The Importance of Being Constant

Mention the word “constant,” and your listeners may think of matrimonial fidelity or financial stability—or maybe they’ll declare that change is the only constant in life. As it happens, the universe has its own constants, in the form of unvarying quantities that endlessly reappear in nature and in mathematics, and whose exact numerical values are of signal importance to the pursuit of science. Some of these constants are physical, grounded in actual measurements. Others, though they illuminate the workings of the universe, are purely numerical, arising from within mathematics itself.

Some constants are local and limited, applicable in just one context, one object, or one subgroup. Others are fundamental and universal, relevant to space, time, matter, and energy everywhere—thereby granting investigators the power to understand and predict the past, present, and future of the universe. Scientists know of only a few fundamental constants. The top three on most people’s lists are the speed of light in a vacuum, Newton’s gravitational constant, and Planck’s constant, the foundation of quantum mechanics and the key to Heisenberg’s infamous uncertainty principle. Other universal constants include the charge and mass of each of the fundamental subatomic particles.

Whenever a repeating pattern of cause and effect shows up in the universe, there’s probably a constant at work. But to measure cause and effect, you must sift through what is and is not variable, and you must ensure that a simple correlation, however tempting it may be, is not mistaken for a cause. In the 1990s the stork population of Germany increased, and the German at-home birth rate rose as well. Shall we credit storks for airlifting the babies? I don’t think so.

But once you’re certain that the constant exists, and you’ve measured its value, you can make predictions about places and things and phenomena yet to be discovered or imagined.

Johannes Kepler, a German mathematician and occasional mystic, made the first-ever discovery of an unchanging physical quantity in the universe. In 1618, after a decade of engaging in mystical drivel, Kepler figured out that if you square the time it takes a planet to go around the Sun, then that quantity is always proportional to the cube of the planet’s average distance from the Sun. Turns out, this amazing relation holds not only for each planet in our solar system, but also for each star in orbit around the center of its galaxy, and for each galaxy in orbit around the center of its galactic cluster. As you might suspect, though, unbeknownst to Kepler, a constant was at work: Newton’s gravitational constant lurked within Kepler’s formulas, not to be revealed as such for another seventy years.

Probably the first constant you learned in school was pi—a mathematical constant denoted, since the early eighteenth century, by the Greek letter π. Pi is, quite simply, the ratio of the circumference of a circle to its diameter. In other words, pi is the multiplier if you want to go from a circle’s diameter to its circumference. Pi also pops up in plenty of other places, including the areas of circles and ellipses, the volumes of certain solids, the motions of pendulums, the vibrations of strings, and electrical circuits.

Not a whole number, pi instead has an unlimited succession of nonrepeating decimal digits; when truncated to include every Arabic numeral, pi looks like 3.141592653589793238­46264338327950. No matter when or where you live, no matter your nationality or age or aesthetic proclivities, no matter whether you vote Democrat or Republican, if you calculate the value of pi you will get the same answer as everybody else in the universe. Thus constants such as pi enjoy a level of internationality that politics does not, never did, and never will—which is why, if people ever do communicate with aliens, they’re likely to talk in mathematics, the lingua francaof the cosmos, and not English.

Because pi is an “irrational” number, you can’t represent it as a fraction made up of two whole numbers—⅔ or ¹⁸⁄₁₁, for instance. But the earliest mathematicians, who had no clue about the existence of irrational numbers, didn’t get much beyond representing it as ²⁵⁄₈ (the Babylonians, about 2000 BC) or ²⁵⁶⁄₈₁ (the Egyptians, about 1650 BC). Then, in about 250 BC, the Greek mathematician Archimedes—by engaging in a laborious geometric exercise—came up with not one fraction but two, ²²³⁄₇₁ and ²²⁄₇. Archimedes realized that the exact value of pi, a value he himself did not claim to have found, had to lie somewhere in between.

Given the progress of the day, a rather poor estimate of pi also appears in the Bible, in a passage describing the furnishings of King Solomon’s temple: a molten sea, ten cubits from the one brim to the other: it was round all about … and a line of thirty cubits did compass it round about (1 Kings 7:23). That is, the diameter was ten units, and the circumference thirty, which can only be true if pi were equal to 3. Three millennia later, in 1897, the lower house of the Indiana State Legislature passed a bill announcing that, henceforth in the Hoosier state, the ratio of the diameter and circumference is as five-fourths to four—in other words, exactly 3.2.

