arXiv:gr-qc/9909009v1 2 Sep 1999 tities quan- cosmological Numbers the Large that tities the states, called It is Hypothesis. conjecture this then aypoete ihcranlrenumbers large elemen- certain multiplying with by properties elementary tary Universe presumable of the a and properties particles 1937 the in between mentioned connection first Hypothesis 5] Numbers Dirac[4, Large The 1.1 Introduction 1 Dirac. and and Einstein of Hypothesis conjectures Numbers expla- the Large consistent combines Dirac’s a frac- gives to suggested Universe nation herein the of The model be issues. tal could this theory the final to no through derived still alive but stayed par- decades elementary conjectures following and Both prop- Universe the visible ticles. between the the link in of miraculous erties P.A.M. role a 1937, essential found In Dirac an particles. gravita- elementary that play of first structure could suspected fields Einstein tional A. 1919 In Abstract o ihtecsooia osatΛ. constant cosmological the with mon ytmo nt.Asml iesoa analysis dimensional simple A complete a units. construct Mass, of to enough the . system is the for length and example and Universe time for the verified of be properties easily may which ∗ 1 oabn:TeLreNme a ohn ncom- in nothing has Λ Number Large The bene: Nota Email:[email protected] ,r t r, m, ,R T R, M, h ag ubr yohss uln faself-similar a of Outline Hypothesis: Numbers Large The T t hog h cl relation[1] scale the through = a erltdt h atcequan- particle the to related be can R r = r M m Λ = ntttfu epyi n Meteorologie und f¨ur Geophysik Institut unu-omlgclModel quantum-cosmological ∼ 10 38 − e nvri¨tz K¨oln Universit¨at zu der 41 .GENREITH H. 1 Since . (1) 1 a oetmo about angu- an of have momentum elementary should lar an Universe like the Universe Hence the particle. handle to order in ed oasae unu faction of quantum scaled a to leads hs mlet’ris fmte.Btti explana- this But of composed matter. of be ’grains’ should smallest matter these all theory, experimenta- to and by tries tion particles one elementary question, most this the answer find To is. parti- really elementary as cles, such matter, what is, answered infinite of would kind problem recur. embedding a the an- be otherwise correct as must the regress, explanation Whatever the question 18]). is, this [17, swer to Linde explanation A. an (e.g. give scientists to Numerous tried relationship them? have be between what be not and there there could Universes, should other was why Universe countless the ’nothing’, if from that does problem created it real the plausible, solve sounds Al- not explanation zero. of this fluctua- overall-energy quantum though an have of should type cre- and a was tion by Universe rea- ’nothing’ the no from that and ated was creation admissible. there the that not for son assumption is the creation, takes This the what the before or embedded therefore space happened is Universe and and the big-bang where time of the question that The with is created the this from. were comes for example, Universe explanation for the main raises, where of Universe question the of destiny Relativity. 10], General so- of 35, cosmological equations rotating Einstein’s 2, the to [1, lution for G¨odel’s value spin the to its is to close which close or black-hole rotating limit, huge Kerr a be to pected nadtoa usinta sas unsatisfactory also is that question additional An and of origin the regarding discussion recent The ∗ H Λ = H/ 3 · 2 h π n tcnb sus- be can it and (2) tion always raises the question that these ’grains’ the Oppenheimer/Snyder-solution may be taken could be composed of something that could be de- for a speculation of a Friedmann-Universe inside scribed by an even more elementary description. a , shrinking down to a singularity and This is where the so-called string-theory comes from there expanding back to its maximum expan- in; this gives a topological explanation, with the sion again. strings as the origin of matter. But what are these If someone does not like such an interpretation, strings composed of, they should be pure topology, there is the problem to explain how the expand- as space-time curvatures are [6, 12]. ing Universe let behind its own event-horizon. As the standard