1

Neutron stars and non-singular black-holes as solutions of the

TOV-equationJan Helm

Table of contents

1. Neutron stars: observational and simulation data, LIGO-results2. Black-holes: observational and simulation data, LIGO-results3. Theoretical background: Einstein equations, orbit equations4. Black-hole paradoxes5. TOV equation and generalized Kerr-spacetime and a solution ansatz6. Equation-of-state for neutron stars7. Neutron stars: TOV results and examples8. Stellar shell-stars: TOV results and examples9. Supermassive shell-stars: TOV results and examples10. Newtonian stars: white dwarves, sun-like stars, and supermassive stars11. Conclusion

2

Introduction

1. Hawking-Penrose black-holeHP-black-hole has an exceptional status among astrophysical objects:- it is a quantum object described by a wave function, not a classical object- the no-hair-theorem is valid, so it has no internal structure- the escape velocity ve=c : whatever comes in, never comes out- it violates the conservation-of-information theorem, which is valid in quantum mechanics- the external (remote non-moving observer) fall time to the horizon is infinite (log-divergent)

2. New ansatz for TOV (non-rotating General-Relativistic-stars) and extended Kerr-spacetime (rotating GR-stars)-boundary condition at r=R outside boundary, instead of r=0 at the center-results for (compact) neutron stars: the same as before Mmax≈3Msun eos=interacting neutron-fluid-results for stars 5 Msun <=M<=80 Msun : shell-star=non-singular black-hole with R>rs thin shell outsidehorizon, equation-of-state(eos)= (non-interacting) neutron-Fermi-gas-results for supermassive objects M>106 Msun : : shell-star=non-singular supermassive black-hole with R>rs

very thin shell outside horizon eos= electron-Fermi-gas

3. New paradigma for black-holesReal black-holes are classical objects, with a well-defined eos- escape velocity ve<c but ve≈c , radiation possible, although strongly red-shifted- all black-holes GR-effects still there: gravitational lensing, accretion, ergosphere, horizon= rs

-paradoxes resolved: no information loss, ve<c , finite fall time, temperature like neutron star (T>>THawking)-Hawking-Bekenstein entropy and the holographic (membrane) principle still valid

4. Overall scheme for astrophysical objects now valid without exception- all astrophysical objects are classical objects (sharp orbits, no wave function, no quantum-uncertainty)-they obey an eos P=P(ρ,T) and the pressure-balance condition Peos=Pgr

neutron stars: compact, eos(neutron fluid)stellar black-holes: shell star, eos(neutron Fermi-gas) P=K* ργ

white dwarves: eos(electron Fermi-gas) P=K* ργ

sun-like stars: eos(ideal H-gas) P=K*kB T ρ supermassive (giant) star: eos(radiation pressure)quasar (supermassive black-hole): membrane, eos(electron Fermi-gas) P=K* ργ

3

1. Neutron stars: observational and simulation data, LIGO-results

A neutron star is a type of stellar remnant that can result from the gravitational collapse of a massive starduring a Type II, Type Ib or Type Ic supernova event. Such stars are composed almost entirely of neutrons.Neutron stars are very hot and are supported against further collapse by the Fermi-pressure of a neutron gas orfluid.

Neutron star

parameter value formula commentmass M 1 – 3 Msun compact

3 –4 Msun shell starobservation,eTOV

radius R ≈ 12km density ρ 106g/cm3 (surface)

8*1014g/cm3 (core)=0.045 ρs(Msun) comp. ρs(Msun)=1.76*1016 g/cm3

surface gravity g 7*1012m/s2 comp. g(Earth)=9.81 m/s2

escape velocity ≈ c/3 bind.en. Egr Egr (2 Msun)= 0.6 Msun

magnetic field B ≈ 108 Tesla comp. B(Earth)=60μT temperature T Tinit=1011K, stable Tcool=106Krotation f=ω/2π f=1Hz – 700Hz eos neutron fluid APR Akmal-Pandharipande-Ravenhallradiation radio (magnetic pulsars)

x-ray accretion (x-ray pulsars)binary star 5%progenitor star M≈ 10 Msun supernova II

4Neutron stars

Structure

Accreting X-ray pulsarAn X-ray pulsar consists of a magnetized neutron star in orbit with a normal stellar companion and is a type ofbinary star system. The magnetic-field strength at the surface of the neutron star is typically about 108 Tesla.

Magnetic radio pulsarRadio pulsars (rotation-powered pulsars) and X-ray pulsars exhibit very different spin behaviors and havedifferent mechanisms producing their characteristic pulses although it is accepted that both kinds of pulsar aremanifestations of a rotating magnetized neutron star. The rotation cycle of the neutron star in both cases isidentified with the pulse period.

5Neutron stars

Magnetar – soft gamma repeater

A rupture in the crust of a magnetar can trigger high-energy eruptions, radiated as soft gamma rays.

6Neutron stars

PSR B1913+16: binary neutron-star-pulsar, decreasing period due to gravitational waves

Orbital decay of PSR B1913+16.[3] The data points indicate the observed change in the epoch of periastron withdate while the parabola illustrates the theoretically expected change in epoch according to general relativity.( R.A.Hulse & J.H.Taylor 1993 Nobel Prize )

7Neutron star merger (LIGO GW170817)

GW170817

The GW170817 signal as measured by the LIGO and Virgogravitational wave detectors

Event typeGravitational wave event, neutron starmerger

Date 17 August 2017

Instrument LIGO, Virgo

Constellation Hydra

Right ascension 13h 09m 48.08s[1]

Declination −23° 22′ 53.3″[1]

Epoch J2000.0

Distance 40 megaparsecs (130 Mly)

Redshift 0.009

Preceded by GW170814

8Neutron star merger (LIGOGW170817)Technically, there were three separate observations, and strong evidence that they came from the sameastronomical source:

GW170817, which had a duration of approximately 100 seconds, and shows the characteristics inintensity and frequency expected of the inspiral of two neutron stars. Analysis of the slight variation inarrival time of the GW at the three detector locations (two LIGO and one Virgo) yielded an approximateangular direction to the source.

GRB 170817A, a short (~ 2 seconds duration) gamma-ray burst detected by the Fermi and INTEGRALspacecraft beginning 1.7 seconds after the GW merger signal.[1][13][14] These detectors have very limiteddirectional sensitivity, but indicated a large area of the sky which overlapped the gravitational waveposition. It has long been theorized that short gamma-ray bursts are caused by neutron star mergers.

AT 2017gfo (originally, SSS17a), an optical astronomical transient found 11 hours later in the galaxyNGC 4993[15] during a search of the region indicated by the GW detection. This was observed bynumerous telescopes, from radio to X-ray wavelengths, over the following days and weeks, and showedthe characteristics (a fast-moving, rapidly-cooling cloud of neutron-rich material) expected of debrisejected from a neutron-star merger.