Decimal-challenged lawmakers notwithstanding, the greatest mathematicians—including Muhammad ibn Musa al-Khwarizmi, a ninth-century Iraqi whose name lives on in the word “algorithm,” and even Newton—steadily labored to increase the precision of pi. The advent of electronic computers, of course, blew the roof right off that exercise. As of the early twenty-first century, the number of known digits of pi has passed the one-trillion mark, surpassing any physical application except the study by pi-people of whether or not the sequence of numerals will ever look random.

Of far more importance than Newton’s contribution to the calculation of pi are his three universal laws of motion and his single universal law of gravitation. All four laws were first presented in his masterwork, Philosophiæ Naturalis Principia Mathematica, or the Principia, for short, published in 1687.

Before Newton published the Principia, scientists (concerned with what was then called mechanics, and later called physics) would simply describe what they saw, and hope that the next time around it would happen the same way. But armed with Newton’s laws of motion, they could describe the relations among force, mass, and acceleration under all conditions. Predictability had entered science. Predictability had entered life.

Unlike his first and third laws, Newton’s second law of motion is an equation: F = ma. Translated into English, that means a net force (F) applied to an object of a given mass (m) will result in the acceleration (a) of that object. In even plainer English, a big force yields a big acceleration. And they change in lockstep: double the force on an object, and you double its acceleration. The object’s mass serves as the equation’s constant, enabling you to calculate exactly how much acceleration you can expect from a given force.

But suppose an object’s mass is not constant? Launch a rocket, and its mass drops continuously until the fuel tanks run out of fuel. And now, just for grins, suppose the mass changes even though you neither add nor subtract material from the object. That’s what happens in Einstein’s special theory of relativity. In the Newtonian universe, every object has a mass that is always and forever its mass. In the Einsteinian, relativistic universe, by contrast, objects have an unchanging “rest mass” (the same as the “mass” in Newton’s equations), to which you add more mass according to the object’s speed. What’s going on is that as you accelerate an object in Einstein’s universe, its resistance to that acceleration increases, showing up in the equation as an increase in the object’s mass. Newton could not have known about these “relativistic” effects, because they become significant only at speeds comparable to the speed of light. To Einstein, they meant some other constant was at work: the speed of light, a subject worthy of its own essay at another time.

As is true for many physical laws, Newton’s laws of motion are plain and simple. His universal law of gravitation is somewhat more complicated. It declares that the strength of the gravitational attraction between two objects—whether between an airborne cannonball and Earth, or the Moon and Earth, or two atoms, or two galaxies—depends only on the two masses and the distance between them. More precisely, the force of gravity is directly proportional to the mass of one object times the mass of the other, and inversely proportional to the square of the distance between them. Those proportionalities give deep insight into how nature works: if the strength of the gravitational attraction between two bodies happens to be some force F at one distance, it becomes one-fourth F at double the distance and one-ninth F when the distance is tripled.

But that information by itself is not enough to calculate the exact values of the forces at work. For that, the relation requires a constant—in this case, a term known as the gravitational constant, labeled G, or, among people on the friendliest terms with the equation, “big G.”

Recognizing the correspondence between distance and mass was one of Newton’s many brilliant insights, but Newton had no way to measure the value of G. To do so, he would have had to know everything else in the equation, leaving G fully determined. In Newton’s day, however, you could not know the whole equation. Although you could easily measure the mass of two cannonballs and their distance from each other, their mutual force of gravity would be so small that no available apparatus could have detected it. You could easily measure the force of gravity between Earth and a cannonball, but you had no way to measure the mass of the Earth itself. Not until 1798, more than a century after the Principia, did the English chemist and physicist Henry Cavendish come up with a reliable measure of G.

