of the moon around the Earth—based on extrapolating the measured downward acceleration of bodies at the earth’s surface with the actual measured motion: namely, that it takes about 28 days for
the moon to orbit the Earth. The predictions and the observations agreed perfectly. Finally, the fact that the moons of Jupiter, which Galileo had first discovered with his telescope, also obeyed Kepler’s law of orbital motion, this time vis-à-vis their orbit around Jupiter, made the universality of Newton’s Law difficult to question.

Now, I mention this story not just to reiterate how the mere observation of
how
things move—in this case, the planets—led to an understanding of
why
they move. Rather, it is to show you how we have been able to exploit these results even in modern research. I begin with a wonderful precedent created by the British scientist Henry Cavendish, about 150 years after Newton discovered the Law of Gravity.

When I graduated and became a postdoctoral fellow at Harvard University, I quickly learned a valuable lesson there: Before writing a scientific paper, it is essential to come up with a catchy title. I thought at the time that this was a recent discovery in science, but I have since learned that it has a distinguished tradition, going back at least as far as Cavendish in 1798.

Cavendish is remembered for performing the first experiment that measured directly in the laboratory the gravitational attraction between two known masses, thus allowing him to measure,
for the first time,
the strength of gravity and determine the value of
G.
In reporting his results before the Royal Society he didn’t entitle his paper “On Measuring the Strength of Gravity” or “A Determination of Newton’s Constant
G.”
No, he called it “Weighing the Earth.”

There was a good reason for this sexy title. By this time Newton’s Law of Gravity was universally accepted, and so was the premise that this force of gravity was responsible for the observed motion of the moon around the Earth. By measuring the distance
to the moon (which was easily done, even in the seventeenth century, by observing the change in the angle of the moon with respect to the horizon when observed at the same time from two different locations—the same technique surveyors use when measuring distances on Earth), and knowing the period of the moon’s orbit—about 28 days—one could easily calculate the moon’s velocity around the Earth. Let me reiterate that Newton’s great success involved not just his explanation of Kepler’s Law that the velocity of objects orbiting the sun was inversely proportional to the square root of their distance from the sun. He also showed that this same law could apply to the motion of the moon and to objects falling at the Earth’s surface. His Law of Gravity implied that the constant of proportionality was equal to the product of
G
times the mass of the sun in the former case, and
G
times the mass of the Earth in the latter. (He never actually proved that the value of
G
in the two cases is, in fact, the same. This was a guess, based on the assumption of simplicity and on the experimental observation that the value of
G
seemed to be the same for objects falling at the Earth’s surface as for the moon, and that the value of
G
that applied to the planets orbiting the sun appeared uniform for all these planets. Thus, a simple extrapolation suggested that a single value of
G
might suffice for everything.)

In any case, by knowing the distance of the moon from the Earth, and the velocity of the moon around the Earth, you could plug in Newton’s Law and determine the
product
of
G
times the mass of the Earth. Until you knew independently the value of
G,
however, you could not extract from this the mass of the Earth. Thus, Cavendish, who was the first person to determine the value of
G,
150 years after Newton had proposed it, was also the first to be able to determine the mass of the Earth. (The latter sounds a lot more exciting, and so his title.)

We have benefited not only from Cavendish’s astute recognition of the value of good press but also from the technique he pioneered for weighing the Earth by pushing Newton’s Law as far as he could. It remains in use today. Our best measurement of the mass of the sun comes from exactly the same procedure, using the known distances and orbital velocities of each of the planets. In fact, this procedure is so good that we could, in principle, measure the mass of the sun to one part in a million, based on the existing planetary data. Unfortunately, Newton’s constant
G
is the poorest measured fundamental constant in nature. We know it to an accuracy of only 1 part in 100,000 or so. Thus, our knowledge of the sun’s mass is limited to this accuracy.

Incidentally, not only does our uncertainty of the precise strength of gravity limit our ability to measure the sun’s mass, it also limits our ability to probe the smallest and largest scales in the universe. For, as we shall see, it turns out that on small scales, less than about one millionth of an inch, we have no direct probe that gravity behaves as Newton described it, and whether deviations from Newton’s law of gravity might exist on these scales that could shed light on the possible existence of extra dimensions, beyond the three dimensions of space we know and love.

Nevertheless when you have a good thing going, don’t stop. Our sun (and thus our solar system) orbits around the outer edge of the Milky Way galaxy, and so we can use the sun’s known distance from the center of the galaxy (about 25,000 light years) and its known orbital velocity (about 150 miles/second), to “weigh” the galaxy. When we do this, we find that the mass of material enclosed by our orbit corresponds to roughly a hundred billion solar masses. This is heartening, since the total light emitted by our galaxy is roughly equivalent to that emitted by about a hundred
billion stars more or less like our Sun. (Both of these observations provide the rationale for my previous statement that there are about a hundred billion stars in our galaxy.)

A remarkable thing happens when we try to extend this measurement by observing the velocity of objects located farther and farther from the center of our galaxy. Instead of falling off, as it should if all the mass of our galaxy is concentrated in the region where the observed stars are, this velocity remains constant. This suggests that, instead, there is ever more mass located outside the region where the stars shine. In fact, current estimates suggest that there is at least ten times more stuff out there than meets the eye! Moreover, similar observations of the motions of stars in other galaxies all suggest the same thing. Extending Newton’s Law further to use the observed motion of galaxies themselves amid groups and clusters of galaxies confirms this notion. When we use Newton’s Law to weigh the universe, we find that at least 90 percent of the matter in it is “dark.”