Gravitational wave detection

9Neutron star merger (LIGO GW170817)Electromagnetic follow-up

Hubble picture of NGC 4993 with inset showing GRB 170817A over 6 days. Credit: NASA and ESAOptical lightcurves

The change in optical and near-infrared spectra

10

2. Black-holes: observational and simulation data, LIGO-results

Black hole

A black hole is a region of space from which nothing, not even light, can escape. It is the result of thedeformation of spacetime caused by a very compact mass. Around a black hole there is an undetectable surfacewhich marks the point of no return. This surface is called an event horizon. It is called "black" because itabsorbs all the light that hits it, reflecting nothing, just like a perfect black body in thermodynamics. [1]

Stellar black-holeparameter value formula commentmass M 5 - 80 Msun shell eTOVradius R R≈rs d(shell) ≈ 0.02rs R≈rs(1+ε) ε≈ 0.05 , eTOV density ρ (0.02-0.04) ρs(Msun) eTOV ρs(Msun)=1.76*1016 g/cm3

surface gravity g 3*1012m/s2 for M=10 Msun ≈ c2/ rs comp. g(nstar)= 7*1012m/s2

escape velocity ≈ c v=c(1- ε/2) eTOV bind.en. Egr M c2 (1- ε /(1+ε) ) eTOV magnetic field B probably weak neutron gas, no neutron fluidtemperature T Tinit=1011K, stable Tcool=106K probably like neutron starsrotation α/ rs ≈ 0.6 ω≈ (α/ rs)ωmax ωmax=c/R , fmax= 1.6kHzeos neutron Fermi gas P=k *ρ5/3 Fermi gas ρ< ρc

radiation x-ray accretion , fat ≈10 red-shifted x-ray, binary star probably like nstar: 5%progenitor star M≈ 30 Msun massive supernova II

Supermassive black-holeparameter value formula commentmass M (106 – 109 ) Msun

radius R R≈rs d(shell) ≈ 0.001 rs R≈rs(1+ε) ε≈ (0.0014-0.04) , eTOV density ρ 10-12 ρs(Msun)= 104 g/cm3 eTOV ρ=ρ(white dwarf) surface gravity g 3*106m/s2 for M=106 Msun ≈ c2/ rs comp. g(nstar)= 7*1012m/s2

escape velocity ≈ c v=c(1- ε/2) eTOV bind.en. Egr M c2 (1- ε /(1+ε) ) eTOV magnetic field B strong ionized gastemperature T T=1010K ?rotation α/ rs ≈ 0.6 ? ω≈ (α/ rs)ωmax ωmax=c/R , fmax≈ mHz eos electron Fermi gas P=k *ρ5/3 Fermi gas ρ< ρc

radiation x-ray accretion , fat ≈700 red-shifted x-ray

11Stellar black-hole

Simulated view of a black hole in front of the Large Magellanic Cloud. The ratio between the black hole Schwarzschild radius and theobserver distance to it is 1:9. Of note is the gravitational lensing effect known as an Einstein ring, which produces a set of two fairlybright and large but highly distorted images of the Cloud as compared to its actual angular size.

Rotating (Kerr) stellar black-hole

The boundaries of a Kerr black hole relevant to astrophysics. Note that there are no physical "surfaces" as such. The boundaries aremathematical surfaces, or sets of points in space-time, relevant to analysis of the black hole's properties and interactions .At close enough distances, all objects – even light itself – must rotate with the black-hole; the region where this holds is called theergosphere.

12Stellar black-hole candidatesOur Milky Way galaxy contains several stellar-mass Black Hole Candidates (BHCs) which are closer to us thanthe supermassive black hole in the Galactic center region. These candidates are all members of X-ray binarysystems in which the compact object draws matter from its partner via an accretion disk.

NameBHC Mass

(solarmasses)

CompanionMass (solar

masses)

Orbitalperiod(days)

Distance fromEarth (light

years)

Location[10]

A0620-00/V616Mon

11 ± 2 2.6−2.8 0.33 about 350006:22:44 -00:20:45

GRO J1655-40/V1033 Sco

6.3 ± 0.3 2.6−2.8 2.8 5000−1000016:54:00 -39:50:45

XTEJ1118+480/KVUMa

6.8 ± 0.4 6−6.5 0.17 620011:18:11+48:02:13

Cyg X-1 11 ± 2 ≥18 5.6 6000−800019:58:22+35:12:06

GROJ0422+32/V518 Per

4 ± 1 1.1 0.21 about 850004:21:43+32:54:27

GS 2000+25/QZVul

7.5 ± 0.3 4.9−5.1 0.35 about 880020:02:50+25:14:11

V404 Cyg 12 ± 2 6.0 6.5 about 1000020:24:04+33:52:03

GX 339-4/V821Ara

5−6 1.75 about 1500017:02:50 -48:47:23

GRS 1124-683/GUMus

7.0 ± 0.6 0.43 about 1700011:26:27 -68:40:32

XTE J1550-564/V381 Nor

9.6 ± 1.2 6.0−7.5 1.5 about 1700015:50:59 -56:28:36

4U 1543-475/ILLupi

9.4 ± 1.0 0.25 1.1 about 2400015:47:09 -47:40:10

XTE J1819-254/V4641 Sgr

7.1 ± 0.3 5-8 2.82 24000 - 40000[11] 18:19:22 -25:24:25

GRS1915+105/V1487Aql

14 ± 4.0 ~1 33.5 about 4000019:15;12+10:56:44

XTE J1650-5003.8 ± 0.5 [12]

smallest. 0.32[13] n-star?

16:50:01 -49:57:45

13Supermassive black-hole (quasar)A supermassive black hole is the largest type of black hole in a galaxy, on the order of hundreds of thousands tobillions of solar masses. Most, and possibly all galaxies, including the Milky Way[2] (see Sagittarius A*), arebelieved to contain supermassive black holes at their centers

An artist's conception of a supermassive black hole and accretion disk.

Formation of extragalactic jets from a black hole's accretion disk

14LIGO GW1508914

Gravitational waves from a binary black-hole merger LIGO GW1508914

Top: Estimated gravitational-wave strain amplitude from GW150914 projected onto H1. This shows the full bandwidth of thewaveforms, without the filtering used for Fig. 1. The inset images show numerical relativity models of the black hole horizons as theblack holes coalesce. Bottom: The Keplerian effective black hole separation in units of Schwarzschild radii ( RS=2GM/c2) and the

effective relative velocity given by the post-Newtonian parameter v/c=(GMπf/c3)1/3, where f is the gravitational-wave frequency

calculated with numerical relativity and M is the total mass (value from Table I).

15LIGO GW1508914

Gravitational waves are anticipated to have a range of different wavelengths and sit on a broad spectrum, analogous toelectromagnetic waves. The spectrum spans all the way from wavelengths so long that a single one of them would span the entireuniverse, with a wave period the same as the age of the universe (left), to smaller wavelengths of hundreds of thousands of kilometreswith wave periods of only milliseconds (right). Shown above the spectrum are possible sources for gravitational waves, with theexperiments aiming to make detections, and the methods they use, below.

