New Solutions of Tolman-Oppenheimer-Volkov-Equation and of Kerr Spacetime with Matter and the Corresponding Star Models

The Tolman-Oppenheimer-Volkov (TOV) equation is solved with a new ansatz: the external boundary condition with mass M 0 and radius R 1 is dual to the internal boundary condition with density ρ bc and inner radius r i , and the two boundary conditions yield the same result. The inner boundary condition is imposed with a density ρ bc and an inner radius r i , which is zero for the compact neutron stars, but non-zero for the shell-stars: stellar shell-star and galactic (supermassive) shell-star. Parametric solutions are calculated for neutron stars, stellar shell-stars, and galactic shell-stars. From the results, an M-R-relation and mass limits for these star models can be extracted. A new method is found for solving the Einstein equations for Kerr space-time with matter (extended Kerr space-time), i.e. rotating matter distribution in its own gravitational field. Then numerical solutions are calculated for several astrophysical models: white dwarf, neutron star, stellar shell-star, and galactic shell-star. The results are that shell-star star models closely resemble the behaviour of abstract black holes, including the Bekenstein-Hawking entropy, but have finite redshifts and escape velocity v < c and no singularity.


Introduction
In General Relativity, one of the most important applications is to calculate the mass distribution and the space-time metric for a given equation-of-state of a stellar model. , and can be transformed into one ordinary differential equation of degree 2 for M(r) by eliminating ρ(r). The boundary condition is imposed normally at r = 0 with M(0) = 0 and ( ) 0 0 ρ ρ = , where ρ 0 is the maximal density. Then the TOV-equation for M(r) is solved with this boundary condition at r = 0 for M(r) and M'(r), which gives the total mass M 0 (ρ 0 ) and the total radius R(ρ 0 ), and a mass-radius relation M 0 (R).
The predominant view of the neutron stars and stellar black-holes is, that neutron stars obey an equation-of state (eos) of an interacting-fluid model [2], which solutions of the TOV equation up to about M = 3M sun . For larger masses, it is assumed that only a black-hole solution remains. This is based on the so-called Oppenheimer limit for the radius of a compact mass The 2 parameters R and M 0 in the dual outer boundary condition correspond uniquely to the 2 parameters r i and ρ 0 in the inner boundary condition.
With rotation, one has an axisymmetric model in the variables r and θ (azimuthal angle), and has to solve the Einstein equations in these 2 coordinates. In and R 1x and R 1y are the equatorial and the polar radius. As in the TOV-case, here to the 3 parameters R 1y , M 0 and ΔR 1 correspond the 3 inner parameters r iy , ρ 0 and Δr i .
So here we get a 3-parametric solution manifold, and as in the spherical case, for a given total mass M 0 we have to find the stable physical solution. As before, these will be the ones with minimal riy and among them the one with minimal mean energy density: this defines the inner ellipticity Δr i . In all considered cases, it can be shown numerically, that such a (non-trivial) minimum exists.
The paper is organized as follows.

The Kerr Space-Time, Schwarzschild Space-Time, Einstein Equations
Using the Minkowski metric µν η ( ) that the (apparent) singularity at s r r = is missing.
The same is valid for the original Kerr space-time: the denominator 12 ρ has no zeros, there is no singularity in ab g , which makes it more well-behaved numerically.
Alternatively, in Boyer-Lindquist-coordinates: [3]   In the limit 0 a → the Schwarzschild space-time in the standard form (4) emerges.
The Einstein field equations with the above Minkowski metric are: where R µν is the Ricci tensor, R 0 the Ricci curvature, 4 8 G c κ = π , T µν is the energy-momentum tensor, Λ is the cosmological constant (in the following neglected, i.e. set 0), with the Christoffel symbols (second kind) and the Ricci tensor The crucial part of the extended Kerr solution is the expression for the energy-momentum tensor T µν . As usual, one uses the formula for the perfect fluid where P and ρ is the pressure and density, u µ is the covariant velocity 4-vector.
In the Schwarzschild case, when deriving the TOV-equation, one sets the spatial contravariant velocity components to 0: 0 i u = , in the Kerr case the tangential velocity 3 0 u u ϕ = ≠ .
For the velocity one has:  i.e.
If we make the obvious assumption that the star rotates as a whole, i.e. with constant angular velocity, then the moment of inertia I becomes r-dependent, like the mass M: The factor 3 in the integral instead of the usual 4π comes from the dimensionless calculation in "sun units" (see below).