Big-Bang Universe starts to expand at 1.2 Can the Universe be described vanishing spatial dimensions, this leads to the fol- as a Black Hole ? lowing seemingly paradoxical context: Either the Schwarzschild-radius is greater-equal to the world The critical mass density of the Universe is the radius 2 of the Universe, as large estimations of amount of mass which is needed to bring the uni- the Universe mass assume, then the Universe may versal expansion to a halt in the future and to a be called a black hole 3. Or, as small estimates collapse in a final . This mass is, depend- of the mass of the Universe assume, the Schwarz- ing on the world model, about 1053 kg. The visible schild-radius is about some 10 percent of the today mass of the Universe is sufficient for only about world radius of the Universe. But then the Universe some percent of the critical mass density. On the should have expanded beyond 4 its Schwarzschild- other hand the mass density of the baryons should radius in former times when it crossed a radius of be 10 to 12 percent of the critical mass density [3] as at least 1.5 billion light-years, which is an event can be derived from the theory of nucleosynthesis that is supposed to be not in casual harmony with and the measured photon density. Therefore the so- . By this one could guess that our called dark matter should be about 10 times larger Universe should be indeed a black hole. than the visible bright mass. This matter shows The theory of inflation [18] tries to avoid such itself in the of light on its way through a paradox by a kind of extremely fast inflationary the Universe as well as by its gravitational force expansion. But finally the standard inflationary on galaxies and galaxy clusters. Including this, the model also demands a critical mass density for the mass density of baryonic matter seems to be not Universe. Recent models prefer cosmologies with large enough to close the Universe. Besides the a Λ = 0 and small densities, classical baryonic matter therefore also exotic mat- which are also in agreement6 with the interpretation ter is taken into consideration as far as the overall of the luminosities of far away Type-Ia-Supernovae mass of the Universe is concerned. which seem to be about 0.3m darker than expected. 2GM The Schwarzschild-radius Rss = c2 of a given These more complex Λ = 0-models open a large mass M is derived from the Einstein field equations 6 2 considering a point-like mass. The Schwarzschild Although the dimension of the world radius and the Schwarzschild-radius may be equal, there is a considerable solution is a static, homogeneous and isotropic so- difference: The world radius is the ’visible’ dimension which lution for the region outside the Schwarzschild ra- is defined by an infinite red shift. On the other hand, the dius. However, the inside solution may look other. Schwarzschild-radius can never be seen. Even when an ob- A very interesting solution is the solution of Op- server is placed close to this border (as seen from ’outside’): In this case the line of sight would be curved when seen from penheimer and Snyder [31, 26] which shows the as- outside but an inside observer would observe it as free of any tonishing result that the inside solution must be forces and straight. Because of this any number of world ra- a Friedmann-Universe [9]. This result is an out- dius’ can be placed into the area of the Schwarzschild- radius come from the fitting of the outside to the in- even if both have the same dimension. 3A black hole, if watched from inside, may be called a side solution of a collapsing star: after the burn- because it is guessed that from its inside singu- ing out of the star every pressure vanishes when larity everything can come into existence [13, 9]. A black the star shrinks to 9/8-th of its Schwarzschild- hole actually needs an outer space which it is defined on, radius. When the collapsing star reaches the event- but the existence or non-existence of an outer space is first of all a question of belief. horizon one has to set p = 0 and one gets a 4A ’pushing in front’ of the event- horizon is easily pos- Friedmann-Universe at its maximum expansion. So sible due to the increasing scale-factor R(t) of the Universe.