16

New calculations LIGO GW15089141. calculation of final distance d (of bh-centers) from angular frequency ω

s

s

m

m

d

cr3

22

2 rs=rs(m), m=m1+m2

m=60ms , rs=60rss=180km , ω=250Hz*2π =15701/s rs * ω/c=0.942 , d/ rs =33.77^(1/3)=3.23, d=581km

2. final angular momentum and initial Kerr-rotation-parameter αspin

Kerr-rotation-parameter αf= 0.67 measured,mc

J

I(sphere)=(3/5)M R2 , IJ

orbit L1≈ –L2 cancel approximatelyω1= ω2=250Hz*2π , orbit ang. momentum from radiationmass defect:

2

2

dmL f , Δm=5ms d/2=290.5km, ω/c=5.23*10-3 km-1

c

d

m

morb

2

2

= 0.19

2122

11

spinspinspinspinspinm

m

m

m

spinorbitspin = 0.67-0.19=0.48

3. assessment of the radius of resulting black-holedimensionless fall time tf from r=r0 to r=rs(1+ε) in {rs ,c} units is in the limit ε->0

0

02/3

00

1)(

2),1(

r

rLogrrt f

LIGO measured black-hole collapse time is approximately tf = 1.5 ms=2.5rs/c , r0 =(581km-2rs(30Msun))/2=200km=1.11 rs(60Msun)

so 2/3

02

r

=1.83, so we get

0

0 1)(

r

rLog

=2.5-1.83=0.67 , Log(ε)=-2.12 , ε=0.119 and for the radius of the formed shell star:

R= 1.119 rs , TOV-result for M=60 Msun is ε=dRsrel≈0.222 see chapter 8. and below

4. assessment of gravitation wave energy from mass defect

grav. energy original single BH)(1

)(

)(1

11

1

121

1

211

MdR

MdRcM

MdRcME

srel

srel

srel

gr

grav. energy original merged BH)2(1

)2(2

1

121

MdR

MdRcME

srel

srelgr

energy difference

)2(1

)2(

)(1

)(2

1

1

1

121

MdR

MdR

MdR

MdRcME

srel

srel

srel

srelgr =

sgr ME 405.000674.0*600152.01

0152.0

0222.01

0222.0*60

i.e. 0.6% mass burned

where 0.0152)60(,0.0222)30( srelsrel dRdR

measured mass difference ssgr MME 43 i.e. 5% , simulation: up to 4% mass burned

17final period sT 01.01 , final distance )(2 11 Mrd s , final velocity

cskms

km

T

dv 188.0/10*6.5

01.0

180 4

1

11

, so final kinetic energy 11

2

0355.02

1*2 MM

c

vEkin

, i.e.

1.77% total mass , so the final assessment 0244.02)00674.00177.0(2 11 MMEEE grkin i.e. 2.4%

total mass burned

18

3. Theoretical background: Einstein equations, orbit equations, TOV equation

The basics: Schwarzschild and Kerr spacetime and the Newtonian and GR energy equationWe start with exact solutions of Einstein equations in spherical coordinates for the non-rotating (Schwarzschild)and rotating (Kerr) black-hole.The Kerr line element reads [3]

222222

222

2222

2

22

222

222

22

222

2

cossincos

sin

cos

cos

sin2

cos1

drdr

rrr

drrrr

r

ddtr

rrdt

r

rrds

s

s

ss

(1)

where2

2

c

GMrs is the Schwarzschild radius, and

Mc

J is the angular momentum radius (amr) , has

the dimension of a distance: r][ , and J is the angular momentum.

In the limit α→0 the Kerr line element becomes the standard Schwarzschild line element

ddr

r

r

drdtc

r

rds

s

s 2222

222 sin

1

1

(2)The total energy for a mass m in Newtonian gravitation field of a mass M is:

2222

22cmE

r

MmGrmrmtt

(3)

where Et is the total energy and εt the relative total energy. We use in the following the terminology of [2] for

the GR energy and radial orbit equation:2

12

Ft , where 122 tF is the (dimensionless) relativistic

velocity factor.From the first (time t) Schwarzschild orbit equation (see below) we get

Fconstr

t

11 [2] , where F is the above relativistic velocity factor .

In the general relativistic Schwarzschild case the Newtonian approximation (3) becomes the exact relativisticenergy equation [2] :

2

121

22

2

22

22

F

r

GmM

rc

GM

r

lmrmt

(4)

The Einstein field equations are [2,4,5]:

TgRgR 02

1(9)

where R is the Ricci tensor, R0 the Ricci curvature,4

8

c

G , T is the energy-momentum tensor, is

the cosmological constant (in the following neglected, i.e. set 0),with the Christoffel symbols (second kind)

x

g

x

g

x

gg

2

1

(10)

and the Ricci tensor

19

xxR

(11)The orbit equations O1…O4 in vacuum ( 0T ) are:

02

2

d

dx

d

dx

d

xd

κ=0…3 (12)with the usual setting λ=τ = proper timeFor λ=τ we get for the line-element ds=c dλ= dλ and therefore trivially:

01

d

dx

d

dxg

(13)This relation yields for the Kerr- and Schwarzschild-spacetimes the GR energy relation.

20The TOV equationIn the Schwarzschild spacetime =0 and a=0, we have spherical symmetry, no dependence on , then theTOV-equation can be derived from the remaining non-trivial Einstein equations eqR00, eqR11, eqR22, eqR41.

The TOV-equation is in the standard form:

1

22

3

22)

)(21()

)(

)(41)(

)(

)(1()

)()(()('

rc

rGM

crM

rPr

cr

rP

r

rrGMrP

(13)

and using sr

1

2

3

22

2

))(

1())(

)(41)(

)(

)(1()

2

)(()('

t

ss

rM

rMr

crM

rPr

cr

rP

r

rrcrP

, where

tM is the total mass, furthermore

)(')(4 2 rMrr , )()( 1 rkrP

In order to make the variables dimensionless, one introduces ‘sun units’

2

3

16

32,1076.1

3/4,3

2)( cP

cm

g

r

Mkm

c

GMsunrr ss

ss

suns

sunsss

where ssr Schwarzschild-radius of the sun, s the corresponding Schwarzschild-density and sP the

corresponding Schwarzschild-pressure.In ‘sun units’ TOV-equation transforms into

))(3)()()(3

)('(

2

1))(()('

3

111011

2

111011

0111

3

111 rrPMrMrrPMrM

MrMrrrP (14)

with the normalized mass M1(r1) , and 1)( 11 RM ,or

))(3)()()(3

)('(

2

1)(()('

3

1111

2

1111

11

3

111 rrPrMrrPrM

rMrrrP

wheresun

t

M

MM 0 , )( 1rM is the mass within the radius r, M(r1)= M0 M1(r1)

in dimensionless variables )(,,3

)(')(, 1112

1

111 rkPM

r

rMrr

and 1R is the dimensionless radius of the star.