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The amr a also becomes r-dependent: In the relativistic axisymmetric case with rotation with angular velocity ω, u μ has the form [4]: The state equation for the pressure P for the nucleon gas has the form 1 P c γ ρ = or in the dimensionless form with a critical density c ρ and dimensionless pressure 1 P and density 1 ρ ( ) For the horizon, with rotation there is the inner and the outer horizon (M = M 0 )
From now on we skip the index of the dimensionless variables and use the original notation, e.g. r instead of r 1 .
Furthermore, we adopt the Boyer-Lindquist coordinates and the metric tensor (12).
We impose an r-θ-analytic boundary condition for Ai, ∂ r Ai, at r = R 1 (R 1 is the star radius): Ai = 1, ∂ r A0 = 0, ∂ r A2 = 0, ∂ r A3 = 0, ∂ r A4 = 0. For A1, there is no differential boundary condition, as ∂ r A1 is the highest r-derivative, for ρ there is no boundary condition at all, because r is algebraic in the equations, but there is an integral condition:

The Solving Process for the Extended Kerr Space-Time
In addition to the fundamental dual parameters {r i , ρ i } corresponding to {R 1 , M 0 } in the rotation-free TOV-case, in the Kerr-case there is the new fundamental parameter Δr i (inner ellipticity for inner boundary condition), resp. ΔR 1 (outer ellipticity for outer boundary condition), and the angular velocity ω. The outer radii are the latter equality arising from the fact that centrifugal distortion acts only in the x-direction (the y-axis being the rotation axis). The inner radii are correspondingly xi i The r-θ-slicing algorithm with an Euler-step obeys the iterative procedure with slice step size h 1 in r, and step size h 2 in θ, starting with the r-boundary at r = R 1 (slice n = 0).
The transition from slice n to n + 1 proceeds as follows. At slice n all variables and 1-derivatives are known from the previous step, 2-derivatives ∂ rr Ai, and ρ are calculated from the 6 equations.
At slice n + 1 the variables and 1-derivatives are calculated by Euler-formula (or Runge-Kutta) The 2-derivatives ∂ rr Ai, ∂ rr Bi and ρ are again calculated from the 6 significant equations with variables and 1-derivatives inserted from above.
The θ-slicingr-backward algorithm with an Euler-step obeys the iterative procedure with slice step size h 1 in θ as above for r, starting with θ = 0, and solves an ordinary differential equation in r in each θ -step. The boundary condition for the r-odeq is set at r = R 1 (θ) (the outer ellipse radius) with Ai = 1, M = M 0 My0(θ), , where ρ bc is the outer boundary value for the density, ρ bc = 0 for the (non-interacting) neutron-gas in a shell-star and ρ bc > 0, ρ bc = ρ equilibrium for the (interacting) neutron fluid in a neutron star. My0(θ) is the mass-form-factor with the condition ( ) ( ) , where ρ bc = ρ i is the inner boundary value for the density, ρ i is approximately the inner (maximum) density ρ(r i ) from the corresponding TOV-equation, the value must be adapted, so that the resulting total mass is M 0 . For the compact neutron star the inner radius r i (θ) is zero.
In the θ-slicingr-backward algorithm one starts with the outer boundary being the ellipsoid r = R(θ, ΔR 1 ), where ΔR 1 is the outer ellipticity of the star. In the θ-slicingr-forward algorithm one starts with the inner boundary being the ellipsoid r = r i (θ, Δr i ), where Δr i is the inner ellipticity of the star.
At the inner boundary the tangential pressure is uniform, so the density is also uniform and equal to the maximum density, ρ(θ) = ρ i .
In the actual calculation we were using the θ-slicingr-backward algorithm, because here the boundary condition M = M 0 is achieved automatically, when one starts with My 0 (θ) = M 0 .
The odeqs in rconsist of the 6 significant Einstein equations eqR00, eqR11, eqR22, eqR33, eqR03, eqR41 for the six variables A0(r, θ), A1(r, θ), A2(r, θ), A3(r, θ), A4(r, θ), M(r, θ) with θ = θ i and θ-derivatives calculated by Euler-step from the preceding q-slice. For i = 0 i.e. θ = 0 the θ-derivatives are taken from start values for all variables, which normally represent the corresponding TOV-solution (here only A0(r), A1(r), M(r) are non-trivial and do not depend on θ). The odeqs are highly non-linear algebraic differential equations and hard to solve numerically with classical methods for linear odeqs extended by an algebraic equation solver. In the case of a nonlinear odeq-system one uses an Euler or Runge-Kutta method and calculates in each step the highest derivatives with a numerical algebraic equation solver. As an alternative one can use minimization of the least-squares-error in the highest derivatives instead of a numerical algebraic equation solver. Minimization has also the advantage that one can minim-Journal of High Energy Physics, Gravitation and Cosmology ize the complete set of Einstein equations plus the 2 additional continuity equations eqR41, eqR42 in the error goal function instead of the 6 significant equations, which improves the stability of the solution (e.g. in case of degeneracy).
The numerical error of the algorithm is calculated from i.e. the Euclidean norm of the equation values (the right side of the Einstein equations being 0). The error is calculated over the lattice {r i , θ j } as median, mean or maximum. In the internal loop of the algorithm over r i at fixed θ j , the solution of the algebraic discretised Einstein-equation is achieved by square-root error minimization, so it is essential to avoid singularities, e.g. at the horizon and the pseudo-singularity at θ = 0. This is achieved by selecting appropriate analytic convergence factors for the (left side of) the Einstein equations. As the equations are to be zeroed for the solution, the convergence factors do not change the solution of course, but they cancel the numerical singularities, which could otherwise jeopardize the numerical convergence of the algorithm.
The actual calculation was carried out in Mathematica using its symbolic and numerical procedures. In the first stage, the Einstein equations were derived from the ansatz for g μν from section 2 and simplified automatically. The arising complexity of the equations is such, that it is practically impossible to handle For the second numerical stage we tried several slicing algorithms, and the best alternative proved to be the θ-slicingr-forward algorithm implemented by hand in Mathematica. The solution of the resulting odeq in each r-step was calculated using NDSolve. Also, for every star model and parameter set, the TOV solution with ω = 0 a = 0 was calculated first with the algorithm and compared with the exact TOV solution.