2 variety of possible large scale structures of the Uni- is a solution of RR˙ 2 = Const., which gives verse. It shall be emphasized, that especial the 2/3 non-critical, so-called open (hyperbolic) Universes, te R(te)= R0 ( ) . (3) must not have an infinite volume in any case[27]. · t0 The correspondance between an open and an in- Therein t is the time of emission of a signal and finite Universe is only true for the very special e t = 0 is the time of the big bang and t is the case of a zero cosmological constant in a simple- e 0 present time. With the Hubble equation dR = connected topology of the Universe [21]. Also there dt H(t)R(t) follow the relations for the age and size exist competing theories to explain the luminosi- of the Universe: ties of the Type-Ia-, such as taking into account the nonhomogenity[20] of the Universe on 2 2c tU = and RU = medium scales and others. And the work of Ga- 3H0 H0 wiser and Silk [7], which numerically calculates the 10 most discussed cosmological models and relates The actual distance r between two points with pre- them to the variation of the observed cosmic mi- vious distance ρ is derived from the equation: crowave background (CMB), shows the astonishing r(t)= R(t) ρ result that the critical standard model fits the ob- · servation of the CMB by far best. The standardization is given by reference to the Therefore, as a working-hypothesis, herein a present time R(t0)= R0 =: 1 . black-hole-Universe is defined through a Universe General Relativity demands a maximal velocity of critical mass density. The possibility of a black- c only for the peculiar movement. The variation of hole Universe was already taken into consideration the scale-factor in the literature. Just as examples, out of the un- counted works about this idea, are cited here the dR 2R0 = 2/3 1/3 books of J-P.Luminet [19]and Smolin [36]. In the dte 3t0 te works of Carneiro [1, 2] the Universe is considered to be a huge [1] with remark- runs to infinity for small times of emission. There- able links between particle and cosmology. fore the overall-velocity of the Expansion A model of a Universe as a white hole is also pro- dr dR dρ vided by Gibbs [9]. = ρ + R (4) dt dt · · dt can be much greater than the . Hence, 2 Simple solutions for a black- considering events for which the speed of light is a hole-Universe given limit, one has to look at variations of ρ. The fact that the EdS-model is simplified with The main sections of the model[8] are repeated5 a curvature parameter κ = 0 seems to point out here for convenience. The coordinate frame used (as (4) never turns its sign to minus) that it is not herein is, if not mentioned differently, for an ob- suitable to treat a black-hole Universe which should server at spatial infinity. have an κ = +1 and should collapse in a finite time. The critical EdS-Universe is, in the scope of this simple model, a black-hole-Universe which 2.1 The Einstein-deSitter model collapses in an infinite time. The Einstein-deSitter model (EdS model) follows from the Friedmann-models which are simplified 2.2 The Planck dimensions with a curvature parameter κ = 0 and a cosmo- If one equates the Planck energy E =¯hω with the logical constant Λ = 0. Therefore the Universe is Einstein energy E = m c2 of a mass charged parti- approximated with an euclidic, isotropic and homo- 0 cle one gets the de-Broglie wavelength of a resting geneous world model. Hence the scale-factor R(t) particle m0, which is also known from the theory 5You may skip to section 3.4. of photon scattering on as the Compton