With the replacement 1kP for the pressure from the equation of state (non-interacting Fermi-gas)

and )(PP (interacting neutron fluid)

and2

0

3

'

r

MM we obtain a diff. equation for M degree 2 in r and we impose the boundary condition in r=R1:

M(R1)=M0 , M’(R1)=0 for non-interacting Fermi-gasand

for an interacting Fermi-gas : M(R1)=M0 , eRRRRM )(,3)()(' 1

2

111 , where e is the equilibrium

density in the minimum of Vnn and P1’(e)=0 (here an equivalent boundary condition is )(' 1R ).

In the Newtonian limit we get for TOV

)()(

2r

r

rGM

dr

dP

21

4. Black-hole paradoxes

Singularity paradoxes in the Hawking-Penrose model of a black-hole

1.The (external) fall time is infinite

The proper fall time τf from r0 to rs (1+ε)in the limit r0 >>1 in dimensionless coordinates: 2/3

002

),1( rrf

,

the external fall time i.e. measured by an external remote observer is

2/3

0

0

00

2

1)(),1( r

r

rLogrt f

that is, tf diverges logarithmically for r->1 .In the shell-star-model the (external) fall time becomes finite, the deviation from horizon ΔR ≈rs exp(-Δtf) canbe calculated from the fall time dilation, e.g. for LIGO GW1508914: ΔR = 0.173 rs

2. The information lossThe HP-black-hole obeys the no-hair-theorem : the only parameters of the HP-black-hole are rs and the Kerr-rotation parameter α . There is no internal structure: no particles, no equation-of-state. This violates the conservation of information theorem in quantum mechanics.In the shell-star model the information loss in HP-black-holes is eliminated , as the escape velocity v<c : realblack-holes are classical objects, they consist of nucleon-Fermi-gas and the no-hair-theorem is not valid3. The uncertainty of the horizonThe HP-black-hole is a quantum object, therefore the horizon rs must have an uncertainty Δrs .According to the no-hair-theorem the only parameters of the HP-black-hole are rs and the Kerr-rotationparameter α . Therefore the only possible value for the uncertainty is the minimal Planck length Lp , so thecorresponding energy uncertainty is ΔE=Ep the Planck energy = 1.21*1019GeV . This is the energy required forpair-production in order to overcome the information loss paradox, and too high for any sensible interaction. Sothere is no way out of the information loss paradox.In the shell-star model the black-hole surface is outside the horizon and pair-production as well as radiation ispossible.4. There is no radiation and no particle emissionThe HP-black-hole does not emit radiation or particles: the escape velocity is c. Whatever comes in, nevercomes out: this is the reason for the information loss.Shell-star black-holes emit thermal and accretion radiation, although with a very strong red-shift.

5. The Hawking temperature

The Hawking-temperature as the limit for a cooled-down black-hole without accretion remainsvalid, as well as the Hawking-radiation, but this is no more needed as an explanation for the information loss.Shell-star black-holes are classical objects and have an equation-of-state of a Fermi neutron gas and atemperature similar to neutron stars.6. The Oppenheimer collapse criterion

The collapse criterion given by the Oppenheimer limit R<2lim

4

9

8

9

c

MGrR s is invalid in the shell-star model,

as the maximum density is reached at the center for a compact star only, so there is no collapse to a singularity.

22

5. TOV equation and generalized Kerr-spacetime and a solution ansatz

The equations for the extended Kerr space-time.

The solution process starts with the metric tensor g in Boyer-Lindquist-coordinates, corresponding

correction-factor functions A0,…,A4, and additive correction functions B0,…B4 for the zero components(12)

)sin

(sin3

42

32/1

4sin

1001

12

22222

12

1212

12

2

12

arrarA

BA

BBA

Aarr

BBArr

g

s

ss

The equations are the 10 Einstein equationseqR00,eqR11,eqR22,eqR33,eqR12,eqR23,eqR31,eqR01,eqR02,eqR03 in the (dimensionless) variables relative

radiusssr

rr 1 and complementary azimuth angle

21 with energy tensor T from (8) and the state

equation 11 1kP for the relative pressure 1P and the relative density 1 . We are using the so called

“sun units” )(sunrr sss , )(sunMM s ,3

4 ss

ss

r

M

, 2cP ss for radius r, mass M, density , and

pressure P, respectively.

In “sun units” the original angle differential ddrrd 2sin4 is transformed into ddrrd 2cos3 ,

as for =0.r=0..1: 1 d .

Also, all equations and variables are symmetric (even) in : Ai(-)=Ai() .From now on we skip the index of the dimensionless variables and use the original notation, e.g. r instead of

1r .

Furthermore, we adopt the Boyer-Lindquist coordinates and the metric tensor (12).In sun units, the Boyer-Lindquist metric tensor becomes: (12a)

)cos

(cos3

42

32/1

4cos

1001

12

220222

12

1212

12

20

12

0

aMrarA

BA

BBA

AarM

BBAMr

g

)sin( 22212 ar

20

212 aMrr

where M0 is the mass in sun units.If we make the ansatz Bi=0, several of the eqRij become identically 0, and we get the 6 equations eqR00,eqR11, eqR22, eqR33, eqR03, eqR12 for the 6 variables Ai and , with the highest derivatives resp. 0Arr ,

1A , 2Arr , 3Arr , 4Arr , ( 2Arr , 1A ).

Thus, we are left with the 6 differential equations degree 2 in r, non-linear (quartic) in variables Ai and

their 1-derivatives and linear in , .

In total, we have 6 algebro-differential eqs for 6 variables Ai and ( enters only algebraically).

23The solving process for the extended Kerr space-time.In addition to the fundamental dual parameters {ri ,i } corresponding to{ R1 , M0 } in the rotation-free TOV-case, in the Kerr-case there is the new fundamental parameter ri (innerellipticity for inner boundary condition) , resp. R1 (outer ellipticity for outer boundary condition) , and theangular velocity . The outer radii areRx1 =R1-R1 and Ry1 =R1 , 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 rxi =ri-ri , ryi =ri .

The r--slicing algorithm with an Euler-step obeys the iterative procedure with slice step size 1h in r , and step

size 2h in , starting with the r-boundary at r= 1R (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 Airr , Birr and

are calculated from the 6 equations.At slice n+1 the variables and 1-derivatives are calculated by Euler-formula (or Runge-Kutta)

nrnn AihAiAi 11

nrrnrnr AihAiAi 11

The 2-derivatives Airr , and are again calculated from the 6 significant equations with variables and 1-

derivatives inserted from above.

