I get how the Planck units are formed. (Take a number of physical constants, set the “nice” ones to unity, and keep things consistent from there.) But what I don’t get is why the units that result are sometimes claimed to have physical significance.
For example, the Planck length is 1.6 × 10[SUP]−35[/SUP] meters, and some people say that this is the smallest length that has any physical meaning? Why is this so? What does that even mean? Why can’t things be smaller than that? Same goes with the Planck time of 5.4 × 10[SUP]−44[/SUP] seconds as the smallest amount of physically meaningful time? Why is that special?
And why, if this is the case for space and time, is this not true for the Planck mass of 2.2 × 10[SUP]−8[/SUP] kilograms (about 22 micrograms), a mass that seems to have no physical significance whatsoever?
ETA: This question is inspired by this thread: What is the shortest time measurable... - Factual Questions - Straight Dope Message Board
Well, there’s no hard, solid evidence that there’s nothing smaller than the Planck length/time/etc.—however, this falls out from several theoretical lines of reasoning. Loop quantum gravity, for instance, postulates a smallest unit of area, roughly about the Planck length squared; in string theory, the Planck length is roughly the string scale, and excited strings give rise to particles of about Planck mass.
Perhaps more intuitive, if you try to measure shorter and shorter distances, you need to use higher and higher energies, because you need smaller and smaller wavelengths to resolve these distances. But eventually, you end up cramming so much energy into such a small space that you create a microscopic black hole—and this happens around the Planck length. If you then attempt to resolve any smaller distances, you’ll just create a larger black hole; and since you can’t resolve anything within the black hole, you just ‘bounce back’ to larger distances. This is at the heart of an attempt to make sense of the infinities that crop up when trying to quantize gravity, headed by Gia Dvali: if you go to sufficiently high energies, gravity just bounces back to macroscopic length scales (it ‘classicalizes’), and the unphysical divergences (hopefully) never arise. Interestingly, something similar also happens in string theory (‘UV-IR duality’).
Interesting. I didn’t know that. But it seems to me that it’s just a coincidence then that these little black holes appear at the Planck length. Or is there something deeper going on here that relates this energy to √([DEL]h[/DEL]G/c[SUP]3[/SUP])?
You can think of the various constants at measuring the strength of various physical effects—ℏ tells you where quantum effects become significant, c tells you where special relativistic effects become dominant, G determines the strength of gravitational/general relativistic effects, etc. So at Planck dimensions, all of these influences are roughly of the same order, so for the physics there, you need to take them all into account; in particular, you need to consider both quantum and gravitational effects. So that these dimensions crop up wherever you consider the relevant physics isn’t all that surprising.
For the black hole example, you need to concentrate an energy of E > 1/L into a box of size L in order to resolve anything of size L. Here, in giving energy in units of inverse length, I have already made use of the Planck units; reintroducing for the moment the physical constants, energy is related to length by E = ℏc/L. The Schwarzschild radius for this energy works out to R = l[sub]p[/sub]²/L, where l[sub]p[/sub] is the Planck length*. So for L = l[sub]p[/sub], we have R = l[sub]p[/sub]—trying to probe the Planck length scale creates precisely a Planck length black hole; trying to get lower creates a bigger one.
Showing my work, the Schwarzschild radius for a black hole of mass m is R = 2Gm/c²; with m = E/c² and E = ℏc/L, we get R = 2Gℏ/c³1/L = l[sub]p[/sub]²/L.