## Tuesday, December 12, 2023

### 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 mp is M/mp (here mp≈ mn 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 gtt from flat Minkowski metric is given by Δ gtt = - Φgr, where one has

Φgr (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

Egr= -∫ρ(r)Φgr (r)dV=-6GM2/5R .

For R= rS= 2GM this gives

Egr=- 6GM2/10GM = -(3/5)M .

Therefore the observed mass Mobs using mp as a unit is given by

Mobs=Etot/m=(2/5) (M/mp) .

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 Lp = h/mp.

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

Lp(heff)= y Lp .

4. The total mass M using mp 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 Vp taken by a single proton:

M/mp= V/Vp= (4/3z) × (rS/Lp)3) y-3 .

Taking into account gravitational potential energy, one obtains

Mobs/m= (2/5)(V/Vp)=(8/15z) × (rS/Lp)3 y-3 .

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

In the above formula Mobs/m on r.h.s decreases slightly in mp→ mn and 1/Lp3 on l.h.s increases slightly in mp→ mn. 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= (mn-mp)/mp&asymp: 1.811× 10-3. This requires Δ (1/z) ≈-4ε (1/z) so that z=.992 is replaced with znew=z(1+ 4ε)≈ .9992, which deviates from unity by -8× 10-4.

The conclusion is that the simple flux tube model for heff=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.