The -slicing r-backward algorithm with an Euler-step obeys the iterative procedure with slice step size 1h in

as above for r, starting with =0, and solves an ordinary differential equation in r in each -step . Theboundary condition for the r-odeq is set at r=R1() (the outer ellipse radius) with Ai=1, M=M0 My0() ,

bcrr RMAi 21)(3,0 ,

where bc is the outer boundary value for the density, bc =0 for the (non-interacting) neutron-gas in a shell-starand bc >0 , bc =equilibrium for the (interacting) neutron fluid in a neutron star. My0() is the mass-form-factor

with the condition 1)()(02/

0

dCosMy , i.e. the overall mass at the outer boundary is M0 . The inner

radius ri() is reached, when My0()=0.The actual calculation was carried out in Mathematica using its symbolic and numerical procedures. In thefirst stage, the Einstein equations were derived from the ansatz for g from section 2 and simplifiedautomatically. The arising complexity of the equations is such, that it is practically impossible to handle themmanually: the Mathematica function LeafCount, which returns the number of terms in the equation, gives thecomplexity ofLeafCount[eqR00]=17408 , LeafCount[eqR11]=27528 , LeafCount[eqR22]=134929for the first 3 equations. To verify the equations, the TOV equation was derived by symbolic manipulation for=0 a=0 from eqR00, eqR11, eqR22, eqR41 .Also, for every star model and parameter set, the TOV solution with =0 a=0 was calculated first with thealgorithm and compared with the exact TOV solution.

24

The TOV-equation: a new ansatzGenerally speaking, the parameters of the solution are :angular momentum radius a (=alpha1, =0 for TOV), the factor in the state equation k1, the power in the stateequation (=gam),radius R, mass M0, the relative radius uncertainty reldr02 (=dr02rel), the moment of inertia

factor If (=infac, 0 for TOV), the singularity smoothing parameter (=epsi, see below), and the boundary

factor nrmax (=nrmax). Here the boundary factor enters the upper boundary of the TOV differential equation as)021( maxmax relr drnRr .

The dimensionless TOV-equation is an differential equation in the mass M(r) of degree 2, and is highly non-

linear, the dimensionless mass-density relation is23

'

r

M .

The customary way of solving the TOV equation is to impose the boundary condition at r=0 with M(0)=0,

M’(0)= 023 r where 0 the maximum central density .

In the new ansatz for the mass M(r) we impose the outer boundary condition at r=R1:

for a pure Fermi-gas without interaction: M(R1)=M0 , 0)(,3)()(' 1

2

111 RRRRM ;

for an interacting Fermi-gas : M(R1)=M0 , eRRRRM )(,3)()(' 1

2

111 , where e is the equilibrium

density in the minimum of Vnn and P1’(e)=0 (here an equivalent boundary condition is )(' 1R ).

The star parameters mass M0 and radius R1 , which enter the outer boundary condition determine completely

the solution. In general, there will be an inner radius ri >0 with the maximum density )('3 20 irMr and

M(ri)=0. The corresponding ‘dual’ parameters are the inner radius ri and the maximum density 0 . One canshow that for 0>>c (where c is the critical density of the equation of state) there is no solution with acompact star ri =0, i.e. there is a maximum mass Mc for the TOV equation, in case of compact neutron stars Mc

= 3.04Msun (see below). As we will see, there is in general a solution, if we allow ri >0 and impose an outerboundary condition at r=R1 , as long as R1 is not too close to the Schwarzschild radius rs = M0 of the star. Inthe limit R1 -> rs there will be no positive zero of M(r) ,i.e. ri <0 and the resulting (mathematical) TOV-solution will be no physical solution. But in general, speaking naively, the gravitational collapse of the star isavoided for large masses (M0 > Mc), if it has a shell structure with the inner radius ri and the outer radius R1 >M0 .As we will see, this outer boundary condition together with allowing ri >0 changes dramatically the resultingmanifold of physical solutions.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 (M0, R1) in the outer boundary condition at r1= R1 orthe dual parameter pair (ri , bc ) in the inner boundary condition r1= ri .The pure neutron Fermi- gas model yields for compact neutron stars a maximum mass of Mmaxc=0.93Msun ,which is in disagreement with observations. Therefore, at least for compact neutron stars, a model ofinteracting neutron fluid must be used. In 6.2 above we have described a Saxon-Wood-potential model for thenucleon-nucleon interaction, which seems to fit the experiment and the theory in the best way. There will be acritical density (dependent on temperature of course), where a transition from interacting fluid to Fermi-gastakes place, it is plausible to set this density equal to the Saxon-Wood critical density c =0.0417 .We made calculations with the TOV-equation using these two models for neutron-based stars and we came tothe conclusion that compact neutron stars with mass M0<=3.04 Msun consist of interacting neutron fluid andneutron shell-stars for M0 >=5Msun obey the Fermi-gas model. The underlying calculation is theMathematica-notebook [18].

25

6. Equation-of-state for neutron stars

The equation of state for an (non-interacting) nucleon gas

Here, 1kP is the equation of state of the star, derived from the thermodynamic Fermi gas equation at T=0

([2], chap. 48).

))(13

(82

3

0 FFF xfx

xP

V

EP

(15)

3

54

3

2

0h

cmmcP

c

, where c is the de-Broglie wavelength of the Fermi gas with particle mass m,mc

hc

3/13/1)3(2

nmc

px cF

F

, where xF is the Fermi-angular-momentum, n the particle density

Fx

F xxdxxf0

22 1)(

The resulting approximate equations of state for P are

c

c

F

F

K

K

x

x

PP

3/42

3/51

4

5

0

12

158 (16)

valid for the density ρ and the critical density ρc

3

83

c

c

m =6.9*1015 g/cm3 =0.392 ρs(Msun)

The full formula including temperature T is given below.

The chem. potential 1 can be calculated, an approximation formula is

)(12

11

1

2

1

2

11 nF

F

Finally, the resulting pressure (=energy density) p1(1,n1):

)))((exp(13

4),(

1111

2/3

1

012

2/3

111n

dnp

(17)

Below a 3D-diagram of p1(β 1, n1) in dimensionless variables for a nucleon gas (m=mn, density2

0

c

nE in sun units,

E0=149.4MeV ) is depicted: Here β1= kT is in E0 units, and one sees the dependence1kP except on the left side,

when kT reaches the magnitude of 1Gev (T=10^10K).

26The equation of state for an (interacting) nucleon fluidFor 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:Vsw[r_,V0_,r0_,dr0_]=V0/(1+Exp[(r-r0)/dr0])

Vnn[r_]= Vsw[r,Va,ra,dra]+ Vsw[r,Vc,rc,drc] where Vnn is the nucleon-nucleon-potential with anattractive part Vsw[r,Va,ra,dra] and a repulsive core Vsw[r,Vc,rc,drc] , the distance r between the nucleonsis

3/1)/( nEr , where En=23/12 4.149))2/(( cmMeVcmn is the nuclear energy scale m=pion mass =

140MeV, mn=neutron mass = 140MeV.