The TOV Equation as the Limit ω → 0 for the Extended Kerr Space-Time
In the Schwarzschild spacetime ω = 0 and a = 0, we have spherical symmetry, no dependence on θ, then the TOV-equation can be derived from the remaining non-trivial Einstein equations eqR00, eqR11, eqR22, eqR41.
The TOV-equation is in the standard form: Journal of High Energy Physics, Gravitation and Cosmology (13) and using r s where M t is the total mass, furthermore In order to make the variables dimensionless, one introduces "sun units" ( ) 16  where r ss Schwarzschild-radius of the sun, ρ s the corresponding Schwarzschild-density and P s the corresponding Schwarzschild-pressure.
In "sun units" TOV-equation transforms into with the normalized mass M 1 (r 1 ), and 0 M R = for non-interacting Fermi-gas and for an interacting Fermi-gas:

The Equation of State for an (Non-Interacting) Nucleon Gas
Here,

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( ) ( ) , where x F is the Fermi-angular-momentum, n the particle density ( ) The resulting approximate equations of state for P are valid for the density ρ and the critical density ρ c The full expression for P, including temperature T, is as follows ( [5], chap.15). Here, we use dimensionless variables (r 1 distance unit de-Broglie-wavelength λ c , V 1 volume unit 3 c λ , n 1 particle density unit , the resulting particle density is From this relation the chem. potential μ 1 can be calculated, an approximation formula is Finally, the resulting pressure (=energy density) ( ) Below a 3D-diagram of ( ) Here kT is in E 0 units, and one sees the dependence 1 P k γ ρ = except on the left side, when kT reaches the magnitude of 1 Gev (T = 10 10 K).

