I have discussed a model of quasars earlier (see this) . The model is inspired by the notion of MECO and proposes that quasar has a core region analogous to black hole in the sense that the radius is apart from numerical factor near unit r_{S}=2GM. This comes from mere dimensional analysis.

**1. Blackholes in TGD framework**

In TGD the metric of blackhole exterior makes sense and also part of interior is embeddable but there is not much point to consider TGD counterpart of blackhole interior, which represents failure of GRT as a theory of gravitation: the applicability of GRT ends at r_{S}. The following picture is an attempt to combine ideas about hierarchy of Planck constant and from the model of solar interior (see this) deriving from the 10 year old nuclear physics anomaly.

- The TGD counterpart of blackhole would be maximally dense spaghetti formed from monopole flux tube. Stars would not be so dense spaghettis. A still open challenge is to formulate precise conditions giving the condition r
_{S}= 2GM. The fact that condition is "stringy" with T= 1/2G taking formally the role of string tension encourages the spaghetti idea with length of cosmic string/flux tube proportional to r_{S}.

- The maximal string tension allowed by TGD is determined by CP
_{2}radius and estimate for Kähler coupling strength as 1/α_{K}≈ 1/137 and is roughly T_{max}∼ 10^{-7.5}/G suggesting that in blackhole about 10^{7.5}parallel flux tubes with maximal string tension and with length of about r_{S}give rise to blackhole like entity. Kind of dipole core consisting of monopole flux tubes formed by these flux tubes comes in mind. The flux tubes could close to short flux tubes or flux tubes could continue like flux lines of dipole magnetic field and thicken so that the energy density would be reduced.

- This picture conforms with the proposal that the integer n appearing in effective Planck constant h
_{eff}=n× h_{0}can be decomposed to a product n=m× r associated to space-time surface which is m-fold covering of CP_{2}and r-fold covering of M^{4}. For r=1 m-fold covering property could be interpreted as a coherent structure consisting of m almost similar regions projecting to M^{4}: one could say that one has field theory in CP_{2}with m-valued fields represented by M^{4}coordinates. For r=1 each region would correspond to r-valued field in CP_{2}.

This suggests that Newton's constant corresponds apart from numerical factors 1/G= mℏ/R

^{2}, where R is CP_{2}radius (the radius of geodesic circle). This gives m∼ 10^{7.5}for gravitational flux tubes. The deviations of m from this value would have interpretation in term of observed deviations of gravitational constant from its nominal value. In the fountain effect of super-fluidity the deviation could be quite large (see this) .

Smaller values of h

_{eff}are assigned in the applications of TGD with the flux tubes mediating other than gravitational interactions, which are screened and should have shorter scale of quantum coherence. Could one identify corresponding Planck constant in terms of the factor r of m: h_{eff}= rhbar_{0}? TGD leads also to the notion of gravitational Planck constant hbar_{gr}= GMm/v_{0}assigned to the flux tubes mediating gravitational interactions - presumably these flux tubes do not carry monopole flux.

- Length scale dependent cosmological constant should characterize also blackholes and the natural first guess is that the radius of the blackhole corresponds to the scaled defined by the value of cosmological constant. This allows to estimate the thickness of the flux tube by a scaling argument. The cosmological constant of Universe corresponds to length scale L=1/Λ
^{1/2}∼ 10^{26}m and the density ρ of dark energy corresponds to length scale r= ρ^{-1/4}∼ 10^{-4}m. One has r= (8π r)^{1/4}Ll_{P}^{1/2}giving the scaling law (r/r_{1})= (L/L_{1})^{1/2}. By taking L_{1}=r_{s}(Sun)=3 km one obtains r_{1}= .7× 10^{-15}m rather near to proton Compton length 1.3× 10^{-15}m and even nearer to proton charge radius .87× × 10^{-15}m. This suggests that the nuclei arrange into flux tubes with thickness of order proton size, kind of giant nucleus. Neutron star would be already analogous structure but the flux tubes tangled would not be so dense.

Denoting the number of protons by N, the length of flux tube would be L

_{1}≈ Nl_{p}== xr_{S}(l_{p}denotes proton Compton length) and the mass would be Nm_{p}. This would give x as x= (l_{p}/l_{Pl})^{2}∼ 10^{38}. Note that the ratio of the volume filled by the flux tube to the M^{4}volume V_{S}defined by r_{S}is

V

_{tube}/V_{S}= (3/8) (l_{P}/l_{Pl})^{2}× (l_{p}/r_{S})^{2}∼ 10 (r_{S}(Sun)/r_{S})^{2}.

