Does the notion of gravastar make sense in the TGD Universe?

Mark McWilliams asked for my TGD based opinion about gravastar as a competing candidate for the blackhole (see this and this). The metric of the Gravastar model would be the de-Sitter metric in the interior of the gravastar. The density would be constant and there would be no singularity at the origin. The condition ρ=-p would be true for de-Sitter and there would be analogy with dark energy, which in TGD framework contributes to galactic dark matter identified as classical volume and magnetic energies for what I call cosmic strings, which are 4-surfaces with string world sheet as M⁴ projection. The condition ρ=p would hold true for ultrarelativistic matter at the surface, which indeed as a light-like metric if the infinite value of the radial component g_rr of the Schwartschild metric is deformed to a finite value at the horizon. In the exterior one would have ρ=p=0.

TGD suggests a model of a blackhole-like object as volume filling monopole flux tube tangle, which carries a constant mass density, which can be interpreted as dark energy as a sum of classical magnetic and volume energies. Quantum classical correspondence forces us to ask whether the description in terms of sequences of nucleons and in terms of classical energy are equivalent or whether the possibly dark nucleons must be added as a separate contribution. I have discussed the TGD based model of blackhole-like objects in cite{btart{Haramein.

It became as a surprise for me that the gravastar could serve as a simple model for this structure and describe the space-time sheet at which the monopole flux tube tangle is topologically condensed. TGD also suggests that the surface of the star carries a layer of M₈₉ matter consisting of scaled variants of ordinary hadrons with mass scale which is 512 times higher than that for ordinary hadrons. This would be the counterpart for the exotic matter and the surface of the gravastar cite{btart{Haramein. This model predicts that the nuclear fusion at the core of the star is replaced with a transformation of M₈₉ hadrons to ordinary hadrons. This would explain the energy production of the star and also the stellar wind and question the structure of the interior. I have proposed that it could be a quantum coherent system analogous to a cell.

Consider now the TGD counterpart of the gravastar model at quantitative level.

1. The metric of AdS_n (anti de-Sitter) resp. dS_n (de-Sitter) can be represented as space-like resp. time-like hyperboloid resp. of n+1-dimensional Minkowski space with one time-like dimension. The metric is induced metric

dx₀²-∑ _i=1ⁿdx_i² ,

with metric tensor deducible from the representation

x₀²-∑ _i=1ⁿx_i²= ε α ² ,

as a surface. Here one has ε =-1 AdS_n and ε=1 for dS_n.

It should be warned that the Wikipedia definition of the dS_n (see this) contains the right-hand side with a wrong sign (there is ε=-1 instead of ε=1) whereas the definition of AdS_n (see this) is correct. For n=4 this could realize AdS₄ resp. dS₄ as a space-like resp. time-like hyperboloid of 5-D Minkowski space.

In TGD this representation as surface is not possible as such. One can however compactify the 5:th space-like dimension and represent it as a geodesic circle of CP₂. dx₅² is replaced with R²dφ² and x₅² with R²φ². The contribution of S¹ to the induced metric is very small since R corresponds to CP₂ radius. The space-time surface would be defined by the condition

a²= R²φ²+ε α² ,

where a²=t²-x²-y²-z² defines light-cone proper time a. In TGD it would be associated with the second half of the causal diamond (CD). A more convenient form is following

R²φ²= a²-ε α ² ,

where a is the light-cone proper time coordinate of M⁴. This requires a²≥ε α². For ε=1 this implies a²≥ α². For ε=1 one has a²≥ -α² so that also space-like hyperboloids are possible.

If the embedding is possible, one obtains an infinite covering of S¹ by mass shells a²= R²φ_n²+ε α², where one has φ_n= φ +n2π. For φ → ∞ one has a → nR. Hyperboloids associated with φ_n define a lattice of hyperboloids at this limit, a kind of time crystal.

If the classical action is Kähler action of CP₂, this surface is a vacuum extremal since the CP₂ projection is 1-dimensional. If also the contribution M⁴ Kähler action to Kähler action suggested by the twistor lift of TGD is allowed, the situation the action is instanton action and vanishes although the induced M⁴ Kähler form does not vanish and defines self dual abelian field. It is not quite clear whether this is vacuum extremal anymore.

If the Kähler action vanishes, volume action is the natural guess for the classical action and minimal surface equations are indeed satisfied if S¹ is a geodesic circle. The mass density associated with this action would be constant in accordance with the de-Sitter solution.

Consider next the induced metric. One has

φ_n= n2π + [(a/R)²-ε (α/R)²]^1/2 .

This gives Rdφ_n/da= +/- a/[a²-ε α²]^1/2. Note that a²≥ ε α² is required to guarantee the reality of dφ/da. The g_aa component of the induced metric (Robertson-Walker metric with k=-1 sub-critical mass density) is

g_aa=1-R²(dφ_n/da)²= 1- a²/(a²+ε α²)= εα²/(a²+εα²) . It is useful to consider AdS₄ and dS₄ separately.

1. For AdS₄ with ε=-1, the reality of dφ/da implies a²>-α² implying g_aa<0 so that the induced metric has an Euclidean signature. This is mathematically possible and CP₂ type extremals with Euclidean signature are in an important role in the TGD based model of elementary particles. What Euclidian cosmology could mean physically, is however not clear.
2. For dS₄ with ε=1, dφ/da is real for a²+α²>0 implying a²≥ -α². This allows all time-like hyperboloids and also some space-like hyperboloids. One has

g_aa=1-R²(dφ_n/da)²= 1- a²/(a²+α²)= α²/(a²+α²) .

g_aa is positive in the range allowed by the reality of dφ/da.

The mass density of Robertson-Walker cosmology is obtained from the standard expression of the metric (note that one has dt²=g_aada²)is given by

ρ =(3/8πG)[[(da/dt)/a)²-1/a²]= (3/8πG)[1/(g_aaa²) -1/a²]=(3/8πG α²) .

The mass density is constant and could be interpreted in terms of a dynamically generated cosmological constant in GRT framework. This is not what happens usually in the Big Bang cosmology but would conform with a model of a star in an expanding Universe. Somewhat surprisingly, 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 cite{allb/tgdgrt} 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 cite{btart/Haramein}.

See the article Does the notion of gravastar make sense in the TGD Universe?.

For a summary of the 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.

Source: https://matpitka.blogspot.com/2025/04/does-notion-of-gravastar-make-sense-in.html