# What is the significance of Planck charge?

It seems for me that Planck units are somehow connected to limits where our current knowledge breaks down because of (quantum) gravitational effects. Please correct me if I'm wrong.

For example Planck mass is the maximum mass allowed for point particle. Had a particle had mass greater than Planck mass, it would have formed black hole, because its Compton wavelength would have been less than its Schwarzschild radius.

Does similar physical significance exist for Planck charge?

• I'm no expert, but I don't think we know enough about quantum gravity (if we even know anything about it at all!) to be so confident as to say that the Planck mass is the maximum mass of a point particle. Anyway, wouldn't the Planck charge be just the electron charge? – Javier May 29 '15 at 16:14
• So far you can put every Planck scale related quantity into the "it's not even false" bin which is also known as "complete intellectual nonsense". It's all fun stuff, but it's not science. The "Planck charge" is an even funnier and even less likely to be relevant number. – CuriousOne May 29 '15 at 16:16
• "The Planck charge is $\alpha^{-1/2} \approx 11.706$ times larger than the elementary charge e carried by an electron". Disappointing, right? – user66432 May 29 '15 at 19:28
• Yes you are wrong. The Planck charge has nothing to do with gravitational effects and knows nothing of Newton's constant. It is just a normalization of the elementary charge involving Planck's constant. – Cosmas Zachos Nov 18 '16 at 2:46
• Related: 147603. – Cosmas Zachos Feb 11 '17 at 23:26

My understanding of the Planck charge is that it is the unit charge necessary to normalize

1. the speed of light (and other "instantaneous" interactions e.g. strong force and gravity), $c=1$
2. reduced Planck constant, $\hbar=1$
3. Coulomb constant, $\frac{1}{4 \pi \epsilon_0}=1$

So it's a natural definition of charge that has no reference to the elementary charge. So you can represent the value of the elementary charge in terms of this Planck charge and it comes out as

$$e = \sqrt{\alpha}$$

where $\alpha$ is the fine-structure constnat.

So the significance of the Planck charge is, like the other Planck units, they are defined in such a way that no object or particle (not even any subatomic particles) are needed nor used to define the base units. Only free space. Then you can go about asking what the properties of these elementary sub-atomic particles are, in terms of these units that only normalize properties of free space. For the charge of an electron, it is $-e = -\sqrt{\alpha} \approx -0.0854245$, in terms of Planck units.

I prefer rationalized units over unrationalized units because I think it is useful to equate the concepts of flux density and field strength and that is done if you normalize the electric constant rather than the Coulomb constant. If you do that, the elementary charge comes out as $e=\sqrt{4 \pi \alpha} \approx 0.302822$. This is done in a few books on QFT or something and makes it more clear that the elementary charge, while not exactly the natural unit of charge, is in the ball park of the natural unit of charge. Well within an order of magnitude.

The curvature scale of a Reissner-Nördstrom solution (the metric for a charged, massive black hole) goes to the Planck length in the limit that the mass goes to zero and the charge goes to the Planck charge. Of course, this is a physically unreasonable limit, since it has a naked singularity; and it does have the problem of relating one quantity of dubious utility to another quantity whose utility is still not un-dubious.

Other than that, I got nothin'.

The plank charge is the maximum charge of a black hole of plank mass. Once the charge exceed the plank charge for a mass equal to the plank mass, an event horizon will not be able to form, according to the Reissner-Nördstrom solution.

The significance of the odd value of the Planck charge, is that it is just an indicator of an oversight in physics which occurred over 150 years ago. This oversight creates a problem in dimensions which gives the wrong relationship between mass and charge. As a result we get the wrong value for Planck charge. When corrections are made to the SI system of units which includes dimensions for the fine structure constant, Planck charge turns out to be nothing more than elementary charge.