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A emerged Universe

H. GENREITH ∗ Institut f¨ur Geophysik und Meteorologie der Universit¨at zu K¨oln

Abstract there was no reason for the creation and that the Universe was created from ’nothing’ by a kind of A simple model of a Universe is presented 1 com- quantum fluctuation and should have a zero overall- posed of black holes and black branes. It uses the energy. Although this explanation sounds well, it most simplest approximations and models of Gen- doesn’t solve the problem really: If the Universe eral Relativity and Quantum Dynamics to offer an was created from ’nothing’, why shouldn’t be there idea of an unification and gives a possible answer uncountable more Universes; and what relationship to the quantization and entropy of Black Holes. could there be between them ? Numerous scientists It proposes a mass spectra for elementary parti- tried to give an explanation to this question (e.g. cles and gives a vivid interpretation of the particle- A. Linde [14]). Whatever the correct answer to this wave-dualism. question is, the explanation must be a kind of an infinite regress as otherwise the embedding problem would recur. 1 Introduction

A simple model for the construction of fermionic 1.2 The ’matter-grain’ problem matter by socalled black-holes will be suggested. A still unsatisfactory answered question is, what This paper gives a short overview on cosmology in matter, like elementary particles, really is. To an- its first part. In its second part it proposes a pos- swer this question, one tries to find the most el- sibly closed model for fermionic matter from neu- ementary particles by experiment and theory. All trinos to the Universe itself, based on the concept matter should be composed of this smallest ’grains’ of black-holes/black-branes. of matter. But this explanation always raises the question that these ’grains’ could be composed of 1.1 The ’embedding’ problem something, that can be described by a more ele- mentary description. Here comes in the socalled The recent discussion on the origin and destiny of string-theory, which gives a topologic explanation, the Universe raises the question where the Universe the strings, as the origin of matter. But what are comes from. The main explanation is that time and these strings composed of ? They should be pure space were created with the big-bang and therefore topology, like space-time curvatures are [23, 11]. the question, where the Universe is embedded in or what happened before the creation, is not per- missible [9] 2. This implies the assumption that 1.3 The mass of the known Universe The mass3 of the bright matter lying in the reach of ∗Email:[email protected], WWW:http://www.uni-koeln.de/math-nat- modern observation techniques may be derived for fak/geomet/geo/mitarb/genr.html example from the outcome of the Hubble deep field: 1Contribution-desk paper on the 8th International Con- On this image made by the Hubble space telescope, ference on the Structure of Baryons, BARYONS’98, Bonn, Sept. 22-26, 1998. 3The ’mass’ of the Universe can be defined only if there 2The nonpermissibility of the question follows from the is a border and an corresponding volume. So the Universe fact, that Riemannian Geometry can describe the metric mass can be defined in the volume defined by the world intrinsic, so an outer space is mathematically not demanded. radius (which is defined by an infinte redshift) or by the But it also doesn’t exclude the possibility. (hypothetical) Schwarzschild-radius.