To make his now-famous measurement, Cavendish used an apparatus whose central feature was a dumbbell, made with a pair of two-inch-diameter lead balls. A thin, vertical wire suspended the dumbbell from its middle, allowing the apparatus to twist back and forth. Cavendish enclosed the entire gizmo in an airtight case, and placed two twelve-inch-diameter lead balls kitty-corner outside the case. The gravitational pull of the outside balls would tug on the dumbbell and twist the wire from which it was suspended. Cavendish’s best value for G was barely accurate to four decimal places at the end of a string of zeroes: in units of cubic meters per kilogram per second squared, the value was 0.00000000006754.

Coming up with a good design for an apparatus wasn’t exactly easy. Gravity is such a weak force that practically anything, even gentle air currents within the case, could swamp gravity’s signature in the experiment. In the late nineteenth century the Hungarian physicist Loránd Eötvös, using a new and improved Cavendish-type apparatus, made mild improvements in G’s precision. This experiment is so hard to do that, even today, G has acquired only a few additional decimal places. Recent experiments conducted at the University of Washington in Seattle by Jens H. Gundlach and Stephen M. Merkowitz, who redesigned the experiment, derive the value 0.000000000066742. Talk about weak: as Gundlach and Merkowitz note, the gravitational force they had to measure is equivalent to the weight of a single bacterium.

Once you know G, you can derive all kinds of things, such as Earth’s mass, which had been Cavendish’s ultimate goal. Gundlach and Merkowitz’s best value for that is just about 5.9722 × 10²⁴ kilograms.

Many physical constants discovered in the past century link with forces that influence subatomic particles—a realm ruled by probability rather than precision. The most important constant among them was promulgated in 1900 by the German physicist Max Planck. Planck’s constant, represented by the letter h, was the founding discovery of quantum mechanics, but Planck came up with it while investigating what sounds mundane: the relation between the temperature of an object and the range of energy it emits.

An object’s temperature directly measures the average energy of motion of its jiggling atoms or molecules. Of course, within this average some of the particles jiggle very fast, whereas others jiggle relatively slow. All this activity emits a sea of light, spread over a range of energies, just like the particles that emitted them. When the temperature gets high enough, the object begins to glow visibly. In Planck’s day, one of the biggest challenges in physics was to explain the full spectrum of this light, particularly the bands with the highest energy.

Planck’s insight was that you could account for the full sweep of the emitted spectrum in one equation only if you assume that energy itself is quantized, or divided up into itty-bitty units that cannot further be subdivided: quanta.

Once Planck introduced h into his equation for an energy spectrum, his constant began to appear everywhere. One good place to find h is in the quantum description and understanding of light. The higher the frequency of light, the higher its energy: Gamma rays, the band with the highest frequencies, are maximally hostile to life. Radio waves, the band with the lowest frequencies, pass through you every second of every day, no harm done. High-frequency radiation can harm you precisely because it carries more energy. How much more? In direct proportion to the frequency. What reveals the proportionality? Planck’s constant, h. And if you think G is a minuscule constant of proportionality, take a look at the current best value for h (in kilogram-meters squared per second): 0.000000000000000000000000­00000000066260693.

One of the most provocative and wondrous ways h appears in nature arises from the so-called uncertainty principle, first articulated in 1927 by the German physicist Werner Heisenberg. The uncertainty principle sets forth the terms of an inescapable cosmic trade-off: for various related pairs of fundamental, variable physical attributes—location and speed, energy and time—it is impossible to measure both quantities exactly. In other words, if you reduce the indeterminacy for one member of the pair (location, for instance), you’re going to have to settle for a looser approximation of its partner (speed). And it’s h that sets the limit on the precision you can attain. The trade-offs don’t have much practical effect when you’re measuring things in ordinary life. But when you get down to atomic dimensions, h rears its profound little head all around you.

It may sound more than a bit contradictory, or even perverse, but in recent decades a lot of physicists have been looking for evidence that constants don’t hold for all eternity. In 1938 the English physicist Paul A. M. Dirac proposed that the value of no less a constant than Newton’s Gmight decrease in proportion to the age of the universe. Today there’s practically a cottage industry of physicists desperately seeking fickle constants. Some are looking for a change across time; others, for the effects of a change in location; still others are exploring how the equations operate in previously untested domains. Sooner or later, they’re going to get some real results. So stay tuned: news of inconstancy may lie ahead.