Quantitative support for the model of blackhole-like object as flux tube spaghetti

The TGD based model for blackhole-like object is as monopole flux tube spaghetti (see this) containing one proton per proton Compton length and filling the entire volume. There is no need to emphasize that the model means giving up the standard view of blackhole-like objects and that the model does not differ very much from the model for neutron stars.

Consider now the estimation of the total mass of the flux tube spaghetti.

1. Assuming additivity and neglecting self-gravitation, the total mass in units of m_p is M/m_p (here m_p≈ m_n is proton mass, the star would consist of neutrons).
2. Self gravitation for a spherically symmetric mass constant distribution inside sphere of radius R and given as ρ= M/Vol(R) created by the flux tube spaghetti gives to the stationary metric the deviation of g_tt from flat Minkowski metric is given by Δ g_tt = - Φ_gr, where one has

   Φ_gr (r)= 2G(2M(r)/r= (8π/3)×(GM/Vol(R)) r²= 2GM(r²/R³) .

   The gravitational potential energy of the mass distribution is in the Newtonian appproximation given by

   E_gr= -∫ρ(r)Φ_gr (r)dV=-6GM²/5R .

   For R= r_S= 2GM this gives

   E_gr=- 6GM²/10GM = -(3/5)M .

   Therefore the observed mass M_obs using m_p as a unit is given by

   M_obs=E_tot/m=(2/5) (M/m_p) .

3. Suppose that the flux radius of thickness R contains a single proton per length zR so that one proton fills the volume π× zR³. Suppose R corresponds to the proton Compton length L_p = h/m_p.

   Assume that h_eff ≠ h is possible so that L_p is scaled by y= h_eff/h. One would have

   L_p(h_eff)= y L_p .

4. The total mass M using m_p as unit and neglecting gravitational potential energy is given by the ratio of the volume V of the blackhole regarded as region of Minkowski space to the volume V_p taken by a single proton:

   M/m_p= V/V_p= (4/3z) × (r_S/L_p)³) y^{-3 .}

   Taking into account gravitational potential energy, one obtains

   M_obs/m= (2/5)(V/V_p)=(8/15z) × (r_S/L_p)³ y^{-3 .}

One can test the model for the Sun. One has M_S=2× 10³⁰ kg and r_S= 3 km. Proton has mass m_p= 1.6× 10^-27 kg and Compton length L_p= 1.3× 10^-15 m. Substituting the values to the above formula, one obtains (y,z)= (1,.992)≈ (1,1).

In the above formula M_obs/m on r.h.s decreases slightly in m_p→ m_n and 1/L_p³ on l.h.s increases slightly in m_p→ m_n. The changes of l.h.s and r.h.s are proportional to -ε × l.h.s and 3ε × r.h.s, where one has ε= (m_n-mp)/m_p&asymp: 1.811× 10^-3. This requires Δ (1/z) ≈-4ε (1/z) so that z=.992 is replaced with z_new=z(1+ 4ε)≈ .9992, which deviates from unity by -8× 10^-4.

The conclusion is that the simple flux tube model for h_eff=h and neutron taking a volume of Compton length, which is definitely different from the general relatistic model, is surprisingly realistic.

See the article Cosmic string model for the formation of galaxies and stars 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.