*h*and the speed of light

*c*—that are now commonly employed in studying the submicroscopic world (“down there”) and the cosmological world (“out there”). in 1899, Max Planck had noticed that one could combine the two constants

*h*and

*c*with the already known gravitational constant

*G*to produce a quantity with the dimension of length, and a corresponding quantity with the dimension of time.

^{1}The so-called “Planck length” is some twenty orders of magnitude smaller than the diameter of a single proton, and the corresponding “Planck time”—the time it takes light to travel one Planck length—is twenty orders of magnitude less than the shortest time about which we have any certain knowledge.

^{2}This leads many physicists, including this author, to imagine that an as-yet-undiscovered additional constant is lurking in the shadows, waiting to reveal a fundamental distance more in line with the scale of the presently studied parts of the natural world.

The meter was originally defined as one ten-millionth of the distance from either of Earth’s Poles to the Equator. (It is not quite that, because the meter was standardized in the nineteenth century but the knowledge of the size of Earth has been improved since then.) The kilogram was defined initially as the mass of a cube of water ten centimeters on a side. Both the meter and the kilogram thus depend on the size of Earth, and there is no reason to believe that there is anything very special about the size of Earth. The third basic unit, the second, also depends on a property of our Earth, its rate of rotation—again, nothing very special. For no better reasons than that the Egyptians divided the day and the night into twelfths and the Sumerians liked to count in sixties, the hour is one twenty-fourth of a day, the minute a sixtieth of an hour, and the second a sixtieth of a minute.

Neither Planck’s constant, *h,* nor the speed of light, *c,* is directly a mass, a length, or a time, but they are simple combinations of these three. The addition of the gravitational constant *G* produces a triad of units that form a basis of measurement as complete as the kilogram, the meter, and the second, but hampered by the seemingly outlandish implications for fundamental lengths and times. (The thoughtful reader might propose the charge of the electron as an alternative candidate for a third natural unit. Unfortunately, it will not serve, for it is not independent of *c* and *h*—just as speed is not independent of time and distance.)

The fact that light travels at a fixed speed has been known for several centuries, but the central significance of this speed in nature could be appreciated only when the theory of relativity appeared on the scene. Relativity revealed first of all that this speed, *c,* is a natural speed limit, attainable by light and by gravitational waves (but not by any particle with mass). The theory also showed that this constant appeared in various surprising places that have nothing to do with speed, for instance, in the mass-energy relation, *E = mc*^{2}.

Planck’s constant was brand new in 1899, but its significance also mounted over the next few decades as it came to be recognized as the fundamental constant of the quantum theory, governing not only the allowed values of spin, but of every other quantized quantity.

It has to be recognized that every measurement is really the statement of a ratio. If you say you weigh 151 pounds you are, in effect, saying your weight is 151 times greater than the weight of a standard object (a pint of water), which is arbitrarily called one pound. A 50-minute class is 50 times longer than an arbitrarily defined time unit, the minute. When one uses natural units, the ratio is taken with respect to some physically significant unit rather than an arbitrary one. On the natural scale, a jet-plane speed of 10^{–6}*c* is very slow, a particle speed of 0.99*c* is very fast. An angular momentum of 10,000*ħ* is large, an angular momentum of ½*ħ* is small. (Here *ħ*—pronounced h bar—is shorthand for *h*/2π.)

The difficult point to recognize here is that once the speed of light has been adopted as the unit of speed, it no longer makes any sense to ask how fast light travels. The only answer is: Light goes as fast as it goes. Since every measurement is really a comparison, there must always be at least one standard that cannot be compared with anything but itself. This leads to the idea of a “dimensionless physics.” Having agreed on a standard of speed, we can say a jet plane travels at a speed of 10^{–6}, that is, at one one-millionth of the speed of light. The 10^{–6} is a pure number to which no unit need be attached; it is the ratio of the speed of the plane to the speed of light. To make possible a dimensionless physics we need one more independent natural unit. Although the gravitational constant, *G*, can, formally, fulfill this mission, physicists have not embraced it. If an alternate natural unit is discovered, it may be a length, and there is much speculation that such a unit will be connected with a whole new view of the nature of space (and of time) in the realms we now study—leaving the “Planck scale” to be unraveled by a future generation of physicists.

It should be added that a dimensionless physics is not so profound as it may sound. It, too, would rest on an arbitrary agreement among scientists about units. What scientists now agree on is that there is nothing special whatever about the meter, the kilogram, and the second.

And, further, they agree that there is nothing magical about having exactly three basic units. Consider the role of the speed of light. By providing a natural link between time units and distance units, the speed of light makes it possible to lay aside one of these units and express both times and distances in terms of a single unit. In units where the speed of light has the value of 1, the circumference of the Earth is about one-tenth of 1 s, and the distance from Earth to Sun is 500 s. The time required for the Earth to complete one orbit is 3 × 10^{7} s (1 year), but the distance it travels in that year is 3.2 × 10^{3} s. The last figure means that light could cover a distance equal to the circumference of the earth’s orbit in 3.2 × 10^{3} s, about one ten-thousandth the time the earth spends in getting around the Sun. Once the standard speed unit has been adopted and the meaning of distance measured in seconds is understood, there is no ambiguity. The number of fundamental dimensions has been reduced from three to two. Working with fewer than three fundamental dimensions is often convenient for the professional physicist, and is very likely connected with a deeper view of nature. For the beginning student, on the other hand, the use of fewer than three dimensions can sometimes obscure the meaning of new concepts and new laws, and it is not recommended. Nevertheless, it is important to appreciate the meaning, the possibility, and the merit of a dimensionless physics.^{3}

1^{} Why do we use *c* for the speed of light? The answer may lie be in Planck’s 1899 paper. In that paper he listed four constants, calling the first three of them *a, b, *and* c.* The first, *a,* was a thermodynamic constant related to Boltzmann’s constant. The second, *b,* was his own new constant, which we now call Planck’s constant and write *h.* The third, *c,* was the speed of light. So the speed of light may be *c* because it was third in Planck’s list. (Planck chose to call his fourth constant *f,* not *d.* It was the gravitational force constant.)

2^{} The term “Planck length” was coined not by Planck nor by any of his contemporary scientists but by the American physicist John Wheeler many years later. Wheeler is famous for his coinages, including “black hole,” “wormhole,” “geons,” and “quantum foam.”

3^{} Note that in this discussion I am using “dimension” to refer to a physical concept such as length or time or mass, not to a spatial dimension such as length, breadth, or height.