Could Quantum Geons Achieve Stability Like Bohr Atoms?

[johne1618]

Apparently a photon sphere is a region of space where gravity is strong enough to cause light to travel in orbits. The radius of the photon sphere is given by
$$r = \frac{3 G M}{c^2}$$
If one had a photon that was energetic enough then its own gravity would trap it in its own photon sphere. I believe this is John Wheeler's geon idea.

Imagine that a photon is forced into a closed orbit. By analogy with the Bohr atom its orbital angular momentum should be quantized such that:
$$p \ r = n \hbar$$
Using the relation between energy and momentum for photons, $E = p \ c$, we find
$$E = \frac{n \hbar c}{r}$$
Using the Einstein relation, $E = M c^2$, we can substitute the quantized photon mass/energy into the photon sphere equation. By eliminating $r$ we find that the mass of the resulting object is given by:
$$M = \sqrt{\frac{n}{3}} \sqrt{\frac{\hbar c}{G}}$$
$$M = \sqrt{\frac{n}{3}} M_{planck}$$
I know that John Wheeler concluded that classical geons are not stable. Classical photon orbits are not stable.

But could a quantum geon produced by a coherent photon orbit as described above be stable in the same way that a Bohr atom is stable?

 

[K^2]

1) Bohr atom is non-physical. I'm not sure in what sense it could possibly be (un)stable.

2) Quantum and gravity don't mix. Quantum mechanics requires a linear Hamiltonian. What you are proposing is definitively non-linear. Even if there is something in that idea, it would require Quantum Gravity of some sort to resolve.

Edit: Though, it is a photon. You might be able to get away with scrapping QM, and just trying to see if you get quantization you need from pure electrodynamics. It's not going to be pretty, though.

 

[cragar]

when you say get quantization from pure electrodynamics.
Is that like when you solve laplaces equation for like the potential in a box
and you get values based on cos(npi)
Is that what you mean?