The condition V

_{tube}/V_{S}<1 gives a lower bound to the Schwartschild radius of the object and therefore also to its mass: r_{S}>10^{1/2}r_{S}(Sun) and M>10^{1/2}M(Sun). The lower bound means that the flux tube fills the entire M^{4}volume of blackhole. Blackhole would be a volume filling flux tube with maximal mass density of protons (or rather, neutrons -) per length unit and therefore a natural endpoint of stellar evolution. The known lower limit for the mass of stellar blackhole is few stellar masses (see this) so that the estimate makes sense.

- An objection against this picture are very low mass stars with masses below .5M(Sun) (see this) not allowed for k≥ 107. They are formed in the burning of hydrogen and the time to reach white dwarf state is longer than the age of the universe. Could one give up the condition that flux tube volume is not larger than the volume of the star. Could one have dark matter in the sense of n
_{2}-sheeted covering over M^{4}increasing the flux tube volume by factor n_{2}.

- This picture does not exclude star like structure realized in terms of analogs of protons for scaled up variants of hadron physics M
_{89}hadron physics would have mass scale scaled up by a factor 512 with respect to standard hadron physcs characterized by Mersenne prime M_{107}. The mass scale would correspond to LHC energy scale and there is evidence for a handful of bumps having interpretation as M_{89}mesons. It is of course quite possible that M_{89}baryons are unstable against transforming to M_{107}baryons.

- The model for star (see this) inspired by the 10 year old nuclear physics anomaly led to the picture that protons form at least in the core dark proton sequences associated with the flux tube and that the scaled up Compton length of proton is rather near to the Compton length of electron: there would be zooming up of proton by a factor about 2
^{11}∼ m_{p}/m_{e}. The formation of blackhole would mean reduction of h_{eff}by factor about 2^{-11}making dark protons and neutrons ordinary.

**Can one see also stars as blackhole like entities?**

The assignment of blackholes to almost any physical objects is very fashionable, and the universality of the flux tube structures encourages to ask whether the stellar evolution to blackhole as flux tube tangle could involve discrete steps involving blackhole like entities but with larger Planck constant and with larger radius of flux tube.

- Could one regard stellar objects as blackholes labelled by various values of Planck constant h
_{eff}? Note that h_{eff}is determined essentially as the dimension n of the extension of rationals (see this and this). The possible p-adic length scales would correspond to the ramified primes of the extension. p-Adic length scale hypothesis selects preferred length scales as p≈ 2^{k}, with prime values of k preferred. Mersennes and Gaussian Mersennes would be in favoured nearest to powers of 2.

The most general hypothesis is that all values of k in the range [127,107] are allowed: this would give half-octaves spectrum for p-adc length scales. If only odd values of k are allowed, one obtains octave spectrum.

- The counterpart of Schwartchild radius would be r
_{S}(k)= (L(k)/L(107))^{2}r_{S}corresponding to the scaling of maximal string tension proportional to 1/G by L(107)/L(k)^{2}, where k is consistent with p-adic length scale hypothesis.

The flux tube area would be scaled up to L(k)

^{2}= 2^{k-107}L(107)^{2}, and the constant x== x(107) would scale to x(k)=2^{k-107}x. Scaling guarantees that condition V(tube)/V_{S}does not change at all so that the same lower bound to mass is obtained. Note that the argument do not give upper bound on the mass of star and this conforms with the surprisingly large masses participating in the fusion of blackholes producing gravitational radiation detected at LIGO.

- The favoured p-adic length scales between p-adic length scale L
_{107}assignable to black hole and L(127) corresponding to electron Compton length assignable to solar interior are the p-adic length scale L(113)= 8L(127) assignable to nuclei, and the length scale L(109), which corresponds to p near prime power of two.

- For k=109 (assignable to deuteron) the value of the mass would be scaled by factor 4 to a lower about 12 km to be compared with the typical radius of neutron star about 10 km. The masses of neutron stars around about 1.4 solar masses, which is rather near to the lower bound derived for blackholes. Neutron star could be seen the last phase transition in the sequence of p-adic phase transition leading to the formation of blackhole.

- Could k=113 phase precede neutron stars and perhaps appear as an intermediate step in supernova? Assuming that the flux tubes consist of nucleons (rather than nuclei), one would have r
_{S}(113)= 64 r_{S}giving in the case of Sun r_{S}(113)=192 km.