The Equation of State for an (Interacting) Nucleon Fluid
For the interacting nucleon gas we take into account the nucleon-nucleon-potential in the form of a Saxon-Woods-potential modeled on the experimental data: [7]- [13] ( )  MeV.
The Saxon-Wood potential is shown in Figure 2 below.
The pressure of the interacting nucleon fluid becomes then The experimental data used here are those from [7], and are shown in Figure   3.
And the hard-core potential from the lattice calculation Reid93 [10] is shown in Figure 4.
Both potentials are fitted with a double Saxon-Woods-potential V nn in Figure   5:

Maximum Omega-Values in Kerr-Space-Time
We consider here a rotation model with constant angular velocity ω. With this model the resulting 4-velocity u μ has the form [4] [14] [15]: The maximum values for ω are calculated from the minimal zeros in omega of the denominator in u 0 from (9a), minimized over r 1 and th in their respective re- The resulting value is , where α f is the form-factor in the moment of inertia I 1 . The star parameters mass M 0 and radius R 1 , which enter the outer boundary condition determine completely the solution. In general, there will be an inner radius r i > 0 with the maximum density As we will see, this outer boundary condition together with allowing r i > 0 changes dramatically the resulting manifold of physical solutions.

The TOV-Equation: The Parametric Solution and Resulting Star Types
By setting-up a parametric solution of the TOV-equation one gets a map of possible physical solutions, i.e. possible star structures. As parameters one can use either (M 0 , R 1 ) in the outer boundary condition at r 1 = R 1 or the dual parameter pair (r i , ρ bc ) in the inner boundary condition r 1 = r i .
The pure neutron Fermi-gas model yields for compact neutron stars a maximum mass of M maxc = 0.93M sun , which is in disagreement with observations. Therefore, at least for compact neutron stars, a model of interacting neutron fluid must be used. In 6.2 above we have described a Saxon-Wood-potential model for the nucleon-nucleon interaction, which seems to fit the experiment and the theory in the best way. There will be a critical density (dependent on temperature of course), where a transition from interacting fluid to Fermi-gas takes place, it is plausible to set this density equal to the Saxon-Wood critical density 0.0417 We made calculations with the TOV-equation using these two models for neutron-based stars and we came to the conclusion that compact neutron stars with mass M 0 ≤ 3.04M sun consist of interacting neutron fluid and neutron shell-stars for M 0 ≥ 5M sun obey the Fermi-gas model. The underlying calculation is the Mathematica-notebook [6].

J. Helm Journal of High Energy Physics, Gravitation and Cosmology
This approach yield results, which are described below.
The admissible mass range ends, where the thickness of the shell above the Schwarzschild-radius becomes very small (minimum 0.01).

So in total the R-M-relation for neutron stars becomes
The maximum mass for a repulsive-hardcore-model for the equation-of-state DD2 [16] is 2.42M sun , from our mapping we have the maximum compact neutron star mass of M maxc = 3.04M sun .  The actual theoretical limit for neutron star core density is ρ max = 3.5 × 10 15 g/cm 3 = 0.199 in sun-units [8] [9].

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The limit for ρ bc reached in our mapping is only ¼ of this ρ bc = ρ bcmax = 0.0544, due to the subluminal-sound-condition and the use of an (attractive) nucleon-nucleon-potential for the nucleon-fluid instead of a pure repulsive-hardcoremodel.
The classical argument for the collapse of a neutron star to a black-hole for ρ bc > ρ max , dating back to Oppenheimer [1], is invalidated here by the simple introduction of shell-star models, where r i > 0, and therefore there is no mass at the center, which means physically, there is only a very diluted nucleon gas there. Stellar shell-stars (stellar black-holes) Journal of High Energy Physics, Gravitation and Cosmology We assume that the underlying equation-of-state state for stellar shell-stars is the Fermi-gas of nucleons with the low-density limit of ( ) We make a further plausible assumption that the "edge" of the solution mapping are the physically stable solutions, i.e. the R-M-relation for stellar shell-stars. The edge in this case consists for fixed ρ bc < 0.0417 = ρ oc of solutions with maximum r i (because then the average density in the shell is lowest) and for ρ bc = 0.0417 = ρ oc it consists of the solutions (M 0 , r i , R 1 ) at the right boundary ( open, but the "thinning-out" of the solutions for small ρ bc and large r i makes it physically plausible (see Figure 11 [6] below).
The resulting R-M-relation is practically linear and has a maximum mass value of M max = 81.3M sun . (Figure 12(a)).
And the corresponding relative shell thickness dR rel = dR/M is (Figure 12(b)) and the relative Schwarzschild-distance dR srel = (R − M)/M is ( Figure 13) The inverse of dR srel gives roughly the light attenuation factor of {1.7, …, 20.}.
Taken the attenuation factor and the small relative shell thickness of around 0.02, these stellar shell-stars have approximately the properties expected of a genuine black-hole, when measured from a distance 1 r R .   , which is identical to the Bekenstein-Hawking entropy with the factor (ln2)4/π = 0.882.

