Ok so I have two very different questions, both closely tied to the concept of relativistic angular momentum.(adsbygoogle = window.adsbygoogle || []).push({});

Question #1: Why do people claim there is a minimum radius for a classical model of electrons with spin? The typical argument is that if the radius were to fall below some minimum value, the surface velocity of the electron would be moving faster than light.

For simplicity, assume an electron is modeled by a rotating ring of charge (the problem is nearly identical for any other configuration, as any cylindrically symmetric object can be composed of rings). Say that at rest, this ring has mass m, mass density λ, radius r, and charge e. Rotating, the tangential momentum is given as [itex]\vec{p} = \gamma\lambda\vec{v}[/itex], where v is the tangential velocity of the matter. It is easy to show that the energy density along the ring is [itex]e=\gamma\lambda[/itex]. Therefore since the total momentum of the system is 0, and the total energy is [itex]\gamma{m}[/itex], the total mass M of the system is just [itex]\gamma{m}[/itex].

The angular momentum is given as [itex]\int{\vec{r}\times\vec{p}}[/itex], so [itex]L=\gamma m r v[/itex]. Leaving L constant, and rearranging the terms we get:

[itex]r = \dfrac{L}{mv}\sqrt{1-v^2}[/itex]

And it becomes clear that as r→0, v→1! This shows that thereis no lower boundon the radius of a spinning particle! Note however that as r→0, M→∞. While it is clear that the mass of an electron is not infinite, this is a well known problem of point particles and shows up everywhere from classical physics to QFT. Also, there is no reason to believe the electron is actually a point particle. All we know is that its radius is below some experimental bound that has been pushed past the "limit" classical physics would set.

Another interesting result is if you consider the "material" making up the electron as massless. Much of the above treatment is the same, except that you need to rephrase everything in terms of E and p. What you find is that the total energy of the system M is [itex]2\pi p[/itex], and that L=Mr. Again you find that there is no minimum radius, but as r→0, M still diverges.

Question #2: Are the typical dismissals of GR predictions of the interior of a black hole valid? The typical argument is that a singularity is inconsistent with pauli exclusion principle (there are plenty of other arguments of similar nature).

Although the above results are derived in flat space, a very similar argument can be made. We find that the singularity of a rotating black hole is "made" of a massless "material" and the overall mass comes entirely from the rotational energy! What I find interesting about this result is that there isn'taren'tany massless fermions!

Now I realize that you need to be careful when interpreting combined GR and QM results. However what I get out of this is that there must be some mechanism to convert any incoming fermions into massless bosons inside of a black hole. Hawking radiation could easily provide such a mechanism! So if the singularity of blackholes is purely bosonic, what is the contradiction between QM and GR?? As I understand it this would be a kind of Bose-Einstein condensate

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# Relativistic Angular Momentum

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