Blackhole-like object as a gravitational harmonic oscillator?

As described, in the TGD Universe blackhole-like objects are monopole flux tube spaghettis and differ from the ordinary stars only in that for blackholes the entire volume is filled by monopole flux tubes with h_eff=h for which the thickness is minimal and corresponds to a nucleon Compton length. For h_eff>h also the flux tubes could fill the entire volume of the star core.

Just for fun, one can ask what the model of a gravitational harmonic oscillator gives in the case of Schwarzschild blackholes. The formula, r_n= n^1/2r₁, r₁/R= [r_s/2β₀^1/2]×(r_s/R)^1/4, gives for R= r_s the condition r₁/r_s= 1/(2β₀)^1/2. β₀≤ 1/2 gives r₁/r_s≥ 1 so that there would be no other states than the possible S-wave state (n=0). β₀=1/2 gives r₁=r_s and one would have just mass at n=0 S-wave state and n=1 orbital. For β₀=1 (the minimal value), one has r₁/r_s= (1/2)^1/2 and r₂=r_s would correspond to the horizon. There would be an interior orbit with n=1 and the S-wave state could correspond to n=0.

The model can be criticized for the fact that the harmonic oscillator property follows from the assumption of a constant mass density. This criticism applies also in the model for stars. The constant density assumption could be true in the sense that the mass difference M(n+1)-M(n) at orbitals r_n+1 and r_n for n≥ 1 is proportional to the volume difference V_n+1-V_n proportional to r_n+1³-r_n³= (n+1)³-n³= 3n²+3n+1. This would give M= m₀+m(n_max+1)³ leaving only the ratio of the parameters m₀ and m free. This could be fixed by assigning to the S-wave state a radius and constant density. This condition would give an estimate for the number of particles, say neutrons, associated with the oscillator Bohr orbits. If a more realistic description in terms of wave functions, this condition would fix the total amount of matter at various orbitals associated with a given value of n.

See the article About the recent TGD based view concerning cosmology and astrophysics or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.