0.5 1.0 1.5 2.0

200

200

400

600

800

1000

Vnn(r) with energy(MeV), r(fm)[]The pressure of the interacting nucleon fluid becomes then

)))(/(()()( 3/11111111 rEVrcrP nnn (18)

The experimental data used here are those from [7]

And the hard-core potential from the lattice calculation Reid93 from [5]both fitted with a double Saxon-Woods-potential Vnn

0.5 1.0 1.5 2.0

200

200

400

600

800

1000

with r(fm) , V(MeV).From the nucleon-nucleon-potential the pressure is calculated taking into account the low-density Fermi-pressure of thenucleons K1*rho^(5/3)Pnn[rho]=Vnn[1/((rho))^(1/3)]*rho

27

0.05 0.10 0.15 0.20rho

0.10

0.05

0.05

0.10

0.15

0.20Pnn

Pfg[rho]=K1*rho^(5/3)+Pnn[rho]

0.02 0.04 0.06 0.08 0.10rho

0.10

0.05

0.05

0.10

0.15Pfgrho

total pressure Pfg(r) , pressure P and density r shown in sun-units.This equation-of-state has a minimum at =c=0.0417 and P’()=1 at =m=0.0544 .

As the sound velocity

d

dPv

)( , v>0 and v<1 (i.e. subluminal), the admissible density range in the neutron-fluid

model is c<=m .

28

7. Neutron stars: results and examples

Neutron stars without rotationNeutron stars consist of interacting neutron fluid and are compact stars with(M0, R1)=(0.14,1.49)...(3.04,3.95) and the maximum density 0.048<= bc <=0.0544=bcmax in sun-units,neutron star R-M-relation follows approximately a cubic-root-law: R~M1/3 .The LIGO cooperation published in November 2018 a confirmed mass range for neutron stars in the range [26]

sunnssun MMM 8.29.0 , which confirms our result for compact neutron stars, the transition from neutron

fluid to neutron Fermi gas leads from a compact neutron star to a shell stellar (quasi) black hole.

The actual theoretical limit for neutron star core density is max =3.5 1015 g/cm3=0.199 in sun-units [8,9].The limit for bc reached in our mapping is only ¼ of this bc =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 purerepulsive-hardcore-model.The parametric mapping of the solutions results in the following dependence for M0(ri,bc) , R1(ri,bc) (ri,bc ,

M0, R1 in sun-units):

For ri =0 the mapping describes the compact neutron stars, resulting in R1(M0) function:

0.5 1.0 1.5 2.0 2.5 3.0MMsun

1

2

3

4

Rrssunri0.01

The R-M-relation follows approximately a cubic-root-law: R~M1/3 , with a range of(M0,R1)=(0.14,1.49)...(3.04,3.95) , i.e. the resulting maximum compact mass is Mmaxc=3.04Msun .For M0>= Mmaxc the function R1(ri =const,bc) is flat or slightly decreasing with bc , son one expects thestable configuration to be the one with maximum bc =bcmax :

29Compact non-rotating neutron starparameters= { k1=0.40,gam=5/3,M0=0.932,R1=2.76,rhobc=0.0456,ri=0.01};

The mean density is here3

1

0

R

Mmean = 0.04447 .

The critical density of the neutron Fermi gas with neutron mass mn is32

34

3

cmncn = 0.35 (see [2]) , so the low-

density approximation with =5/3 can be used.Results TOV:rho, M:

0.5 1.0 1.5 2.0 2.5r1

1.0

0.5

0.5

M M0 ,R1 ,K1,gamact 0.932 ,2.767 ,0.4 ,53

0.0 0.5 1.0 1.5 2.0 2.5r1

0.043

0.044

0.045

0.046

rho M0 ,R1 ,K1,gamact 0.932 ,2.767 ,0.4 ,53

As can be seen in the -diagram, the derivative )(' 1R , because there the equilibrium density c with

P’(c)=0 in the pressure is reached.

30Rotating compact neutron starThe underlying star model here is a compact (ri=0) neutron star of neutron liquid (i.e. strongly interactingneutrons), mass M0=0.932 sun-masses, radius R1=2.76 sun-Schwarzschild-radii rss (rss =3.0km), =0.108688.The solution for the Kerr-case starts with this corrected TOV-solution and yields the values:outer radius R1()={2.83912...2.83722},mean=2.83897, dthrel=0.00118total mass M02eff=0.932error: med(err)=0.0639 wavefront, =0.0491 interpolation, =0.0492 Fourier fitmean energy density=0.0791

density over x=radius r1, y=angle th

(ring) mass profile for th=0.1 (equatorial) and th=1.4708 (polar), the outer edge is reached when M1(r1)=M0

effective radius r02e over angle th

The rotation results in the very small flattening in the polar direction of dthrel=0.00118. The neutron starbehaves like a fluid because of its “viscosity”, that is, its nuclear interaction and becomes “pumpkin-like”.

31

8. Stellar shell-stars: results and examples

Stellar shell-starsWe assume that the underlying equation-of-state state for stellar shell-stars is the Fermi-gas of nucleons withthe low-density limit of

3/51)( KP .

In the resulting solution map only the “ridge” yields stable solutions (maximum ρbc=minimum gravitationalenergy for fixed ri).The “edge” of the mapping yields the (M0-, ri ,R1-)-range of (M0, ri ,R1)= (5.35,7,.8.49)…(81.3,89.6, 91.2) ,where the upper limit is in fact mathematically open, but the ” thinning-out” of the solutions for small bc andlarge ri makes it physically plausible (see the images below).

The resulting R-M-relation is practically linear and has a maximum mass value of Mmax=81.3 Msun.

20 40 60 80M Msun

20

40

60

80

R rssunR M

And the corresponding relative shell thickness dRrel=dR/M is

20 40 60 80M Msun

0.05

0.10

0.15

0.20

0.25

dR rssunrel shell thickness

and the relative Schwarzschild-distance dRsrel=(R-M)/M is

32

20 40 60 80M Msun

0.1

0.2

0.3

0.4

0.5

0.6

R rs rs

The LIGO cooperation published in November 2018 a mass range for stellar black holes in the range [26]

sunbhsun MMM 8035.5 , which confirms our result with an error margin of sunbh MM 5.0)( for

sunbh MM 5

33Entropy of a thin shell starThe celebrated Hawking- Bekenstein formula for the entropy of a black hole reads [1]:

24 P

B

L

AkS , where A is the surface area, kB the Boltzmann-constant , and LP the Planck-length.

The entropy of a (cold) shell-star with radius R and thickness dR in the limit R=rs , dR<<rs , with all particlesin the lowest possible energy state, can be easily calculated from the Boltzmann-formula

WkS B ln , where W is the number of possible micro-states.

With the elementary area πLmin2 , where PLL min , W becomes )2

2min/( LAW (each of the )/(

2

minLAN

area elements can be occupied or empty) , so the entropy (thin shell star, T=0)

2ln2

P

B

L

AkS

which is identical to the Hawking- Bekenstein entropy with the factor (ln2)4/π=0.882.