Galactic (supermassive) shell-stars
The mean density of a black-hole scales with its radius R like i.e. for supermassive black-hole with M = 10 6 M sun we have In the following we use the abbreviation MM sun = 10 6 M sun .
Therefore it is plausible to try a parametric mapping with the white-dwarf equa-  MM sun . Third, a stable solution for a fixed mass will have the highest possible maximum density ρ bc and that will lie on the "ridge". So one can calculate the R-M-relation following the "ridge". The resulting R-M-relation is as follows (Figure 15).
And the inner radius is ( Figure 16).
The R-M-relation is almost linear, as expected, and goes up to 50MM sun .

( )
is the relative thickness (Figure 17), and shows, that the shells are very thin indeed, with a minimum of 0.001. The fourth diagram shows the relative Schwarzschild-distance ( Figure 18) has a minimum at {M 0 , dR srel } = {7, 0.00142857}, so that its reciprocal value (approximate light attenuation factor) is around 700. So the overall result is, that the supermassive shell-stars become ever thinner shells, while the distance from the Schwarzschild-horizon is increasing.

The TOV-Equation: A Case Study for Typical Star Types
In the nearly-rotation-free case the solution of the TOV-equation was calculated for 4 models in sun units with r s = Schwarzschild radius J. Helm Journal of High Energy Physics, Gravitation and Cosmology         Here the radius R 1 is reached, when M'(r 1 = R 1 ) = 0, i.e. ρ(R 1 ) = 0.
The "naive" mean density is here

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TOV-solution for rho (in 10 −12 units, Figure 27), M (in 10 6 units, Figure 28) in r (in 10 6 units), is: Here there is an internal "hole" with a radius r i = 4.356 × 10 6 , maximum ρ = 4.934 × 10 −12 at r i . The inner radius r i lies a little below the Schwarzschild-radius r s = M 0 . The relative shell thickness  Furthermore, r i is little sensitive to the temperature up to T = 10 7 K. As for a stellar black hole, when R converges to r s = M 0 , so does the inner radius r i , and there is no physical solution (with positive ρ and M) for a boundary within the horizon.