1 which contains galaxies down to about 29 th. mag- of Oppenheimer and Snyder [21, 17] which shows nitude, one can calculate about 1 Million Galaxies the astonishing result, that the inside solution must on an area of 1 square degree. With usual masses be a Friedmann-Universe. This results from the fit- for galaxies this means that the mass of the bright ting of the outside to the inside solution of a col- matter reaches an amount of approximately some lapsing star: After the burning out of a star every 1051 kg (Sexl) [21]. The critical mass density of the pressure vanishes when the star shrinks to 9/8-th Universe is the amount of mass which is needed to of its Schwarzschild-radius. When the collapsing bring the universal expansion to a halt in the future star reaches the event-horizon one has to set p =0 and to a collapse in a final . This mass and one gets a Friedmann-Universe at its maximum is, depending on the world model, about 1053 kg. expansion. So the Oppenheimer/Snyder-solution This means that the visible mass of the Universe is maybe taken for a speculation of a Friedmann- sufficient for only about some percent of the critical Universe inside a black hole, shrinking down to a mass density. singularity and from there expanding back to its On the other hand the mass density of the maximum expansion again. baryons derived from the theory of nucleosynthe- If one don’t like such an interpretation, there is sis and the measured photon density should be 10 the problem to explain how the expanding Uni- to 12 percent of the critical mass density [3]. There- verse let behind its own event-horizon. As the fore the so-called dark matter should be about 10 Universe expands starting at vanishing dimensions, times greater than the visible bright mass. This this leads to the a paradox looking context: Ei- matter shows itself in the of light on its ther the Schwarzschild-radius is greater-equal to way through the Universe as well as by its gravita- the world radius4 of the Universe, as large estima- tional force on galaxies and galaxy clusters [6]. tions of the Universe mass assume, then the Uni- Nevertheless the mass density of baryonic matter verse may be called a black hole or , be- is with that only about a tenth of the mass density cause of the time-reversal of the dynamics. Or, needed to close the Universe. Besides of the clas- as small estimates of the mass of the Universe as- sical baryonic matter therefore also exotic matter sume, the Schwarzschild-radius is about 10 percent gets into considerations on the overall mass of the of the today world radius of the Universe. Now Universe. These are, for example, a not vanishing then the Universe should have expanded beyond5 rest mass of the neutrino, super-symetric elemen- its Schwarzschild-radius in former times after the tary particles as e.g. gravitinos, extremely weak big bang when it crossed a radius of about 1.5 bil- interfering particles (WIMPS), and black holes of lion light-years which is an event that is supposed various sizes. So there exist a lot of estimations of to be not in harmony with . the Universe mass density up to several times the By this one could guess that our Universe should 6 critical mass density as is ρU = 0.01 2.0 ρcritical be indeed a black hole . The theory of inflation [14] ∼ − · in Coughlan [3] or ρU = 0.01 10.0 ρcritical in ∼ − · 4Although the dimension of the world radius and the Kaku [11]. Schwarzschild-radius may be equal, there is a considerable difference: The world radius is the ’visible’ dimension which is defined by an infinite red shift. On the other hand, the 1.4 The Schwarzschild-metric and Schwarzschild-radius can never be seen even by an observer placed close to this border as seen from ’outside’. In this the primordial expansion of the case the line of sight would be curved when seen from out- Universe side but an inside observer would observe it as free of any forces and straight. By this any number of world radius’ can 2GM be placed into the area of the Schwarzschild- radius even if The Schwarzschild solution Rss = c2 for the Schwarzschild-radius of a given mass M is de- both have the same dimension. 5A ’pushing in front’ of the event- horizon is easilly pos- rived from the Einstein field equations considering sible due to the increasing scale-factor R(t) of the Universe. a point-like mass. The Schwarzschild solution is a 6A black hole, if watched from inside, may be called a static, homogenous and isotropic solution for the white hole because it is guessed that from its inside singu- region outside the (’A black larity everything can come into existence [12]. A black hole actually needs an outer space which it is defined on, but the hole has no hairs’). The inside solution may have existence or not of an outer space is first of all a question of other solutions, the most interesting is the solution belief.

2 tries to avoid such a paradox by a kind of extremely 2 Simple solutions for a black- fast inflationary expansion. But the inflation phase hole-Universe ends when the Universe is some cm in size, much smaller than its least possible event-radius. And 2.1 The Einstein-deSitter model last the inflationary model also demands a critical mass density for the Universe. The Einstein-deSitter model (EdS model) follows from the Friedmann-model simplified with κ =0 But even if one believes in a Universe which is and Λ = 0 which means an euclidic, isotropic and not of critical mass density in its visible worldradius homogenous world model. Then the scale-factor ˙ 2 and which should expand over this range up to in- R(t) is a solution of RR = Const. which gives

