Is God a Mathematician? (29 page)

Since strings are closed loops moving through space, as time progresses, they sweep areas (known as
world sheets
) in the form of cylinders (as in figure 60). If a string emits other strings, this cylinder
forks to form wishbone-shaped structures. When many strings interact, they form an intricate network of fused donutlike shells. While studying these types of complex topological structures, string theorists Hirosi Ooguri and Cumrun Vafa discovered a surprising connection between the number of donut shells, the intrinsic geometric properties of knots, and the Jones polynomial. Even earlier, Ed Witten—one of the key players in string theory—created an unexpected relation between the Jones polynomial and the very foundation of string theory (known as
quantum field theory
). Witten’s model was later rethought from a purely mathematical perspective by the mathematician Michael Atiyah. So string theory and knot theory live in perfect symbiosis. On one hand, string theory has benefited from results in knot theory; on the other, string theory has actually led to new insights in knot theory.

Figure 60

With a much broader scope, string theory searches for explanations for the most basic constituents of matter, much in the same way that Thomson originally searched for a theory of atoms. Thomson (mistakenly) thought that knots could provide the answer. By a surprising twist, string theorists find that knots can indeed provide at least some answers.

The story of knot theory demonstrates beautifully the unexpected powers of mathematics. As I have mentioned earlier, even the “active” side of the effectiveness of mathematics alone—when scientists generate the mathematics they need to describe observable science—presents some baffling surprises when it comes to accuracy. Let me
describe briefly one topic in physics in which both the active and the passive aspects played a role, but which is particularly remarkable because of the obtained accuracy.

A Weighty Accuracy

Newton took the laws of falling bodies discovered by Galileo and other Italian experimentalists, combined them with the laws of planetary motion determined by Kepler, and used this unified scheme to put forth a universal, mathematical law of gravitation. Along the way, Newton had to formulate an entirely new branch of mathematics—calculus—that allowed him to capture concisely and coherently all the properties of his proposed laws of motion and gravitation. The accuracy to which Newton himself could verify his law of gravity, given the experimental and observational results of his day, was no better than about 4 percent. Yet the law proved to be accurate beyond all reasonable expectations. By the 1950s the experimental accuracy was better than one ten-thousandth of a percent. But this is not all. A few recent, speculative theories, aimed at explaining the fact that the expansion of our universe seems to be speeding up, suggested that gravity may change its behavior on very small distance scales. Recall that Newton’s law states that the gravitational attraction decreases as the inverse square of the distance. That is, if you double the distance between two masses, the gravitational force each mass feels becomes four times weaker. The new scenarios predicted deviations from this behavior at distances smaller than one millimeter (the twenty-fifth part of an inch). Eric Adelberger, Daniel Kapner, and their collaborators at the University of Washington, Seattle, conducted a series of ingenious experiments to test this predicted change in the dependence on the separation. Their most recent results, published in January 2007, show that the inverse-square law holds down to a distance of fifty-six thousandths of a millimeter! So a mathematical law that was proposed more than three hundred years ago on the basis of very scanty observations not only turned out to be phenomenally accurate, but also proved to hold in a range that couldn’t even be probed until very recently.

There was one major question that Newton left completely unanswered: How does gravity really work? How does the Earth, a quarter million miles away from the Moon, affect the Moon’s motion? Newton was aware of this deficiency in his theory, and he openly admitted it in the
Principia:

The person who decided to meet the challenge posed by Newton’s omission was Albert Einstein (1879–1955). In 1907 in particular, Einstein had a very strong reason to be interested in gravity—his new theory of
special relativity
appeared to be in direct conflict with Newton’s law of gravitation.

Newton believed that gravity’s action was instantaneous. He assumed that it took no time at all for planets to feel the Sun’s gravitational force, or for an apple to feel the Earth’s attraction. On the other hand, the central pillar of Einstein’s special relativity was the statement that no object, energy, or information could travel faster than the speed of light. So how could gravity work instantaneously? As the following example will show, the consequences of this contradiction could be disastrous to concepts as fundamental as our perception of cause and effect.

Imagine that the Sun were to somehow suddenly disappear. Robbed of the force holding it to its orbit, the Earth would (according to Newton) immediately start moving along a straight line (apart from small deviations due to the gravity of the other planets). However, the Sun would actually disappear from view to the Earth’s inhabitants only about eight minutes later, since this is the time it takes light to traverse the distance from the Sun to the Earth. In other
words, the change in the Earth’s motion would precede the Sun’s disappearance.

To remove this conflict, and at the same time to tackle Newton’s unanswered question, Einstein engaged almost obsessively in a search for a new theory of gravity. This was a formidable task. Any new theory had not only to preserve all the remarkable successes of Newton’s theory, but also to explain how gravity works, and to do so in a way that is compatible with special relativity. After a number of false starts and long wanderings down blind alleys, Einstein finally reached his goal in 1915. His
theory of general relativity
is still regarded by many as one of the most beautiful theories ever formulated.

