Help me understand the Planck charge

I understand a lot of the Planck units, e.g. the Planck time is the time it takes light to travel one Planck length, the Planck temperature is the temp where the blackbody photon wavelength is one Planck length, the Planck mass is the mass at which an elementary particle will collapse into a black hole, but what is the Planck charge? I understand that its about 11 times the charge of a proton, but wherefore the “naturalness” of it?

Also, how are Planck units not arbitrary? I could also say that the sweeteviljesus time is the time it takes light to travel 1 sweeteviljesus length. Does it come down to the Planck mass?

I think they’re all mostly estimates, although some may be coupled to one another (e.g. Planck length to Planck time by a factor of c).
And the Planck charge is the charge which would make a black hole if it were concentrated in a volume with a radius of one Planck length.
Or sumfink like dat…

The Planck length, mass, and time are all derived from the fundamental constants c, G, and hbar taken to various powers. They’re the only units with those dimensions you can get from just those constants. From those, you can similarly get a momentum, energy, etc. Toss in Boltzman’s constant, and you can also get a temperature. Toss in Coulomb’s constant, and you can get a charge. In fact, for the charge, you don’t even need to use G at all, just hbar, c, and Coulomb’s constant.

The Planck charge isn’t used much by itself. Where it usually shows up is that if you divide the electron charge by it and square it, you get the Fine Structure Constant, AKA alpha, which is approximately equal to 1/137. This shows up pretty much whenever you have both electromagnetism and quantum mechanics.

Can someone give us a Planck-for-Dummies Primer on all things Planck, from the first principles on up, for those of us who don’t even know what to ask?

OK, so, there are three really important fundamental constants in physics: G, c, and hbar. G describes the strength of the gravitational force, and so if you’re working with gravity, it shows up all over the place. It shows up so much that, if you’re working with gravity, it’s easiest to pick units where G = 1. hbar (written as a letter h with a line through the upper part) is the fundamental quantum of angular momentum, and shows up all over the place in quantum mechanics. So, like the gravity people, quantum folks usually pick units such that hbar = 1. And c is the fundamental speed of the Universe (among other things, it’s the speed of massless particles like photons). If you’re doing relativity, then c shows up all over the place, and so, again, it’s customary to choose units where c = 1.

Now, it’s quite common to be working with relativity and gravity at the same time, in which case you use units where both G and c are equal to 1. Likewise, it’s common to be working with both relativity and quantum mechanics, in which case you use units where c and hbar are both 1. In either of these cases, setting those constants equal to 1 means that everything can be measured in the same units. Typically, the gravity people end up using length units for everything and the quantum people end up using energy units for everything, but that’s a minor detail: You could use any units.

But what if you’re doing something where you’re working with gravity, relativity, and quantum mechanics all at once? In practice, that’s quite rare, because we know almost nothing about the interaction of gravity with quantum mechanics. But if you were doing that, you’d want to use units where all three of those fundamental constants are equal to 1. And that means that you effectively don’t have any units left at all: All quantities are just measured in dimensionless numbers. Instead of a time of 1 day, or 1 second, or 1 meter/c, you can have a time that’s just 1. That time is called the Planck time. Similarly, a length of just 1 is the Planck length, and a mass of just 1 is the Planck mass, and so on. And from mass, length, and time, you can construct the other quantities of interest in physics, and so get a Planck energy and so on.

What do these quantities mean physically? Well, in some cases, the answer is obvious: The “Planck speed”, for instance, just brings us back to c, and the “Planck angular momentum” is just hbar. But in most cases, we don’t know of any particular physical significance: It’s just a notational convenience. It might be the case that, once we develop a theory of quantum gravity, it’ll turn out that we find significance for the others: For instance, if spacetime is quantized, and if the quantization is such that there’s a smallest length possible, then our best guess for what that smallest length would be would be somewhere in the vicinity of the Planck length. But nobody would be particularly surprised if it was something like half the Planck length or pi times the Planck length, and it might be that our best guess is just completely off. Or, of course, it might be that there is no such thing as a “smallest length”: That’s just a guess, too.

Nor are all of the Planck quantities even all that extreme. The Planck mass, for instance, is far greater than the mass of any known subatomic particle, but far less than the mass of any object you’re familiar with. Some bacteria have a mass about equal to the Planck mass. And the Planck momentum is a perfectly reasonable amount of momentum to encounter in a high school physics lab.

And the reason people talk about the Planck length in popular physics books and what not, is
there’s a fair amount of feeling among physicists that physics will look very different at super small distances (much much smaller than an atom). This isn’t a strange concept: physics looks very different for things the size of baseballs and for things the size of electrons. [It’s not that the basic physical laws are different, but some effects are very big for electrons, but so small they can be ignored for soccer balls, and vice versa, so in practice things look very different].

Now, there’s also a fair amount of, maybe ‘suspicion’ is the right word, that if it’s true that physics looks different at super small distances, then what ‘super small distances’ means is something around the Planck length. So physicists trying to work out theories of quantum gravity and what-not are often trying to think about what their theory says about what happens at Planck-length distances.

We don’t just suspect that physics would be different at the Planck length scale; we know it. At such a length scale, both general relativity and quantum mechanics would be relevant at the same time, and we know that our current physics aren’t adequate for such a situation.

Thanks to Chronos, as usual, for his one-paragraph first post, a jewel, to the long one made after a request from the peanut gallery (where I live).