finity, there will always be a radius where the mass 2/3 te density of the inner region gets critical: as vacuum R(te)=R0 ( ) (1) is never empty of energy and as the Schwarzschil- · t0 dradius grows linearly with the mass, while the in which te is the time of emission of a signal ( mass of a volume grows with the 3rd power of the t = 0 is the time of the big bang) and t the time 2 2 2 e 0 radius: ρc =3c/8πGR Rc = 3c /8πGρ today. The relationship for the age of the world This gives a critical value for⇔ the mass density of 30 g p follows from the Hubble equation the Universe of some 10− cm3 if one considers the area of the visible worldradius which is R =2c/H0 dR 2 = H(t)R(t)whichgives t0= in the socalled EdS-model. So if the mass density dt 3H0 would be greater-equal to this value, our Universe may be explained as a black hole; if the value is less, The actual distance r between two points with dis- one could avoid such an explanation if one claims tance ρ is derived from the equation: that the regions outside the related event-horizon r(t)=R(t) ρ are gravitational incoherent. But the high unifor- · mity of the 3K-background radiation and also the The standardization is given referring to the theory of inflation indicate that they are indeed co- present time by R(t )=R =: 1 . herent. 0 0 General Relativity demands a maximal velocity c only for the peculiar movement. The variation of A lot of numerical simulatuions of Universes with the scale-factor different parameters were done [7, 24] (and will go dR 2R on). A lot of models prefer cosmologies with Λ =0 = 0 2/3 1/3 6 dte and small densities, which are also in agreement 3t0 te with the interpretation of the luminosities of far away Type-Ia-Supernovae, which seem to be about runs to infinity for small times of emission. There- 0.3m darker than expected. But the recent work of fore the overall-velocity of the Expansion Gawiser and Silk [8], which numerically calculates dr dR dρ the 10 most discussed cosmological models and re- = ρ + R (2) dt dt · · dt lates them to the variation of the observed cosmic microwave background (CMB), shows the astonish- can be much greater than the . Con- ing result, that the critical standard model fits the sidering events for which the speed of light is a given observation of the CMB by far best. limit one has to look upon variations of ρ. The red shift z is interpreted as the scale varia- In this sense a black-hole-Universe is defined tion of the wavelength of a photon: through a Universe of critical mass density. The λ0 R0 possibility of a black-hole-Universe was already = =: 1 + z (3) λe Re taken into consideration e.g. in J-P.Luminets [15] book ’Black Holes’ or in a controversal darwinistic The fact that the EdS-model is simplified with a view in the recent work of Smolin [22, 2]. curvature parameter κ = 0 seemes to point out

3 (as (2) never turns its sign to minus) that it is equals to a mass which bends space to a black hole not suitable to treat a black-hole-Universe which of Planck size. should have an κ = +1 and should collapse in a finite time. But in the scope of this simple model 2.3 The mass formula for the it means that a critical EdS-Universe is a black- Einstein-de Sitter model hole-Universe collapsing in an infinite time. A derivation of the EdS model can be gath- The Eds model delivers the coordinate distance of ered from the current literature, e.g. Sexl [21], an event travelling with the speed of light from (2) c dt Schulz [19], B¨orner [1]. Because of its considerable through the integration of dρ = R·(t) : simplifications the EdS model is analytical treat- 3ct t able and therefore is an excellent tool for the the- ρ = 0 1 ( e )1/3 (5) R · − t oretical handling of cosmological phenomena. On 0  0  the other hand this model doesn’t fit for instance This distance should be equal at maximum to any 2 for large values of the Hubble constant H0 which given Schwarzschild-radius ρSS =2GM/c which leads to a smaller age of the world than observa- results in: tional results suggest. So recently world models are 1 3 3 brought into discussion which are determined by a c 3H0te M(te)= 1 (6) H G· − 2 Λ = 0 [24, 18, 7]. These models are better compat- 0 "   # ible6 with large values of the Hubble constant; they This time-dependent mass is the mass to be at least are mathemathically more complex and have to be included by a gravitational spherewave starting at calculated partly numerically. time te and running with velocity c.Otherwise the wave would have to go beyond its own event- 2.2 The Planck dimensions horizon. Inserting te = 0 for the origin of the Uni- verse one gets: If one equates the Planck energy E =¯hω with the 2 3 Einstein energy E = m0c for the rest energy of a c MUmin M(0) = (7) mass charged particle one gets the de-Broglie wave- ≥ H0G length of a resting particle m0. This is also known This is the mass which makes an EdS Universe crit- from the theory of photon scattering on h¯ ical. The value of H0 is still controversal and differs as the Compton wavelength λC = .Thiswave- m0c depending on the source between approximately 50 length is opposite proportional to the mass as the and 100 km . So the mass of the Universe should Schwarzschild-radius ρ = 2Gm .Theidentityof secMpc ss c2 be in the range of [1 248 2 497] 1053kg . both lengths leads to the Planck dimensions 7: MU . , . Since the geometry∈ of space-time· in our Uni- verse yet depends only on the mass included in ch¯ 8 mpl = =1.54 10− kg the Schwarzschild-radius, the equality sign in (7) 2G · r should be right: The mass of the Universe gets h¯ 2¯hG 35 herewith the rank of a constant of nature as it is lpl = = 3 =2.29 10− m mplc r c · expected for a black hole. In this case one may formulate: h¯ 2¯hG 43 t = = =0.762 10− sec pl 2 5 3 3 mplc r c · c = MUH0G =: α MU (8) 5 2 c h¯ 9 which means thatp a black-hole-Universep would re- Epl = mplc = =1.38 10 J(4) r 2G · late the speed of light to the mass and expansion rate of the Universe. One may define a topologic The main property of the Planck dimension is the constant τ for the EdS model as fact that the energy of a wave with wavelength lpl c3 kg := = =4038 1035 (9) 7The usual description is m = ch¯ from just dimen- τ MUH0 . pl G G · sec sional arguments. I will use the definition following from the equality of event-radius to Compton-wavelength.p See which is the product of the Universe expansion rate also footnote following equation (29) and mass and relates c and G as c = √3 τ G. ·

