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D rocket from Wallops Island, Virginia, that carried an atomic clock accurate to about a trillionth of a second per hour. They hoped to show that as the rocket gained altitude (thereby decreasing the effect of the earth's gravitational pull), an identical earthbound atomic clock (still subject to the full force of the earth's gravity) would tick more slowly. Through a two-way stream of microwave signals, the researchers were able to compare the rate of ticking of the two atomic clocks and, indeed, at the rocket's maximum altitude of 6,000 miles, its atomic clock ran fast by about 4 parts per billion relative to its counterpart on earth, agreeing with theoretical predictions to better than a hundredth of a percent.

10. In the mid-1800s, the French scientist Urbain Jean Joseph Le Verrier discovered that the planet Mercury deviates slightly from the orbit around the sun that is predicted by Newton's law of gravity. For more than half a century, explanations for this so-called excess orbital perihelion precession (in plain language, at the end of each orbit, Mercury does not quite wind up where Newton's theory says it should) ran the gamut—the gravitational influence of an undiscovered planet or planetary ring, an undiscovered moon, the effect of interplanetary dust, the oblateness of the sun—but none was sufficiently compelling to win general acceptance. In 1915, Einstein calculated the perihelion precession of Mercury using his newfound equations of general relativity and found an answer that, by his own admission, gave him heart palpitations: The result from general relativity precisely matched observations. This success, certainly, was one significant reason that Einstein had such faith in his theory, but most everyone else awaited confirmation of a prediction, rather than an explanation of a previously known anomaly. For more details, see Abraham Pais, Subtle Is the Lord (New York: Oxford University Press, 1982), p. 253.

11. Robert P. Crease and Charles C. Mann, The Second Creation (New Brunswick, N.J.: Rutgers University Press, 1996), p. 39.

12. Surprisingly, recent research on the detailed rate of cosmic expansion suggests that the universe may in fact incorporate a very small but nonzero cosmological constant.

Chapter 4

1. Richard Feynman, The Character of Physical Law (Cambridge, Mass.: MIT Press, 1965), p. 129.

2. Although Planck's work did solve the infinite energy puzzle, apparently this goal was not what directly motivated his work. Rather, Planck was seeking to understand a closely related issue: the experimental results concerning how energy in a hot oven—a "black body" to be more precise—is distributed over various wavelength ranges. For more details on the history of these developments, the interested reader should consult Thomas S. Kuhn, Black-Body Theory and the Quantum Discontinuity, 1894–1912 (Oxford, Eng.: Clarendon, 1978).

3. A little more precisely, Planck showed that waves whose minimum energy content exceeds their purported average energy contribution (according to nineteenth-century thermodynamics) are exponentially suppressed. This suppression is increasingly sharp as we examine waves of ever larger frequency.

4. Planck's constant is 1.05 × 10-27 grams-centimeters2/second.

5. Timothy Ferris, Coming of Age in the Milky Way (New York: Anchor, 1989), p. 286.

6. Stephen Hawking, lecture at the Amsterdam Symposium on Gravity, Black Holes, and String Theory, June 21, 1997.

7. It is worthwhile to note that Feynman's approach to quantum mechanics can be used to derive the approach based on wave functions, and vice versa; the two approaches, therefore, are fully equivalent. Nevertheless, the concepts, the language, and the interpretation that each approach emphasizes are rather different, even though the answers each gives are absolutely identical.

8. Richard Feynman, QED: The Strange Theory of Light and Matter (Princeton: Princeton University Press, 1988).

Chapter 5

1. Stephen Hawking, A Brief History of Time (New York: Bantam Books, 1988), p. 175.

2. Richard Feynman, as quoted in Timothy Ferris,