The Three Star Models for Kerr-Space-Time with Mass and Rotation
The calculation of Kerr-space-time with mass and rotation was carried out for 3 star models: γ = gam, gam1, gam2 is the exponent, infac is the moment of inertia factor α f , epsi is the singularity cancellation parameter with limit(epsi) = 0 introduced to improve the numerical stability in singularities r i = riact is the polar inner radius R y Δr i the inner ellipticity is the difference between the polar r iy and the equatorial inner radius r ix , Δr i = r iy − r ix ΔR 1 Is the outer ellipticity, with outer radii R x1 = R 1 − ΔR 1 and R y1 = R 1 rilow is the minimal radius r 1 reached in the solution ρ bc = rhobcx is the boundary condition density dthrel is the maximum relative difference of a value dependent on θ, e.g.
The density distribution is similar to the TOV-case but with a decrease in θ-direction.  The rotation results show very small flattening in the polar direction ofdthrel = 0.00118. The neutron star behaves like a fluid because of its "viscosity", that is, its nuclear interaction and becomes "pumpkin-like".
The outer Kerr-horizon is r + = 15.21. The underlying calculation is the Mathematica-notebook [20], the results in [19].
The outer ellipticity ΔR 1 is at first a free parameter and calculated from a case-study of minimal mean energy density to ΔR 1   The two significant non-spherical features are the relative shell thickness variation dthrel(dR 1 ) and the relative inner ellipticity dthrel(r i ). The first depends roughly linearly on the outer ellipticity ΔR 1 , plus the value at ΔR 1 = 0 (dthrel(dR 1 ) = 0.0241), which is results from rotation. The second, dthrel(r i ), is almost equal to the relative outer ellipticity dthrel(r i ), plus the small amount at ΔR 1 = 0 (dthrel(R 1 ) = 0.00123).
The density distribution is shown in Figures 32-34.
The mass distribution is shown in Figure 35, Figure 36.
The physical mass distribution ends at the inner boundary at r i = 16.7, where the density jumps to ρ = 0.    A remarkable result, distinct from the case of the neutron star, is the shape with rotation. The energy-minimal stellar shell-star behaves like a ball of neutron gas (negligible interaction) and decreases slightly its equatorial radius, so that, speaking naively, the increased gravitation counteracts the centrifugal force, the shell-star becomes "cigar-like", with the shell thickness approximately constant.  Typical rotating galactic shell-star This is modelled (approximately) on the central black-hole in the Milky Way with mass M 0 = 4.36 mega-sun-masses (MM s ), radius R 1 = 4.38 mega-sun-Schwarzschild-radii (13.14 × 10 6 km, Mr ss ) [21].
The underlying calculation is the Mathematica-notebook [22], the results in [23].
The outer Kerr-horizon is r + = 4.26Mr ss .
In order to maintain numerical performance, we are using for mass and distance 10 6 (mega) units 10 6 M s and 10 6 r ss and for density 10 −12 (mega −2 ) unit 10 −12 ρ s .
Like in the case of the stellar shell-star, the outer ellipticity ΔR 1 is at first a free parameter and calculated from a case-study of minimal mean energy density to ΔR 1 = −2 dTOV, where dTOV is the shell thickness of the spherical shell-star dTOV = 0.057.
The full parameters are: In contrast to the stellar shell-star, here the relative variation of the shell thickness for the spherical-outer-boundarysolution is smaller by a factor of 20 as compared to the minimal solution with a high outer ellipticity, so here there is a dependence of the shell thickness on the ellipticity.
The density distribution is shown in Figures 37-39. The density distribution increases in th-direction. The mass distribution is shown in Figure 40, Figure 41.
The physical mass distribution ends at the inner boundary at r i = 4.46456, where the density jumps to ρ = 0. The fit extrapolates it to lower r-values.
The maximum distance from the horizon is max(r 02e ) − r + = 0.125, therefore the minimal light energy attenuation is roughly 4.262/0.125 = 34, it means that visible green light of 0.514 μm is shifted to 17 μm into far-infrared.   The galactic shell-star has all its mass concentrated within a thin shell (dR 1 = 0.0362) which has its inner radius inside and its outer radius outside its   M 0 /min(R 1 (θ)) = 0.9971 and the attenuation factor 1/(1 − M 0 /min(R 1 (θ))) = 345, it means that x-ray-radiation from in-falling matter from the accretion disc with an energy of 5 keV and λ = 0.2 nm is shifted to λ = 69 nm, that is into hard UV-radiation.

Experimental Evidence with Recent LIGO and X-Ray Measurements
In November 2018, the LIGO cooperation published the newest statistics of neutron stars and black holes, based on gravitational waves and x-ray measurements [24].
The resulting mass distribution for black-holes and neutron stars is shown in Figure 42 [24].
From these results, we can deduce a confirmed mass range for neutron stars of Journal of High Energy Physics, Gravitation and Cosmology The compact neutron star with M 0 = 0.932M sun , R 1y = 2.8372r ss = 8.51 km, R 1x = 2.8391r ss , ω = 0.1087, has the relative ellipticity of dthrel = 0.00118. The neutron star behaves like a fluid because of its "viscosity", that is, its nuclear interaction, and becomes slightly "pumpkin-like".
The stellar shell-star behaves like a ball of neutron gas (negligible interaction) and decreases slightly its equatorial radius, so that, speaking naively, the increased gravitation counteracts the centrifugal force, the shell-star becomes "cigar-like", with the shell thickness approximately constant.
The redshift is roughly 345. The galactic shell-star is a shell object with a thin mass shell (ΔR = 0.0352Mr ss ) situated close above its outer Kerr horizon r + = 4.26Mr ss . The polar radius is smaller than the equatorial radius, so the outer shape and the inner shape are both pancake-like.
The overall result is, that the introduction of numerical shell-star solutions of the TOV-and Kerr-Einstein-equations creates shell-star star models, which mimic closely the behaviour of abstract black holes and satisfy the Bekens-tein_Hawking entropy formula, but have finite redshifts and escape velocity v < c, no singularity, no information loss paradox, and are classical objects, which need no recourse to quantum gravity to explain their behaviour.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.