About the physical interpretation of gravastar in the TGD framework

The gravastar model for a blackhole-like object describes stellar interior as de-Sitter space and exterior as Schwarscild metric. The surface of the blackhole is predicted to carry exotic phase. In the previous post (see this) I demonstrated de-Sitter metric allows a realization as space-time surface and that blackhole-objects and in fact all stars could be modelled in this way.

In the sequel I will consider the physical interpretation of de-Sitter space-time represented as a 4-surface in the TGD framework.

1. In TGD, twistor lift predicts cosmological constant Λ with a correct sign (see this and this). The twistor lift of TGD predicts that Λ= 3/α², where α is a length scale is dynamical and has a spectrum. The mass density ρ is associated with the volume term of the dimensionally reduced action having 3/(8π Gα²) as coefficient. Also Kähler action is present and contains CP₂ part and possibly also M⁴ part.

   Λ is not a universal constant in TGD but depends on the size scale of the space-time sheet. The naive estimate is that it corresponds to the size scale of the space-time sheet associated with the system or its field body of the system, which can be much larger than the system.

   p-Adic length scale hypothesis suggests that apart from a numerical constant the scale L_Λ=(1/Λ)^1/2 equals to the p-adic length scale L_p characterizing the space-time sheet. If p-adic length scale hypothesis L(k)= p^1/2, where the prime p satisfies p∼ 2^k, it implies L(k)= 2^(k-151)/2 L(151), L(151)∼ 10 nm.

2. How does the average density of an astrophysical object or even smaller object relate to the vacuum energy density determined by Λ. There are two options: vacuum energy density corresponds to an additional contribution to the average energy energy density or determines it completely in which case one must assume quantum classical correspondence stating that the quantal fermionic contributions to the energy and other conserved quantum numbers are identical with the classical contributions so that there would be kind of duality. This would hold true only for eigenvalues of charges of the Cartan algebra.
3. One can assign to the cosmological constant a length scale as the geometric mean

   l_Λ= (l_P L_Λ)^1/2 ,

   where Planck length is defined as l_P= (ℏ G)^1/2. One obtains therefore 3 length scales, Planck length, the big length scales L_Λ and their geometric mean l_Λ.

4. What is the relationship to the spectrum of Planck constants predicted by the number theoretical vision of TGD? If one replaces ℏ with ℏ_eff=nh₀, one obtains a spectrum of gravitational constants G and of Planck length scales. CP₂ size scale R ∼ 10⁴l_P is a fundamental length scale in TGD. One can argue that G is expressible in terms of R=l_P as G_eff=l_P/(ℏ_eff^1/2 and that the CP₂ length scale satisfies R=l_P for the minimal value h₀ of h_eff so that one obtains G_eff= R/h_eff^1/2. For h₀ one obtains the estimate h= (7!)²h₀ in terms of Planck constant h. This would predict a hierarchy of weakening values of G.

   Note that G=l_P/ℏ_eff^1/2 would predict the scaling l_Λ∝ ℏ_eff^1/4. Gravitational Planck constant ℏ_gr= GMm/\beta₀ for the system formed by large mass M and small mass m has very large values.

It is interesting to look at what values of l_Λ are associated with L_Λ , characterizing the size scale of a physical system or possibly of its field body.

1. For the "cosmological" cosmological constant one has L_Λ∼ 10⁶¹l_P giving l_Λ∼ 10^31.5l_P ∼ 2× 10^-4 m. This corresponds to the size scale of a neuron. L_Λ could characterize the largest layer of its field body with a cosmological size scale.
2. A blackhole with the mass of the Sun has Scwartschild radius r_S= 3 km. Λ=r_S gives l_Λ∼ 2.19× 10^-16 m. The Compton length of the proton is l_p=2.1× 10^-16 m. This estimate motivated the proposal that stellar blackholes could correspond to volume filling flux tubes containing a sequence of protons with one proton per Compton length of proton. This monopole flux tube would correspond to a very long nuclear string defining a gigantic nucleus. This result conforms with quantum classical correspondence stating that vacuum energy density corresponds to the density of fermions.
3. One can also look at what one obtains for the Sun with radius R_S= 6.9× 10⁸ m, which is in a good approximation 100 times the radius R_E= 6.4× 10⁶ m of the Earth. l_Λ scales up by the ratio (R_S/r_S)^1/2 to l_Λ ∼ 5.7× 10²× l_P∼ 1.3× 10^-14 m. This corresponds to a nuclear length scale and the corresponding particle would have a mass of about 17 MeV. Is it mere coincidence that there is recent very strong evidence (23 sigmas!) from the so called Ytterbium anomaly (see this) for so called X boson with mass 16-17 MeV (see this and this).

   The corresponding vacuum energy density ℏ/Λ⁴ would be about 8× 10³⁸ m_p/m³. This is 12 orders of magnitude higher than the average density .9× 10²⁷ m_p/m³ of the Sun. Since l_Λ ∝ L_Λ^1/2 and ρ ∝ l_Λ^-4∝ L_Λ^-2 one obtains L_Λ≥ 10¹²R_S∼ 10²⁰ m ∼ 10⁵ ly, which corresponds to the size scale of the Milky Way.

   The only reasonable interpretation seems to be that L_Λ characterizes the lengths of monopole flux tubes which fill the volume only for blackhole-like objects. The TGD based model for the Sun involves monopole flux tubes connecting the Sun with the galactic nucleus or blackhole-like object (see this){Haramein. In this case the density of matter at the flux tubes would be much higher since protons would be replaced with their M₈₉ counterparts 512 higher mass. For this estimate, the vacuum energy density along flux tubes would be the average density of the Sun. At least two kinds of flux tubes would be required and this is consistent with the notion of many-sheeted space-time.

   The proposed solar model in which the solar wind and energy would be produced in the transformation of M₈₉ nuclei to ordinary M₁₀₇ nuclei allows to consider the possibility that the Sun and stars are blackhole-like objects in the sense that the interior correspond contains a volume filling flux tube tangle carrying vacuum energy density which is the average value of the solar mass density. I have considered this kind of model in (see this).

   One can wonder whether the scaling up the value of h to h_eff help to reduce the vacuum energy density assigned to the Sun? From l_Λ∝ ℏ_eff^1/4 the density proportional to ℏ_eff/l_Λ⁴ does not depend on the value of h_eff.

To sum up, TGD could allow the interior of the gravastar solution as a space-time surface and this would correspond to the simplest imaginable model for the star. It is not clear whether Einstein's equations can be satisfied for some action based on the induced geometry but volume action is an excellent candidate even if cosmological constant is not allowed. In the TGD framework, the cosmological constant would correspond to the volume action as a classical action.

Schwartschild metric as exterior metric is representable as a space-time surface (see this) although it need not be consistent with any classical action principle and it could indeed make sense only at the quantum field theory limit when the many-sheeted space-time is replaced with a region of M⁴ made slightly curved. The spherical coordinates for the Schwartschild metric correspond to spherical coordinates for the Minkowski metric and Schwartschild radius is associated with the radial coordinate of M⁴. The exotic matter at the surface of the star as a blackhole-like entity could have a counterpart in the TGD based model of star (see this).

See the article Does the notion of gravastar make sense in the TGD Universe? or the chapter Some Solar Mysteries.

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.