Once Before Time - Martin Bojowald [35]
To obtain the value of the Planck time, we can think of it as the time required for light to pass the distance of one Planck length. This is clearly tiny; the number of Planck time intervals in one second has 42 digits. The Planck mass may sound more down-to-earth, at about a microgram. But this value is also extreme if we think of it as a fundamental mass, such as the mass of an elementary particle rather than of a composite object made from a gazillion constituents.
How extreme the Planck scales are is perhaps best indicated by the size of the Planck density: the density corresponding to a Planck mass in a cubic region of edge length given by the Planck length. This density is the equivalent of a trillion suns compressed to the size of a proton. Since the Planck length contains mainly Planck’s constant, which is crucial for quantum theory, and the gravitationally important Newton’s constant, one expects its magnitude to play a large role in a combination of those two theories. In cosmology, quantum gravity becomes indispensable when the universe near the big bang singularity is compressed to Planckian density. Such dimensional arguments are often reliable, at least approximately, and they provide important expectations for theories still to be developed. For instance, the radius of a hydrogen atom in quantum mechanics can be estimated in a similar way because the electron’s charge and mass, the defining properties of the system, together with Planck’s constant, which is required by the quantum nature of the problem, determine a length parameter in a unique way. Up to a factor of two, this length agrees with the quantum mechanical extension of an electron’s wave function in the ground state of a hydrogen atom, the so-called Bohr radius.
We can easily see why no observations exist so far that would without a doubt require a quantum theory of gravity. After all, the Planck length is much, much smaller than any length measured thus far. Should it one day be possible to resolve lengths about the size of the Planck length, quantum gravity and the space-time structure it implies would become important. Then again, such arguments are to be taken with a grain of salt, for they fail in more complicated situations in which several distinct parameters are involved. Sizes of heavy atoms such as uranium, with a large number of electrons—in this case, 92—cannot be estimated by simple dimensional calculations. Would we have to multiply the Bohr radius by the number of electrons to obtain the radius of a uranium atom? Or divide by it? Or perform some other procedure? More detailed knowledge of the physics at play is needed, a full theory.
Moreover, different effects can, as we will describe later, result in magnifications by adding up many tiny terms. Even though quantum gravitational deviations from general relativity should, at a fixed time during cosmic evolution, be tiny, such corrections could be enhanced in the long duration while the universe expands; they would become detectable sooner than the smallness of the Planck length might indicate. Such indirect effects have often played large roles in the discovery of new physical phenomena. A well-known example is Brownian motion, used to uncover the atomic structure of matter, in which the tiniest impacts of molecules in a liquid cause a vibrating motion of suspended pollen grains, visible by light microscopy. We will return to this phenomenon in the chapter