- For k=127 the p-adic scaling from k=107 would give Schwartschild radius r
_{S}(127) ∼ 2^{20}r_{S}. For Sun this would give r_{S}(127)=3× 10^{9}m is roughly by factor 4 larger than the radius of the solar photosphere radius 7× 10^{8}meters. k=125 gives a correct result. This suggests that k=127 corresponds to the minimal value of temperature for ordinary fusion and corresponds to the value of dark nuclear binding energy at magnetic flux tubes.

The evolution of stars increases the fraction of heavier elements created by hot fusion and also temperatures are higher for stars of later generations. This would suggest that the value of k is gradually reduced in stellar evolution and temperature increases as T∝ 2

^{(127-k)/2}. Sun would be in the second or third step as far the evolution of temperature is considered. Note that the lower bound on radius of star allows also larger radii so that the allowance of smaller values of k does not lead to problems.

- For k=109 (assignable to deuteron) the value of the mass would be scaled by factor 4 to a lower about 12 km to be compared with the typical radius of neutron star about 10 km. The masses of neutron stars around about 1.4 solar masses, which is rather near to the lower bound derived for blackholes. Neutron star could be seen the last phase transition in the sequence of p-adic phase transition leading to the formation of blackhole.

**2. What about blackhole thermodynamics?**

Blackhole thermodynamics is part of the standard blackhole paradigm? What is the fate of this part of theoretical physics in light of the proposed model?

** 2.1. TGD view about blackholes**

Consider first the natural picture implied the vision about blackhole as space-filling flux tube tangle.

- The flux tubes are deformations of cosmic strings characterized by cosmological constant which increases in the sequence of increasing the temperature of stellar core. The vibrational degrees of freedom are excited and characterized by a temperature. The large number of these degrees of freedom suggests the existence of maximal temperature known as Hagedorn temperature at which heat capacity approaches to infinity value so that the pumping of energy does not increase temperature anymore.

The straightforward dimensionally motivated guess for the Hagedorn temperature is suggested by p-adic length scale hypothesis as T= xhbar/L(k) , where x is a numerical factor. For blackholes as k=107 objects this would give temperature of order 224 MeV for x=1. Hadron physics giving experimentally evidence for Hagedorn temperature about T=140 MeV near to pion mass and near to the scale determined by Λ

_{QCD}, which would be naturally relate to the hadronic value of the cosmological constant Λ.

The actual temperature could of course be lower than Hagedorn temperature and it is natural to imagine that blackhole cools down. The Hagedorn temperature and also actual temperature would increase in the phase transition k→ k-1 increasing the value of Λ(k) by a factor of 2.

- The overall view about the situation would be that the thermal excitations of cosmic string die out by emissions assignable perhaps to black hole jets and also going to the cosmic string until a state function reduction decreasing the value of k occurs and the process repeats itself.

The naive idea is that this process eventually leads to ideal cosmic string having Hagedorn temperature T= hbar/R and possible existing at very low temperature: this would conform with the idea that the process is the time reversal of the evolution leading from cosmic strings to astrophysical objects as tangles of flux tube. This would at least require a phase transition replacing M

_{107}hadron physics with M_{89}hadron physics and this with subsequent hadron physics. One must of course consider also all values of k as possible options as in the case of the evolution of star. The hadron physics assignable to Mersenne primes and their Gaussian counterparts could only be especially stable against a phase transition increasing Λ (k).

**2.2. What happens to blackhole thermodynamics in TGD?**

Blackhole thermodynamics (see this) has produced admirable amounts of literature during years. What is the fate of the blackhole thermodynamics in this framework? It turns out that the the dark counterpart of of Hawking radiation makes sense if one accepts the notion of gravitational Planck constant assigned to gravitational flux tube and depending on masses assignable to the flux tube. The condition that dark Hawking radiation and flux tubes at Hagedorn temperature are in thermal radiation implying T_{B,dark}= T_{H}. The emerging prediction T_{H} is consistent with the value of the hadronic Hagedorn temperature.

- In standard blackhole thermodynamics the blackhole temperature T
_{B}identifiable identifiable as the temperature of Hawking radiation (see this) is essentially the surface gravity at horizon and equal to T_{B}= κ/2π= hbar/4π r_{S}is analogous to Hagedorn temperature as far as dimensional analysis is considered. One could think of assigning T_{B}to the radial pulsations of blackhole like object but it is very difficult to understand how the thermal isolation between stringy degrees of freedom and radial oscillation degrees of freedom could be possible.

- The ratio T
_{B}/T_{H}∼ L_{p}/4π r_{S}would be extremely small for ordinary value of Planck constant. Situation however changes if one has

T

_{B}= hbar_{eff}/4π r_{S},

with hbar

_{eff}= nhbar_{0}=hbar_{gr}, where hbar_{gr}is gravitational Planck constant.