34Stellar non-rotating shell-starparameters= { k1=0.40,gam=5/3,M0=15.69,R1=17.89,rhobc=0.0359,ri=17.};The mean density is here mean = 0.0194 .The resulting rho and M are:

17.2 17.4 17.6 17.8r1

5

10

15

M M0 ,R1 ,K1,gamact 15.69 ,17.89 ,0.4 ,53

17.0 17.2 17.4 17.6 17.8r1

0.01

0.02

0.03

0.04

0.05

rho M0 ,R1 ,K1,gamact 15.69 ,17.89 ,0.4 ,53

Here the radius R1 is reached, when M’(r1= R1)=0 , i.e. (R1)=0 .

35Rotating stellar shell-starThe star model here is a shell-star (ri>Schwarzschild-radius) with mass M0=15.69 sun-masses, radius R1=17.89sun-Schwarzschild-radii rss (rss=3.0km), and angular frequency =0.0126 .The outer Kerr-horizon is r+=15.21 . The outer ellipticity R1 is at first a free parameter and calculated from acase-study of minimal mean energy density to R1=0.3=0.0168 R1 .This solution with R1=0.3 yields the values:outer radius R1()={17.59…17.89}, mean=17.739, dthrel=0.0164inner radius ri()={16.704…16.982}, mean=16.823, dthrel=0.0162total mass M02eff=15.69, inner boundary max(bc)=0.035955shell thickness dR1 : mean=0.916, dthrel=0.0448mean energy density=0.0170

density over x=radius r1, y=angle th

density over angle th at the inner boundary r1=16.704

The fitted density profile for th=0.1 (equatorial) The physical mass distribution ends at the inner boundary at ri=16.7 , where the

density jumps to =0 .

Effective inner radius ri over angle thFourier-fitted total massThe shell-star becomes “cigar-like”, the attenuation factor is 16.53.

36

9. Supermassive (galactic) shell-stars

Galactic (supermassive) shell-stars

The mean density of a black-hole scales with its radius R like23 4

3

)3/4()(

RR

R

V

MR

i.e. for supermassive black-hole with M=106Msun we have 1210 su in sun units ,3

610cm

g .

In the following we use the abbreviation M Msun =106Msun .

The density scale of a white-dwarf star is 106g/cm3=5.7 10-11su [2]. Therefore it is plausible to try a parametricmapping with the white-dwarf equation-of-state, where the underlying Fermi-pressure is that of an electron gas

instead of a nucleon gas, i.e. equation-of-state )()( 11111 rkrP for a pure Fermi gas, =5/3 if the density is

below the critical density c .The results for M0(ri,bc) , R1(ri,bc) are shown below:

The actual density is around 10-12, that is well below the critical density for a white-dwarf of c =0.91106g/cm3=5.17 10-11su : =5/3 in the equation-of-state is justified. One calculates the R-M-relation followingthe ”ridge”.The resulting R-M-relation is as follows:

10 20 30 40 50MMsun

10

20

30

40

50

Rrssun

10 20 30 40 50MMsun

10

20

30

40

50

rirssun

10 20 30 40 50MMsun

0.002

0.004

0.006

0.008

0.010

RriM

10 20 30 40 50MMsun

0.01

0.02

0.03

0.04

RMM

The attenuation factor is around 700.So the overall result is, that the supermassive shell-stars become ever thinner shells, while the distance from theSchwarzschild-horizon is increasing.

37Non-rotating galactic shell-starparameters= {M0=4.367*106,R1=4.380*106,rhobc=4.934*10-12,ri=4.356*106};The mean density is here mean = 3.16*10-12 .TOV-solution for rho (in 10^-12 units), M (in 10^6 units) in r (in 10^6 units), is:

4.365 4.370 4.375 4.380r1

1

2

3

4

M M0 ,R1 ,K1,gamact 4.36731477304323 ,4.38 ,0.024300000000000002 ,53

4.360 4.365 4.370 4.375 4.380r1

0.05

0.10

0.15

0.20

0.25

0.30

rhorhocrit M0 ,R1 ,K1 ,gamact 4.36731477304323 ,4.38 ,0.024300000000000002 ,53

Here there is an internal ”hole” with a radius ri= 4.356*106, maximum =4.934*10-12 at ri . The inner radius ri

lies a little below the Schwarzschild-radius rs=M0.dRrel=(R1- ri)/M0 =0.00551 , dRsrel=(R1- M0)/M0 =0.00290 , the attenuation factor is roughly 1/ dRsrel =344 .

38Rotating galactic shell-starThis is modelled (approximately) on the central black-hole in the Milky Way with mass M0=4.36 mega-sun-masses (M Ms), radius R1=4.38 mega-sun-Schwarzschild-radii (13.14 106km, M rss).The outer Kerr-horizon is r+=4.26 M rss .We are using for mass and distance 106 (mega) units 106 Ms and 106 rss and for density 10-12 (mega-2) unit 10-12

s.The solution yields the values:outer radius R1()={4.494…4.38}, mean=4.43654, dthrel=0.025367inner radius ri()={4.46456…4.34109}, mean=4.40029, dthrel=0.02765total mass M02eff=4.3673, inner boundary max(bc)=4.96075shell thickness dR1 : mean=0.035256, dthrel=0.2507,mean energy density=1.73479Here there is a dependence of the shell thickness on the ellipticity.

density over x=radius r1, y=angle thdensity over angle th at the inner boundary r1=4.464

The fitted density profile for th=0.1 (equatorial)Inner radius ri over angle th

Total mass M1(r1) over radius

The attenuation factor is 34.

39

10. Newtonian stars: white dwarves, sun-like stars, and supermassive stars

White-dwarf starparameters= { k1=1.43*106,gam=5/3,M0=0.6,R1=3000,rhobc=2.02*10-11,ri=0.};The underlying state equation is that of a small-momentum electron Fermi-gas with the critical density [2]

32

33

3

cmm necw = 0.517*10^-10su .