At the heart of Einstein’s groundbreaking insight lay the idea that gravity is nothing but warps in the fabric of space and time. According to Einstein, just as golf balls are guided by the warps and curves across an undulating green, planets follow curved paths in the warped space representing the Sun’s gravity. In other words, in the absence of matter or other forms of energy,
spacetime
(the unified fabric of the three dimensions of space and one of time) would be flat. Matter and energy warp spacetime just as a heavy bowling ball causes a trampoline to sag. Planets follow the most direct paths in this curved geometry, which is a manifestation of gravity. By solving the “how it works” problem for gravity, Einstein also provided the framework for addressing the question of how fast it propagates. The latter question boiled down to determining how fast warps in spacetime could travel. This was a bit like calculating the speed of ripples in a pond. Einstein was able to show that in general relativity gravity traveled precisely at the speed of light, which eliminated the discrepancy that existed between Newton’s theory and special relativity. If the Sun were to disappear, the change in the Earth’s orbit would occur eight minutes later, coinciding with our observing the disappearance.

The fact that Einstein had turned warped four-dimensional spacetime into the cornerstone of his new theory of the cosmos meant that he badly needed a mathematical theory of such geometrical entities. In desperation, he turned to his old classmate the mathematician Marcel Grossmann (1878–1936): “I have become imbued with great respect for mathematics, the more subtle parts of which I had
previously regarded as sheer luxury.” Grossmann pointed out that Riemann’s non-Euclidean geometry (described in chapter 6) was precisely the tool that Einstein needed—a geometry of curved spaces in any number of dimensions. This was an incredible demonstration of what I dubbed the “passive” effectiveness of mathematics, which Einstein was quick to acknowledge: “We may in fact regard [geometry] as the most ancient branch of physics,” he declared. “Without it I would have been unable to formulate the theory of relativity.”

General relativity has also been tested with impressive accuracy. These tests are not easy to come by, since the curvature in spacetime introduced by objects such as the Sun is measured only in parts per million. While the original tests were all associated with observations within the solar system (e.g., tiny changes to the orbit of the planet Mercury, as compared to the predictions of Newtonian gravity), more exotic tests have recently become feasible. One of the best verifications uses an astronomical object known as a
double pulsar.

A pulsar is an extraordinarily compact, radio-wave-emitting star, with a mass somewhat larger than the mass of the Sun but a radius of only about six miles. The density of such a star (known as a
neutron star
) is so high that one cubic inch of its matter has a mass of about a billion tons. Many of these neutron stars spin very fast, while emitting radio waves from their magnetic poles. When the magnetic axis is somewhat inclined to the rotation axis (as in figure 61), the radio beam from a given pole may cross our line of sight only once every rotation, like the flash of light from a lighthouse. In this case, the radio emission will appear to be pulsed—hence the name “pulsar.” In one case, two pulsars revolve around their mutual center of gravity in a close orbit, creating a double-pulsar system.

There are two properties that make this double pulsar an excellent laboratory for testing general relativity: (1) Radio pulsars are superb clocks—their rotation rates are so stable that in fact they surpass atomic clocks in accuracy; and (2) Pulsars are so compact that their gravitational fields are very strong, producing significant relativistic effects. These features allow astronomers to measure very precisely changes in the light travel time from the pulsars to Earth caused by the orbital motion of the two pulsars in each other’s gravitational field.

Figure 61

The most recent test was the result of precision timing observations taken over a period of two and a half years on the double-pulsar system known as PSR J0737-3039A/B (the long “telephone number” reflects the coordinates of the system in the sky). The two pulsars in this system complete an orbital revolution in just two hours and twenty-seven minutes, and the system is about two thousand light-years away from Earth (a light-year is the distance light travels in one year in a vacuum; about six trillion miles). A team of astronomers led by Michael Kramer of the University of Manchester measured the relativistic corrections to the Newtonian motion. The results, published in October 2006, agreed with the values predicted by general relativity within an uncertainty of 0.05 percent!

Incidentally, both special relativity and general relativity play an important role in the
Global Positioning System (GPS)
that helps us find our location on the surface of the Earth and our way from place to place, whether in a car, airplane, or on foot. The GPS determines the
current position of the receiver by measuring the time it takes the signal from several satellites to reach it and by triangulating on the known positions of each satellite. Special relativity predicts that the atomic clocks on board the satellites should tick more slowly (falling behind by a few millionths of a second per day) than those on the ground because of their relative motion. At the same time, general relativity predicts that the satellite clocks should tick faster (by a few tens of millionths of a second per day) than those on the ground due to the fact that high above the Earth’s surface the curvature in spacetime resulting from the Earth’s mass is smaller. Without making the necessary corrections for these two effects, errors in global positions could accumulate at a rate of more than five miles in each day.