The gravitational Planck constant hbar

_{gr}was originally introduced by Nottale (see this and this) assignable to gravitational flux tube (presumably non-monopole flux tube) connecting dark mass M_{D}and mass m (M and m touch the flux tubes but do not define its ends as assumed originally) is given by

hbar

_{gr}= GM_{D}m/v_{0},

where v

_{0}<c is velocity parameter. For the Bohr orbit model of inner planets Nottale assumes M_{D}= M(Sun) and β_{0}=v_{0}/c≈ 2^{-11}. For blackholes one expects that one has β_{0}<1 is not too far from β_{0}=1.

The identification of M

_{D}is not quite clear. I have considered the problem how v_{0}and M_{D}are determined in (see this and this). For the inner planets of Sun one would have β_{0}∼ 2^{-11}∼ m_{e}/m_{p}. Note that the size of dark proton would be that of electron, and one could perhaps interpret 1/β_{0}as the h_{eff}/hbar assignable to dark protons in Sun. This would solve the long standing problem about identification of β_{0}.

- One would obtain for the Hawking temperature T
_{B,D}of dark Hawking radiation with h_{eff}=h_{gr}

T

_{B,D}= (ℏ_{gr}/ℏ) T_{B}= (1/8π β_{0})× (M_{D}/M) × m .

For k=107 blackhole one obtains

T

_{B,D}/T_{H}= ( ℏ_{gr}/ℏ)× T_{B}× (L(107)/xℏ)= (1/8π β_{0}(107))× (M_{D}/M) × (L(107)m/xℏ) .

For m=m

_{p}this gives

T

_{B,D}/T_{H}= ℏ_{gr}/ℏ) T_{B}× (L(107)/xℏ)= (1/8π x β_{0}(107))× (M_{D}/M) × (m_{p}/224 MeV) .

The order of magnitude of thermal energy is determined by m

_{p}. The thermal energy of dark Hawking photon would depend on m only and would be gigantic as compared to that of ordinary Hawking photon.

- Thermal equilibrium between flux tubes and dark Hawking radiation looks very natural physically. This would give

T

_{B,D}/T_{H}=1

giving the constraint

(ℏ

_{gr}/ℏ) T_{B}×( L(107)/xℏ)= (1/8π x β_{0})× (M_{D}/M) (m_{p}/224 MeV)=1 .

on the parameters. For M/M

_{D}=1 this would give xβ_{0}≈ 1/6.0 conforming with the expectation that β_{0}is not far from its upper limit.

- If ordinary stars are regarded as blackholes in the proposed sense, one can assign dark Hawking radiation also with them. The temperature is scaled down by L(107)/L(k) and for Sun this would give factor of L(107)/L(125)=2
^{-9}if one requires that r_{S}(k) corresponds to solar radius. This would give

T

_{B}(dark,k)→ (ℏ_{gr}/ℏ)× (L(107)/L(k)) T_{B}= (2^{(k-107)/2}/8π β_{0})× (M_{D}/M) × m .

For k=125 and M

_{D}= M this would give T_{B}(dark,125)= m/2π.

The condition T

_{B,D}= T_{H}for k=125 would require scaling of β_{0}(107) to β(125)= 2^{-9}β_{0}(107) ≈ 2^{-11}. This would give β_{0}(107)≈ 1/4 in turn giving x ≈ .66 implying T_{H}≈ 149 MeV. The replacement of m_{p}=1 GeV with correct value .94 GeV improves the value. This value is consistent with the value of hadronic Hagedorn temperature so that there is remarkable internal consistency involved although a detailed understanding is lacking.

- The flux of ordinary Hawking thermal radiation is T
^{4}_{B}/ℏ^{3}. The flux of dark Hawking photons would be T^{4}_{B,dark}/ℏ_{gr}^{3}= (ℏ_{gr}/ℏ) T_{B}^{4}and therefore extremely low also now also. In principle however the huge energies of the dark Hawking quanta might make them detectable. I have already earlier proposed that T_{B}(h_{gr}) could be assigned with gravitational flux tubes so that thermal radiation from blackhole would make sense as dark thermal radiation having much higher energies.

One can however imagine a radical re-interpretation. BHE is not the thermal object emitting thermal radiation but BHE plus gravitational flux tubes are the object carrying thermal radiation at temperature T

_{H}= T_{B}. For this option dark Hawking radiation could play fundamental role in quantum biology as will be found.

