LOG#183. Bohrlogy: some pocket formulae.

This post continues the Bohrlogy thread I did some time ago. It aims to parallel electric Bohr (atom), the gravitational Bohr atom and the black hole atom (as new!). Note that \alpha= K_Ce^2/\hbar c, \lambda_C=\hbar/mc and L_C=\hbar/Mc, R_S=2GM/c^2.

First of all, the common Bohr atom (electric case):

1. Quantized energy (frequency) via E=n\hbar \omega, f=E/nh.
2. Quantized action/angular momentum: L=n\hbar.
3. Quantized orbital radii:

    \[r_n=n^2 a_0=n^2 \dfrac{\hbar}{Z\alpha mc}=\dfrac{n^2\lambda_C}{Z\alpha}=\dfrac{n^2\hbar^2}{ZK_Cme^2} \]

4. Quantized velocity: v_n=Z\alpha c/n
5. Quantized spectrum:

    \[E_n=-\dfrac{Z^2\alpha^2 mc^2}{2n^2}\]

6. Quantized acceleration:

    \[a_n=\dfrac{Z^3\alpha^3 mc^3}{\hbar n^4}\]

7. Quantized density (\rho=m/V):

    \[\rho_n=\dfrac{3}{4\pi}\dfrac{Z^3\alpha^3}{n^3}\dfrac{m}{\lambda_C^3}\]

Now, the gravitational Bohr atom:

1. Quantized energy (frequency) via E=n\hbar \omega, f=E/nh.
2. Quantized action/angular momentum: L=n\hbar.
3. Quantized orbital radii:

    \[r_n(G)=n^2 a_0^G=n^2 \dfrac{\hbar^2}{NG_NMm^2}=\dfrac{2n^2\lambda_C^2}{NR_S} \]

4. Quantized velocity:

    \[v_n(G)=\dfrac{NG_NMm}{n\hbar}=\dfrac{NR_S}{2n\lambda_C}c\]

5. Quantized spectrum:

    \[E_n(G)=-\dfrac{N^2G_N^2 M^2m^3}{2n^2\hbar^2}=-\dfrac{N^2}{8}\dfrac{R_S^2}{\lambda_C^2}mc^2\]

6. Quantized acceleration:

    \[a_n(G)=\dfrac{N^3R^3_S}{\lambda_C^3} \dfrac{mc^3}{8 \hbar n^4}=\dfrac{N^3R^3_S}{(2\lambda_C)^3} \dfrac{mc^3}{\hbar n^4}\]

7. Quantized density (\rho=m/V):

    \[\rho_n(G)=\dfrac{3}{4\pi}\left(\dfrac{NG_NM}{c^2}\right)^3\dfrac{m}{n^6\lambda_C^6}=\dfrac{3}{32\pi}\left(\dfrac{R_S^3N^3}{\lambda_C^6n^6}\right)m\]

Finally, the Bohr black hole atom (yeah!). It is obtained from the previous one, plugging \lambda_C\sim R_S. It happens when \lambda_C\sim R_S\sim L_P, where L_P is the Planck length. You get:

1. Quantized energy (frequency) via E=n\hbar \omega, f=E/nh.
2. Quantized action/angular momentum: L=n\hbar.
3. Quantized orbital radii:

    \[r_n(BHA)=2n^2 a_0(BH)/N=2n^2R_S/N=2n^2L_P/N \]

4. Quantized velocity:

    \[v_n(BHA)=\dfrac{N}{2n}c\]

5. Quantized spectrum:

    \[E_n(BHA)=-\dfrac{N^2}{8}mc^2\]

6. Quantized acceleration:

    \[a_n(BHA)=\dfrac{N^3}{8n^4}\dfrac{mc^3}{\hbar}\]

7. Quantized density (\rho=m/V):

    \[\rho_n(BHA)=\dfrac{3}{32\pi}\dfrac{N^3m}{R_S^3 n^6}=\dfrac{3}{32\pi}\left(\dfrac{N}{ n^2}\right)^3\dfrac{m}{L_P^3}\]

Black holes have temperature T_{BH}=\hbar c^3/(8\pi G_N M k_B), and frequency, then

    \[f_{BH}=\dfrac{ c^3}{16\pi^2 G_NM k_B}\]

in S.I. units. Note that frequencies do not contain \hbar but energy does via Planck E=hf=\hbar \omega.

Interestingly, electrons can not be black holes. The Schwarzschild radius R_S=2G_NM/c^2 and a Planck length electron has nothing to do with this. There is A CLASSICAL way to have BH electrons, via the Reissner-Nordström solution in General Relativity. You can introduce a charge radius about

    \[R_{Q}=\sqrt{\dfrac{G_NQ^2}{4\pi \varepsilon_0 c^4}}=\sqrt{\dfrac{K_C G_N}{c^4}}Q\]

For the electron, it yields R_Q\approx 10^{-57}m< L_P. Taking a charged rotating BH electron is different. The Kerr parameter for any elementary electron with J=\hbar/2 is a=J/mc\approx 10^{-13}m. Some time ago, Wheeler talked about geons, gravitationally self-sustained electromagnetic fields, very non-linear and intense fields. A similar current concept is the so-called “kugelblitz”. No geons or kugelblitz have been observed in Nature, but they could exist via laser sabers (Jedi, you are?), or they could arise from numerical solutions in the field equations of certain gravitational or extended gravitational theories.

Finally, a similar concept to Bohr BH atoms is known in the literature. They are called holeums. For holeums

    \[E_n=-\dfrac{mc^2 \alpha_g^2}{4n^2}\]

with

    \[\alpha_g=\dfrac{G_Nm^2}{\hbar c}=\dfrac{m^2}{m_P^2}\]

    \[r_n=\dfrac{n^2\pi^2R_S}{8\alpha_g^2}\]

and m=2m(MBH), 2 microblackholes. Even more, if you generalized this to a set of madroholeums. Macroholeums have been considered even as the internal states of BH. Black holeums would be BH with internal structure, indeed quantum black holes. They would be stable. This is very speculative.

As curious comment, I will compare some numbers from the Bohr atom and the gravitational Bohr atom. Bohr radius is about 1 angstrom, 10^{-10} meters. The gravitational analogue is big, very big. It is about 10^{34}m. The Rydberg for the hydrogen atom is 13.6 eV, about 10^{-18} joules. In the case of the gravitational atom is about 10^{-105} joules.

Final exercise, and think about it. Imagine a world where electrons and protons are (electrically) uncharged and they are bound by gravitational forces. What is the minimal radius? What is the minimal energy in the fundamental level? In order to fit the radii to the known values, assuming, e.g., that you have equal proton and electron mass, what should the electron mass be in order to get the usual Bohr atom result? Compare all the numbers and scales you get with your every day experience.

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