4 2.4 The resulting force on elemen- mass. After that the wave hits its selfmade horizon. tary level The rest energy of the so created (virtual) mass is borrowed from the energy of the gravitational wave One may generalize the mass formula (6) to local and is transformed into an potential energy of the gravitational waves running in a local flat space- same quantity represented by a virtual black hole10. time. For that purpose (6) is expanded to a Tay- As a result of S. Hawking the power of radi- lor series at the time t0 transforming the time co- ation [21] of a black hole is approximately P ordinate to t = t t . By this one gets the 6 ≈ 0 e hc¯ . As the energy is E = P t = Mc2 this re- mass formula for small− masses implieing small times G2M 2 ∼ sults in an approximately time for· vaporizing of a t<

4 4 d c 2 c solution for black hole par- F~e = (mv)= (1+3H0t+R(O )) = | | dt 2G ∼ 2G (13) ticles This practical constant force acts on the event over Because the elementary force F~e always obstructs the area of the Schwarzschild-radius: the movement of the gravitational wave the integral 4 of energy on a closed loop is not zero. The constant c 2Gm 2 E F ρ = = mc (14) c4 e ss 2 force F~e = ~e has just a pseudo-potential ≈ · 2G · c − 2G · As it seems a gravitational wave may run unhin- c4 V (r)= r (17) dered just if the mass included in its sphere is zero 2G· or infinite. Every distortion9 of space-time causing a mass creates an event-horizon proportional to this As a simple approximation one can consider the ef- fect of a wave in a rectangular potential well. The 8This force is seen only by a observer from far away. An gravitational wave runs unhindered in a small area observer travelling inside a black hole would never feel to hit against the event-horizon because any forced curvature 10A ’virtual’ black hole differs from a ’real’ black hole by of his line of sight would be sensed to be straight. the fact that the energy for its creation does not come from 9A distortion of such a kind is given in general by the a fatal and by this leaving behind a deviation of the space-time-curvature through the gravita- large potential well but was borrowed from the energy of a tional wave itself if the energy of the wave is not neglectible. gravitational wave.