The mean density is here mean = 2.22*10-11 , the maximum deviation of is max0.21*10-11 , so thedensity is practically constant, as expected.The solution of the TOV-equation becomesrho, M:

500 1000 1500 2000 2500 3000r1

0.2

0.4

0.6

0.8

1.0M K1,gamact ,bfunc ,drmax ,nrmaxact 1.43*̂ 6 ,53,0.032 ,2

0 500 1000 1500 2000 2500 3000r1

2.05 1011

2.1 1011

2.15 1011

2.2 1011

2.25 1011

2.3 1011rho K1,gamact ,bfunc ,drmax ,nrmaxact 1.43*̂ 6 ,53,0.032 ,2

40Sun-like starsPhysical data:R=0.696*106 km , mean density ρ=1,408 g/cm3 , ρ=162.2 g/cm3 (core) , mass M=1 Msun=1,989*1030 kg ,temperature T=5800K (surface), T=15.7*106K (core) , P=265*109 bar (core)

Here the gas pressure of the ideal gas counterbalances the gravitational pressure:

Pm = Pgr , Newtonian TOV )()(

2r

r

rGM

dr

dP

and the equation-of state becomes

n

BBm

m

Tk

V

TNkP

41Supermassive starsPhysical data:R=30-500 Rsun , mean density ρ= 10-7 - 10-9 g/cm3 , mass M=8-20Msun ,temperature T=10000- 40000 K (surface)In supermassive stars the radiation pressure counterbalances the gravitational pressure (Hoyle-Fowler 1963)[2].Radiation pressure from Stefan-Boltzmann

3

42

)(

)(

453 c

TkuP Br

r

, gas pressure of ideal gas

4/3

4/1

3

2)(

45222

c

mm

Tk

V

TNkP

nn

BBm

we get the equation-of-state3/4KP ,

with 4/3

4/1

3

2)(

452

Kc

mP

Pconst

nr

m

ratio of gas pressure and radiation pressure,

so )(452

3/1

2

3/4

cmP

PK

nr

m

42

11. Conclusion

We introduce an eos for the nucleon-fluid in the density range c<=m , wherec=0.0417 s and m=0.0544s ( sun units rss=3km , s=1.76 1016 g/cm3) ,which is based on measurementdata for the nucleon-nucleon-potential.There is a phase transition at =c from the (interacting) nucleon fluid to the (weakly interacting) nucleonFermi-gas .Neutron stars obey the nucleon fluid eos and there arecompact neutron stars in the range(M0, R1)=(0.14Msun,1.49 rss)...(3.04 Msun,3.95 rss) , the R-M-relation follows approximately a cubic-root-law:

R~M1/3 .Neutron shell-stars exist in the range (M0, R1)= (3.04 Msun,3.95 rss)…(4.91 Msun,4.92 rss) .Stellar shell-stars exist in the range of (M0, R1)= (5.5 Msun,9.1 rss)…(81.3 Msun, 91.2 rss) .The underlying equation-of-state is the Fermi-gas of nucleons with the eos

3/51)( KP . The resulting R-M-relation is practically linear and has a maximum mass value of

Mmax=81.3Msun .The light attenuation factor (redshift) is roughly {1.7…20}.

Furthermore, the entropy of a thin spherical shell is proportional to its surface: 2ln2

P

B

L

AkS

,

which approximates the Hawking- Bekenstein black-hole entropy formula S=(kB/LP2)A/4 .,

and confirms the holographic principle for black holes

The supermassive shell-stars (quasars) have the density scale and the eos of a white-dwarf-star, i.e. of anelectron Fermi-gas. The R-M-relation is almost linear and goes from 1MMsun up to 50MMsun

(MMsun =106Msun , Mrss =106rss ).The relative thickness is around 0.001 .The relative Schwarzschild-distance dRsrel= (R1-M0)/M0 has a minimum at 0.00142857, the redshift is around700.

The shell-star model for black-holes resolves singularity paradoxes:-the (external) fall time becomes finite, the deviation from horizon ΔR ≈rs exp(-Δtf) can be calculated from thefall time dilation, e.g. for LIGO GW1508914: ΔR = 0.173 rs

-the information loss in abstract black-holes is eliminated , as the escape velocity v<c : real black-holes areclassical objects, they consist of nucleon-Fermi-gas and the no-hair-theorem is not valid- real black-holes emit thermal and accretion radiation, although with a very strong red-shift

- the Hawking-temperature as the limit for a cooled-down black-hole without accretionremains valid, as well as the Hawking-radiation, but this is no more needed as an explanation for theinformation loss

-the collapse criterion given by the Oppenheimer limit R<2lim

4

9

8

9

c

MGrR s is invalid in the shell-star model,

as the maximum density is reached at the center for a compact star only, so there is no collapse to a singularity.

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 satisfythe Hawking- Bekenstein entropy formula, but have finite redshifts and escape velocity v<c, no singularity , noinformation loss paradox, and are classical objects , which need no recourse to quantum gravity to explain theirbehaviour.

43References[1] S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford University Press 1992[2] T. Fliessbach, Allgemeine Relativitätstheorie, Bibliographisches Institut 1990[3] M. Visser, The Kerr spacetime: a brief introduction [arXiv: gr-qc/07060622],2008[4] A.L.Wasserman, Thermal Physics, Oregon State University, 2011[5] M.Hjorth-Jensen, Models for Nuclear Interactions, Tokyo 2007[6] J.M.Lattimer,M.Prakash,Neutron Star Structure [arXiv: astro-ph/00022320],2000[7] Th.Papenbrock,Physics of Nuclei, Nat.Nucl.Phys. SummerSchool, 2008[8] A.W.Steiner, M.Hempel, T.Fischer Astrophys.J. 2013, 7774, 17[9] M.Hempel, Physik in unserer Zeit 1/2014,Wiley-VCH[10] S.Typel et. al., Phys.Rev.C 2010, 81, 015803[11] K.D.Kokkotas, M.Vavoulidis,Rotating relativistic stars, Journal of Physics, Conference Series 8 (2005),71-80[12] N.Stergioulas, Rotating Stars in Relativity, Living Reviews in Relativity, 06/2003[13] S. Dain, Geometric Inequalities for axially symmetric black holes, [arXiv:1111.3615v2][14] L.Ferrarese, D.Merritt, Supermassive Black Holes, [arXiv: 0206222v1][15] F.Douchin, P.Haensel, A unified equation of state of dense matter and neutron star structure,[arXiv:0111092v2][16] P.Rosenfield, Properties of Rotating Neutron Stars, PhD Thesis, 04/2007, Univ. of Washington[17] M.Urbanec, Equations of State and Structure of Neutron Stars, PhD Thesis, 07/2010, Univ. Opava[18] Mathematica-notebook GRSchwarzTOVOrig3.nb, www.janhelm-works.de: calculation for non-rotatingstar-models[19] Mathematica-notebook KerrBLS0e.nb, www.janhelm-works.de: calculation for rotating shell-starM=15.69Msun

[20] Mathematica-notebook KerrBLS1e.nb, www.janhelm-works.de: calculation for rotating galactic shell-starM=4.37*106 Msun

[21] Mathematica-notebook KerrBLS2e.nb, www.janhelm-works.de: calculation for rotating compact neutron-star M=0.932Msun

[22] KerrBLS_Results.doc, www.janhelm-works.de: concise results for rotating star-models[23] Jan Helm, New solutions of the Tolman-Oppenheimer-Volkov (TOV) equation and of Kerr space-timewith matter and the corresponding star models, viXra 1404.0037, 2017[24] Jan Helm, New solutions of gravitational collapse in General Relativity and in the Newtonian limit, viXra1408.0117, 2014[25] Jan Helm, An exact GR-solution for the relativistic rotator, viXra 1706.0200, 2017[26] LIGO cooperation, Nov. 2018, https://www.ligo.caltech.edu/image/ligo20171016a