- What about the analog of blackhole entropy given by

S

_{B}= A/4G= π l_{Pl}^{2}T_{B}^{2},

where A= 4π r

_{S}^{2}is blackhole surface area. This corresponds intuitively to the holography inspired idea that horizon decomposes to bits with area of order l_{P}^{2}?

The flux tube picture does not support this view. One however ask whether the volume filling property of flux tube could effectively freeze the vibrational degrees of flux tubes. Or whether these degrees of freedom are thermally frozen for ideal blackhole. If so, only the ends of he flux tubes at the surface or their turning points (in case that they are turn back) can oscillate radially. This would give an entropy proportional to the area of the surface but using flux tube transversal area as a unit. This would give apart from numerical constant

S

_{B}= A/4L(k)^{2}.

**2.3. Constraint from ℏ**

_{gr}/ℏ>1 Under what conditions mass m can interact quantum gravitationally and are thus allowed in h_{gr} for given M_{D}?

- The notion of h
_{gr}makes sense only for h_{gr}>h. If one has h_{gr}<h assume h_{gr}=h. An alternative would be h_{gr}=→ h_{0}=h/6 for h_{gr}<h_{0}. This would given GM_{D}m/v_{0}>hbar_{min}(hbar_{min}=hbar or hbar/6) leading

m>( β

_{0}ℏ/2r_{S}(M_{D})) × (ℏ_{min}/ℏ) .

This condition is satisfied in the case of stellar blackholes for all elementary particles.

- One can strengthen this condition so that it would satisfied also for gravitational interactions of two particles with the same mass (M
_{D}=m). This would give

m/m

_{Pl}>β_{0}^{1/2}.

For β

_{0}=1 this would give m=m_{Pl}, which corresponds to a mass scale of a large neuron and to size scale 10^{-4}m. β_{0}(125)=2^{-11}gives mass scale of cell and size scale about 10^{-5}meters. β_{0}(127)≈ 2^{-12}corresponding to minimum temperature making hot fusion possible gives length scale about 10^{-6}m of cell nucleus. A possible interpretation is that the structure in cellular length scale have quantum gravitational interaction via gravitational flux tubes. Biological length scales would be raised in special position from the point of view of quantum gravitation.

- Also interactions of structures smaller than the size of cell nucleus with structures with size larger the size of cell nucleus are possible. By writing the above condition as (m/m
_{Pl})(M_{D}/m_{pl})>β_{0}, one sees that from a given solution to the condition one obtains solutions by scaling m→ xm and M_{D}→ M_{D}/x. For β_{0}(127)≈ 2^{-11}corresponding to the scale of cell nucleus the atomic length scale 10^{-10}m and length scale 10^{-4}m of large neuron would correspond to each other as "mirror" length scales. There would be no quantum gravitational interactions between structures smaller than cell nucleus. There would be master-slave relationship: the smaller the scale of slave, the larger the scale of the master.

**2.4. Quantum biology and dark Hawking radiation**

The scaling formula β_{0}(k)∝ 1/L(k) with flux tube thickness scale given by L(k) allows to estimate β_{0}(k). In this manner one obtains also biologically interesting length scales. An interesting question is whether the scales for the velocities of Ca waves (see this) and nerve pulse conduction velocity could relate to v_{0}.

- The tube thickness about 10
^{-4}m, which corresponds to ordinary cosmological constant being in this sense maximal corresponds to the p-adic length scale k=171. The scaling of β_{0}∝ 1/L(k) gives v_{0}(171)∼ 4.7 μm/s. In eggs the velocity of Ca waves varies in the range 5-14 μm/s, which roughly corresponds to range k∈ {171,170,169,168}.

In other cells Ca wave velocity varies in the range 15-40 μm/s. k=165 corresponds to 37.7 μm/s near the upper bound 40 μm/s. The lower bound corresponds to k=168. k=167, which corresponds to the larges Gaussian Mersenne in the series assignable to k∈{151,157,163,167} the velocity is 75 μm/s.

- For k=127 gives v
_{0}∼ 75 m/s. k=131 corresponds to v_{0}= 18 m/s. These velocities could correspond to conduction velocities for nerve pulses in accordance with the view that the smaller the slave, the larger the master.

To sum up, these considerations suggest that biological length scales defined by flux tube thickness and cosmological length scales defined by cosmological constant are related.

See the article Cosmic string model for the formation of galaxies and stars or the chapter of "Physics in many-sheeted space-time" with the same title.

For a summary of earlier postings see Latest progress in TGD.

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