3 wavelength λ = h¯ . The wavelength of the par- This is the mass which makes an EdS Universe criti- C m0c ticle is in inverse proportion to its mass than the cal. The value of H0 is still controversial and differs Schwarzschild-radius of the corresponding black- depending on the source between approximately 50 2Gm km hole ρss = c2 . The identity of both lengths leads and 100 secMpc . So the mass of the Universe should 6 53 to the Planck dimensions : be in the bounds MU [1.248, 2.497] 10 kg . For any computational purpose∈ herein the· average c¯h 8 km mpl = =1.54 10− kg value of 75 secMpc is used, which agrees also with re- r2G · cently elaborated values (72 6)km/(secMpc) [29] ± ¯h 2¯hG 35 of the Hubble constant. lpl = = 3 =2.29 10− m mplc r c · The mass formula (7) is now generalized to local gravitational waves running in a local flat space- ¯h 2¯hG 43 tpl = 2 = 5 =0.762 10− sec time. For that purpose (7) is expanded to a Tay- mplc c · r lor series at the time t0 transforming the time co- 5 2 c ¯h 9 ordinate to t = t0 te. By this one gets the Epl = mplc = =1.38 10 J (5) − r 2G · mass formula for small masses implying small times t<

4 or infinite. Every distortion 8 of space-time caus- Then the masses of the virtual Schwarzschild areas ing a mass creates an event-horizon proportional to have the eigenvalues 11 : this mass. Then the wave hits its self-made horizon. m± = √n m (17) The rest energy of the so created (virtual) mass n ± · pl is borrowed from the energy of the gravitational The difference of neighbouring positive eigenvalues wave and is transformed into a potential energy of is the same quantity represented by a virtual-black- 1 hole 9. ∆mn = (√n +1 √n)mpl = mpl (18) − ∼ 2√n · for large n. From the equations (17) and (18) fol- 3 An approximate stationary lows 2 2 solution for black hole par- 1 mpl 1 mpl ∆mn = with n = 2 (19) ticles 2 · mn 4 · ∆mn and the relation between stimulated mass ∆mn and As a simple approximation one can consider the ef- positive virtual mass mn is: fect of a wave in a rectangular potential well. The ∆m 1 gravitational wave runs unhindered in a small area n = (20) until it is stopped, as mentioned from outside, by mn 2n its self-made event-horizon which exerts a nearly in- The radius of an elementary particle in this model finite apparent force Fe which hinders the progress is the event-radius of the virtual mass of the wave 10 : 2Gmn ρn := ρSS(mn)= 2 (21) V = 0 for r [0,ρ ] c ∈ SS V = for r>ρ (14) The formula (19) therefore serves the equation SS h¯ h¯ ∞ λC = m0 λC = for the Compton m0c ⇔ · c The common known solution of the time-indepen- wavelength of an elementary particle: dent Schr¨odinger equation for a particle in such a m2 4Gm ¯h rectangular potential well delivers the energy eigen- pl n ∆mn 2ρn = 2 = (22) values · 2mn · c c 2 2 2 n π ¯h with the substitutions m0 = ∆mn and λC =2ρn. En = (15) 2ma2 Heisenberg’s uncertainty relation is fulfilled, dur- with n = 1, 2, 3 ... and a the diameter of the well. ing the creation of the virtual black hole, for the 4Gm stimulated mass: With the substitution of a =2ρSS = c2 and m = En 2 c2 one gets the energy eigenvalues as the real roots ∆x 2Gmn Gmpl ¯h 2 2 2 10 2 4 n π h¯ c ∆E∆t = ∆En = ∆mnc 2 = = of En = 32G2 : · c · c c c 2 · (23) √π ~ √ The angular momentum L = m~v ~r of a wave En = 3/4 nEpl (16) × ±2 with mass ∆mn which circulates at the speed of light c along the horizon of a black hole with mass √π The factor 23/4 = 1.054 origins from the simpli- mn is: fication of the potential (14) and is set equal to 1. L~ = ∆mnc ρn 8A distortion of such a kind is given in general by the | | · m 2G G c¯h ¯h deviation of the space-time-curvature through the gravita- = pl c √nm = = tional wave itself if the energy of the wave is not neglectible. 2√n · c2 pl c · 2G 2 9A ’virtual’ black hole differs from a ’real’ black hole by the fact that the energy for its creation does not come from ¯h sz = (24) a fatal and by this leaving behind a ⇒ ± 2 large potential well but was borrowed from the energy of a 11The same quantization rule for black holes was found for gravitational wave. example by Khriplovich [14] from a totally different point of 10One may also imagine this wave as a wave travelling view. For the handling of negative masses see also the article with c along the event-horizon. of Olavo [25].