5 until it is stopped, as mentioned from outside, by The radius of an elementary particle in this model its selfmade event-horizon which exerts a nearly in- is the event-radius of the virtual mass finite apparent force of hindering the wave11 in Fe 2Gm going on: ρ := ρ (m )= n (26) n SS n c2 V =0 forr [0,ρSS] ∈ The formula (24) therefore serves the equation V = for r>ρSS (18) ∞ h¯ h¯ The common known solution of the time-indepen- λC = m0 λC = m0c ⇔ · c dent Schr¨odinger equation for a particle in such a rectangular potential well delivers the energy eigen- for the Compton wavelength of an elementary par- values ticle: 2 2 2 2 n π h¯ mpl 4Gmn h¯ ∆m 2ρ = = (27) En = 2 (19) n n 2 2ma · 2mn · c c with n =1,2,3... and a the diameter of the well. with the substitutions m0 =∆mn and λC =2ρn. 4Gm With the substitution of a =2ρSS = c2 and m = During the creation of the virtual black hole En c2 one gets the energy eigenvalues as the real roots Heisenberg’s uncertainty relation is fulfilled for the 4 n2π2h¯2c10 stimulated mass: of En = 32G2 : 2 √π ∆x 2 2Gmn Gmpl h¯ En = √nEpl (20) ∆E∆t =∆En =∆mnc = = ±23/4 · c · c2 c c 2 · (28) √π The factor 3/4 =1.054 origins from the simpli- The angular momentum L~ = m~v ~r of a wave 2 × fication of the potential (18) and is set equal to 1. having mass ∆mn circulating on the horizon of a Then the masses of the virtual Schwarzschild areas black hole with mass mn at the speed of light c is: have the eigenvalues 12 13: L~ =∆mncρn mn± = √n mpl (21) | | · ± · mpl 2G G ch¯ h¯ = c √nmpl = = The difference of neighbouring positive eigenvalues 2√n · c2 c · 2G 2 is with this h¯ sz = (29) ⇒ ± 2 ∆mn = mn+1 mn =(√n+1 √n)mpl (22) − − The spin of a stimulated black hole particle is by which can be written for large n : →∞ this of half Planck’s quantum which corresponds 1 toafermion14. Particles relating to the energy of ∆mn = mpl (23) 2√n· stimulation of a virtual-miniature-black-hole (23) will be herein referred to as SBH-particles. From (21) and (23) follows

2 2 1 mpl 1 mpl ∆mn = with n = 2 (24) 4 Discussion 2 · mn 4 · ∆mn and the relation between stimulated mass ∆mn and 4.1 SBH-Particles positive virtual mass mn is: Table (1) shows the values of some selected masses ∆m 1 n = (25) referring to the formulas of chapter 3 and gives mn 2n an idea of an Universe which is build by a frac- 11One may also imagine this wave as a wave travelling tal manner of black-hole-topologies. All masses around the event-horizon with . c 14 G 2 12 However the value of the spin L~ = m relates to The same quantization rule for Black Holes was found | | c pl by Khriplovich [13](1998) from a totally different point of the definition of the Planck mass: If one attaches instead of view. one two Compton wavelengths to one Schwarzschild-radius 13 one gets ch¯ For the handling of negative masses see also the article G for the Planckmass and a Spin ofh ¯,which of Olavo[16] corresponds to a boson. p

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 chapter→∞ 3 to their limits. The value for→ the SBH-massless arises if one gives the mass of the Universe from (7) for the virtual mass mn , and the value of the mass of the SBH-Universe arises if one gives the mass of the Universe for the stimulated mass ∆mn. The mass of a is herein estimated as 10 MeV. ∼ Eigenvalue virtual mass stim. mass SS-diam. Hawk. time Name n mn ∆mn ∆En 2ρn = λC tV of stim. mass ∼ [1] [kg] [kg] [eV] [m] [sec] m0 =∆mn

70 53 88 96 0 8.3 10− 1.4 10 7.8 10 2.5 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- · 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 · ± · · · ∼ 121 53 70 33 26 139 8.7 10 1.4 10 8.3 10− 0.46 10− 4.3 10 10 SBH-massless · ± · · · · ∼

