Gravity coupled to matter requires a coupling constant G that has units of length over mass. One finds the Planck scale if one lets quantum mechanics come into the game. For this, let us consider a quantum particle of a (so far unknown) mass mp with a Compton wavelength lp, the relation between both given by the Planck constant
This is the quantum input. Now consider that particle to be as localized as it is possible taking into account its quantum properties. That is, the mass mp is localized within a space-time region with extensions given by the particle's own Compton wavelength. The higher the mass of that particle, the smaller the wavelength. However, we know that General Relativity says if we push a fixed amount of mass together in a smaller and smaller region, it will eventually form a black hole. More general, one can ask when the perturbation of the metric that this particle causes will be of order one:
which then can be solved for the mass, and subsequently for the length scale we were looking for. If one puts in some numbers one finds
These Planck scales thus indicate the limit in which the quantum properties of our particle will cause a non-negligible perturbation of the space-time metric, and we really have to worry about how to reconcile the classical with the quantum regime. Compared to energies that can be reached at the collider (the LHC will have a center of mass energy of the order 10 TeV), the Planck mass is huge. This reflects the fact that the gravitational force between elementary particles is very weak compared to the the other forces that we know, and this is what makes it so hard to experimentally observe quantum gravitational effect.
Max Planck introduced these quantities in 1899, the paper (it's in German) is available online
- Sitzungsberichte der Königlich Preußischen Akademie der Wisseschaften zu Berlin
1899 - Erster Halbband (Berlin: Verl. d. Kgl. Akad. d. Wiss., 1899)
Über irreversible Strahlungsvorgaenge, von Max Planck.
(Credits to Stefan for finding it). You'll find the natural mass scales introduced on page 479ff. He didn't call them 'Planck' scales then, and it is also interesting why he found them useful to introduce, namely because the aliens would also use them
- "It is interesting to note that with the help of the [above constants] it is possible to introduce units [...] which [...] remain meaningful for all times and also for extraterrestrial and non-human cultures, and therefore can be understood as 'natural units'."
Coincidentally, yesterday I saw a paper on the arxiv
- What is Special About the Planck Mass?
By C. Sivaram
Abstract: Planck introduced his famous units of mass, length and time a hundred years ago. The many interesting facets of the Planck mass and length are explored. The Planck mass ubiquitously occurs in astrophysics, cosmology, quantum gravity, string theory, etc. Current aspects of its implications for unification of fundamental interactions, energy dependence of coupling constants, dark energy, etc. are discussed.
which gives a nice introduction into the appearances of various mass scales in physics, with some historical notes.
* With the speed of light set to be equal 1, in which case a length is the same as a time. It you find that confusing, just define a Planck time by dividing the length through the speed of light.