5 The spin of a stimulated black hole particle is, by and they reach their maximum at √n = 1061 with this, half of Planck’s quantum, which corresponds the Universe as the largest Schwarzschild area. The to a fermion 12. Particles relating to the energy of stimulated mass of this SBH-Universe is a rather stimulation of a virtual-miniature-black-hole (18) massless particle, as are neutrino or photon. An will herein be refered to as SBH-particles. extrapolation of the ∆mn/mn dependence for an elementary SBH-particle to the (not allowed) eigen- 3.1 SBH-Particles value n = 0 gives cause to a speculation of a nearly massless particle of which the stimulated mass is The table (1) shows the values of some selected a Universe with an incredibly small lifetime sat- masses refering to the formulas of chapter 3 and isfieing the uncertainty relation. As follows from gives an idea of an Universe which is built up chapter 3 these quantums may be fermions; the in a fractal manner of black-hole-topologies. All one with the lightest weight is a neutrino and the masses origin from stimulated curvatures of space- one with the heaviest weight is the Planck quan- time and every black hole has its typical quantum. tum. The speculative extrapolation to the eigen- For macroscopic black holes these quantums have value n = 0 gives as the heaviest fermion a Uni- energies much smaller than the rest mass verse. Hence a possible fractal construction of the in the area of (practically) massless 13 particles world is conceivable, based on black-areas of differ- like neutrinos or photons: a macroscopic black hole ent sizes. seems to radiate thermally like a black body. But for virtual-miniature-black-holes the stimulated en- 3.2 The entropy of black holes ergies are in the typical bounds of the well known elementary particle restmasses, matching their typ- A fundamental question in physics is to ical properties as rest energy, Compton wavelength explain the high entropy of black holes. The A and spin. Bekenstein-Hawking-entropy [11] is SBH = 4Gh¯ . 2 The Hawking-times tV are a rough hint for the The surface of the black hole A = 4πρn can be lifetimes of SBH-particles and they show meaning- deployed ful values for stable elementary particles. But ef- 2π S = n (26) fects of quantum gravitation should generate very BH c3 · different values for this lifetimes, namely those ex- and relates the black hole entropy directly to its perimentally seen values which can be much less for excitation level n. instable and much more for stable particles. If one relates the rest-energy of a particle, using The product of virtual mass and stimulated mass the herein derived mass-quantization (17)(18), to is always a constant for every SBH-particle: the Hawking-temperature of a black-hole, which is hc¯ 3 1 2 ch¯ TH = , one gets: C± := ∆m m± = m = 8πkBGM mn n · n ± 2 pl ± 4G 16 2 2 = 1.185 10− kg (25) E0 ∆mnc ± · = =2πkB (27) TH TH(mn) The basic eigenvalue n = 1 of self-curvature is This shows that the SBH-particle is in thermody- given by the Planck mass which has a stimulated namical equilibrium [13] with the Hawking- tem- mass of approximately the same quantity. The sizes perature of the stimulated-black-hole. of the self-curvatures of space-time increase with n 12 ~ G 2 However the value of the spin | L |= c mpl relates to 3.3 The mass of the proton the definition of the Planck mass: If one equates two instead of one Compton wavelength to the Schwarzschild-radius one As pointed out in [8], the gravitational power of the ch¯ gets G for the Planckmass and a spin ofh ¯, which corre- evolving Universe is greater than the power needed sponds to a boson. to create virtual black holes for the first primordial 13Ap massless particle in classical quantum dynamics may also have at least a very small rest mass ≈ 0 according to 1/3 7 1 28 2 3/10 G¯h 1 20 25 the uncertainty relation ∆E∆t = ∆E · t0 ≥ ¯h/2 ⇔ ∆E ≥ tB = ( · ) 5 6 = 2 10− sec ¯h/(2t ) = 3 ¯hH ≈ 10−33eV if one considers a particle 3 c · H ∼ · 0 4 0   0 blurred[27] over the whole Universe. h i (28)

6 Table 1: Chart of SBH-particles: The entries in italics for the ’SBH-Massless’ (n ) and the ’SBH- Universe’ (n 0) are values introduced by an extrapolation of the formulas of cha→pter ∞ 3 to their limits. → The value for the SBH-massless arises if one identifies the virtual mass mn with the mass of the Universe from equation (8), and the value of the mass of the SBH-Universe arises if one identifies the stimulated mass ∆m with the mass of the Universe. The mass of a is estimated here as 10 MeV. n ∼ Eigenvalue virtual mass stim. mass SS-diam. Hawk. time Name n m ∆m ∆E 2ρ = λ t of stim. mass n n n n C ∼ V [1] [kg] [kg] [eV] [m] [sec] m0 = ∆mn

70 53 88 96 0 7.1 10− 1.7 10 9.3 10 2.1 10− 0 SBH-Universe ∼ ± · 8 · 9 · 28 · 35 ≈ 43 1 1.5 10− 7.7 10− 0.4 10 4.6 10− 10− Planck quant ± · · · · ∼ 37 10 27 16 12 2.1 10 7.1 10 1.7 10− 938 [MeV] 2.1 10− 10 SBH-proton · 41 ± · 12 · 29 · 14 ∼ 18 1.9 10 6.6 10 1.8 10− 10 [MeV] 2.0 10− 10 SBH-quark · 43 ± · 14 · 31 · 13 ∼ 22 7.2 10 1.3 10 9.1 10− 0.51 [MeV] 3.9 10− 10 SBH-elektron · ± · · · ∼ 122 53 70 33 26 139 1.2 10 1.7 10 7.1 10− 0.40 10− 5.0 10 10 SBH-massless · ± · · · · ∼

