Scaling Mount Planck I: A View from the Bottom

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G ravity dominates the large-scale structure of the universe, but only by default, so to speak. Matter arranges itself to cancel electromagnetism, and the strong and weak forces are intrinsically short range. At a more fundamental level, gravity is extravagantly feeble. Acting between protons, gravitational attraction is about 10–36 times weaker than electrical repulsion. Where does this outlandish disparity come from? What does it mean? These questions greatly disturbed Richard Feynman. His famous paper on quantizing general relativity,1 in which he first described his discovery of the “ghost particles” that eventually played a crucial role in understanding modern gauge field theories, begins with a discussion of the smallness of gravitational effects on subatomic scales, after which he concludes, There’s a certain irrationality to any work on [quantum] gravitation, so it’s hard to explain why you do any of it. . . . It is therefore clear that the problem we [are] working on is not the correct problem; the correct problem is: What determines the size of gravitation? The same question drove Paul Dirac2 to consider the radical idea that the fundamental “constants” of nature are time dependent, so that the weakness of gravity could be related to the great age of the universe, through the following numerology: The observed expansion rate of the universe suggests that it began with a bang roughly 1017 seconds ago. On the other hand, the time it takes light to traverse the diameter of a proton is roughly 10–24 seconds. Squinting through rose-colored glasses, we can see that the ratio, 10–41, is not so far from our mysterious 10–36. (For what it’s worth, the numbers agree better if we compare gravitational attraction versus electrical repulsion for electrons, instead of protons.) But the age of the universe, of course, changes with time. So if the numerological coincidence is to abide, something else—the relative strength of gravity, or the size of protons—will have to change in proportion. There are powerful experimental constraints on such effects, and Dirac’s idea is not easy to reconcile with our standard modern theories of cosmology and fundamental interactions, which are tremendously successful. In this column, I show that today it is natural to see the problem of why gravity is extravagantly feeble in a new way—upside down and through a distorting lens compared to its superficial appearance. When viewed this way, the feebleness of gravity comes to seem much less enigmatic. In a sequel, I’ll make a case that we’re getting close to understanding it. First let’s quantify the problem. The mass of ordinary matter is dominated by protons (and neutrons), and the force of gravity is proportional to mass squared. Using Newton’s constant, the proton mass, and fundamental constants, we can form the pure dimensionless number