This time may be interpreted as a kind of phase see-saw-mechanism in GUT-theory: change as the Universe stops boiling. On the other 2 hand the uncertainty equation for the Universe mD h¯ mν (31) gives at this time a mass-equivalent of ml = 2 . ≃ M 2c tB R Inserting (28) results in In this case, the Dirac-mass MD could be identified 13 6 with the Planckmass m and the mass of the right- ¯h H0 1 pl m =0.239 ( ) 20 = 1.7 m (29) l · c5G7 ∼ · p handed neutrino MR with the mass of the Universe. If the Universe is assumed to be a kind of a which is a limiting upper value for SBH-masses fermion and the SBH-partner of the neutrino, it close to the mass of the proton. should have an intrinsic rotation of about ¯h = m ω R2 ω = H /2 (32) 3.4 The mass of the neutrino and the 2 ν U U ⇒ U 0 rotation of the Universe by substituting the herein derived mass of the neu- In this model the neutrino is the less weighted trino and the EdS radius and mass of the Universe. fermion, which results from a fermionic excitation As this is just a rough estimation, one should ex- of the Universe as the underlying virtual-mass- pect a rotational ratio of the order of some H0: 2 1 mpl particle. The mass relation (19) ∆mn = 2 m · n ω H (33) gives a minimum mass for the lightest neutrino of U ∼ 0

¯hH0 33 Past works [1, 2, 15, 22, 24] about the claimed ob- m =0.40 10− eV (30) ν ≥ 4c2 · servation of a rotation of the Universe give esti- mates [15] of ω = (6.5 0.5)H . ± 0 which can also be interpreted as the minimum mass From equation (32) follows the scaled quantum for a particle which is blurred over the whole Uni- of action H for the EdS-Universe as verse as shown by Heisenbergs uncertainty relation. The relation (19) shows resemblance to the Dirac- H 4c5 kgm2 = M ω R2 H = 2.5 1088 (34) mass- term [30], which follows from the so-called U U U 2 2 ⇒ GH0 ≈ · sec

7 and with equation (2) the scale-factor of the Large 3.6 The electric charge Numbers Hypothesis can be expressed as: Moreover a black hole has not to gravitate because 1 5 3 as an effect of its event-horizon no gravitational 3 H 4c 2 40 Λ= = = (t H )− 3 =3.3 10 waves or gravitons may leave it. An ordinary black h hGH2 ∼ pl 0 r 0 · hole gravitates because it leaves behind the gravi-   (35) tational field of a collapsing object. If for example a Daemon would cut out a exactly 3.5 The Large Numbers coinci- at the Schwarzschild-border and place it somewhere dences else in the Universe, such a black hole would not gravitate except by the small mass equivalent of We now can combine the formulas (1),(8),(35) to its . This mechanism may define derive the masses of usual matter: elementary particles as the Hawking radiation of M c3 virtual-miniature-black-holes. m = 2 = 2 (36) By this assumption the Compton-wavelength of Λ H0GΛ a particle is assoziated with a virtual black hole, which gives: which means a quantum space-time topology with at least one curvature according to this size. Such 1/3 π2¯h2H a virtual-miniature-black-hole is charged with the m = 0 (37) 4Gc constant Cm±n (25). This charge is independent of   the mass of the virtual black hole and could give rise This formula is equivalent to the empirical Wein- to an electrical charge by effects: berg formula [37, 35] for the mass of the pion the gravitational force between a non-gravitating 2 1/3 virtual-black-hole and a gravitating stimulated- m h¯ H0 . The Large Numbers coincidence π ≃ Gc mass (20) would be 2n-times stronger than the is the fact, that all elementary particles are close gravitational force between two stimulated masses. together considering the large number Λ: The classical ratio between the gravitational m2 3 3 e c ln (c /(mH0G)) force FG = G r2 and the electromagnetic force m = x = (38) 1 e2 x F = 2 for for example an electron in any H0GΛ ⇒ ln Λ Q 4πε0 r given distance· r is: Setting in the masses of the proton, pion and elec- tron, x is clustered 14 at values close to 2, which are F e2 Q = =4.167 1042 (40) 1.97411, 1.99458 and 2.05465 respectively. So all el- 2 FG 4πε0Gme · ementary particles are determined by the same Λ. The mass (36) can be compared with the herein This classical ratio is also a worth mentioning Large derived mass relation (20) ∆mn = 1 . The Number and it lies just between the eigenvalues for mn 2n Compton-radius of the mass is the Schwarzschild SBH- and SBH-electrons. If the virtual mass 2 radius r = 2Gmn/c of the related virtual-mass, of a SBH-particle would be present as a higher order 2 which gives ∆mn = m = rc /4nG. Comparing the effect in any unknown way, the gravitational force 2 values for m gives r/4n = c/H0Λ , and as will be between a SBH-particle and the virtual-SBH-mass shown later Λ = 2n, this results in: of its vis-a-vis would be:

2c R ∆mn mn c¯h r = = (39) FG virtual = G 2· = 2 (41) H0Λ Λ − r 4r So the classical Dirac conjecture (1) is recovered. As this force is independent of n and therefore inde- pendent of the mass of the particle it can be related 14Further analysis shows that massless particles cluster at x ≃ 3, elementary particles at x ≃ 2, Life (cells . . . biotope to an electromagnetic charge: . . . solar-system) around x ∼ 1, Galaxies,Clusters,Walls around x ∼ 1/2 and the Universe itself is x = 0. See also 1 Q2 c¯h [1]. FQ = 2 = FG virtual = 2 (42) 4πε0 · r − 4r

8 which gives SBH-particles have at least two quantum-numbers in this plain model: The spin s = ¯h/2 and the z ± Q = πε0c¯h = 5.853 e (43) virtual-mass-charge C = c¯h/4G, which corre- ± ± · m±n sponds to the electrical charge± e and a virtual- and is about 6 timesp the elementary charge. As mass of m . A SBH- has± the opposite val- this assumption is just a plain one (as it must be n ues of this± quantum-numbers, so SBH-electron and an effect of higher order), this charge is not so far SBH-positron annihilate to a radiation of 2-times away from unity as it seems at first sight. Hence the the energy of the stimulated mass ∆m , just as Large Number (40) represents the ratio of virtual n electron and positron do. When two SBH-electrons mass to particle mass: meet, they will not merge to a double-massive SBH-

FG virtual mn 38 44 electron, as the two ¯h/2-spins would add to ¯h or 0, Λ − = =2n 10 − (44) but the double-massive SBH-electron must have a ≃ FG ∆mn ∼ ¯h/2-spin. ± 3.7 On the quantum-topology of the SBH 4 Conclusion To considere elementary particle as a static sphere The Large Numbers Hypothesis was discovered by may be to much simplified. Especially one could Dirac in 1937 in which he found that cosmological expect a rotating black hole close to its Kerr-limit, quantities can be related to particle quantities by which gives a more complified model [32, 33, 34]. simple scale relations of which he thought could not The equivalence principle of GR provides the as- just be random in nature. sumption that matter should be equal to a re- Gravitation is a sort of energy and in conse- lated space-time curvature. Hence the quantum- quence itself is a source of gravitation. Such a topology of a SBH may be for example a torus, self-relating characteristic is a typical element of with one main-curvature related to the Planck- any fractal structure. Here a blueprint of a fractal length and the other main-curvature related to the Universe composed of black holes was outlined by Compton-length. Then it would appear to be a assuming that a gravitational wave should not over- closed string. Another consideration could be to run its own event-horizon. The model gives possi- imagine the hadrons as a bag model[16, 23] with ble answers to still miraculous issues, such as the bumps in it, so that the overall-curvature might be entropy of black holes, the origin and integer value of Compton-size and the bumps providing curva- of the electric charge, the spin, sizes and weights of tures which relates to the quark masses and frac- elementary particles and the background of Dirac’s tional charges. Large Numbers coincidences. It was A. Einstein[6] himself who first suspected elementary particles to be build up by gravitational fields. An attempt in recent time was done by Re- References cami et al[28], describing hadrons as strong-black- holes by a concept of strong-gravity. The Recami- [1] Carneiro, S.: 1997; The Large Numbers model describes hadrons and its constitutents with Hypothesis and Quantum Mechanics, gr- the, according to the large numbers hypothesis, qc/9712014 scaled Einstein-field equations of GR. This bi-scale theory of gravitational and strong interactions de- [2] Carneiro, S.: 1998; A G¨odel-Friedman cosmol- livers a description of the confinement and asymp- ogy ?, gr-qc/9809043 totic freedom of the quarks, the Yukawa-potential [3] Coughlan, G. and Dodd, J.: 1991, The ideas of of strong interactions and derives a strong coupling particle physics, Vol. 2nd edition, Cambridge constant and a mesonic mass spectra, which fit well University Press the theoretical and experimental values. A crucial requirement for SBH-particles is that [4] Dirac P.A.M., Letters to the Editor: The Cos- this particles should behave as known particles do. mological Constants, Nature Vol. 139, (1937), So what happens if two SBH-particles collide ? 323.

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