are initiated by stimulated curvatures of space-time is always a constant for every SBH-particle: where every black hole has its typical quantum. For 1 2 ch¯ C± := ∆mn m± = m = macroscopic black holes these quantums have ener- mn · n ± 2 pl ± 4G 16 2 gies much smaller than the rest mass in the = 1.185 10− kg (30) area of (practical) massless 15 particles like neutri- ± · nos or photons: a macroskopic black hole seems to So large black holes have small stimulated masses radiate thermal like a black body. But for virtual- and small black holes have large stimulated masses. miniature-black-holes the stimulated energies are in The basic eigenvalue n = 1 of self-curvature is given the typical range of the well known elementary par- by the Planck mass which has a stimulated mass of ticle restmasses matching their typical properties as approximately the same quantity. The sizes of the rest energy, Compton wavelength and spin. self-curvatures of space-time increase with n and 60 16 reach their maximum at √n =9.3 10 with the The Hawking times tV are a rough hint for · the lifetimes of SBH-particles which show meaning- Universe as the largest Schwarzschild area. An ex- ful values for stable particles in the range of typi- trapolation of the ∆mn/mn dependence for an el- cal masses for elementary particles. But effects of ementary SBH-particle to the (not allowed) eigen- quantum gravitation should generate very different value n = 0 gives cause to a speculation of a nearly values for this lifetimes, namely those experimental massless particle of which the stimulated mass is seen values which can be much less for instable and an Universe with an incredible small lifetime that much more for stable particles. satisfies the uncertainty relation. Through the extrapolation of this supposition The product of virtual mass and stimulated mass the stimulated mass of an SBH-Universe is a mass- less particle (like neutrino or photon) with a rather 15 Also in classical quantum dynamics a massless parti- infinite lifetime, and the stimulated mass of a mass- cle may have at least a very small rest mass 0 accord- ≈ less particle is an Universe with a rather incredi- ing to the uncertainty relation∆E∆t =∆E t0 h/¯ 2 ∆ ¯ (2 3 33 · ≥ ⇔ E h/ t0)= 4hH¯ 0 10− eV if one considers a parti- ble short lifetime. As follows from chapter 3 these cle blurred≥ over the whole≈ Universe. quantums may be fermions. The most light-weight 16 S. Hawking proposed ’orderly’ primordial black holes in is a neutrino and the most heaviest a Planck quan- the mass-range in question of 1011 to 1015 kg, which should explode as an effect of the Hawking-radiation just nowadays, tum. The speculative extrapolation to the (not but couldn’t be observed yet. allowed) eigenvalue n = 0 gives as the heaviest

7 fermion an Universe. Here one may imagine a in which r is the distance of the objects and R1 possible fractal construction of the world build up and R2 the Schwarzschild radii of the objects. This by black-areas of different sizes. The observation approximation is valid if R1 = R2 = r as for a col- problem of the Kopenhagen interpretation of quan- lapsing black hole. Therefore the estimated gravi- tum dynamics [4], which means ’who watches the tational radiation power of a collapsing black hole Universe’, could be brought closer to an expla- c5 is P G . This power is independent on the mass nation through the assumption that the Universe of the≈ black hole and will be provided by any black watches itself or the Universes watches themselfs: hole as far as R1 R2 r is valid. For a black- ≈ ≈ ’The snake is eating its own tail’ (S.Glashow in Lu- hole-Universe this time is about t0. minet [15]). The rest energy of the Universe at all is EU = 2 MUc which is converted in the time t0 since the 4.2 The entropy of black holes big bang. The average power over all is with (8):

A fundamental question in is to 2 3 5 5 EU MUc c 2 3H0 3 c c explain the high entropy of black holes. The PU ∼= = = c = > A h i tU t0 H0G · 2 2 G G Bekenstein-Hawking-entropy [10] is SBH = 4Gh¯ . (34) From the above follows by setting in the surface This means that the gravitational power of a black- 2 area of the black hole A =4πρn: hole-Universe at all should be large enough to cre- 2π ate all the mass it contains. SBH = n (31) c3 · The energy density of the ∆mn and mn related which relates the black hole entropy directly to its to the Schwarzschild volume of the related virtual mass is: excitation level n. The herein proposed black area mn for elementary particles should therefore be a black ∆E ∆m c2 3c7 1 membrane as proposed by string theory [10]. Its ε = n = n = ∆ Vol. ∼ 4 3 32πG2h¯ n2 size should be of the dimension like it is related 3 πρn · 2 7 to elementary particles by the Compton/deBroglie- En mn c 3c 1 ε = = = (35) relation and the uncertainity-relation. n ∼ 4 3 2 Vol. 3 πρn 16πG h¯ · n If one relates the rest-energy of a particle, using the herein derived mass-quantization (21)(23), to The gravitational power P falls off with the fifth the Hawking-temperatur of a Black-Hole, which is power of the dimension r of the object as (33) hc¯ 3 TH = ,onegets: shows and close to the singularity P (t) diverges. So 8πkBGM the following integration is made for a black-hole- 2 E0 ∆mnc Universe starting at the Planck dimension with = =2πkB (32) TH TH(mn) h¯ 43 tpl = 2 =0.76 10− sec. The time-dependent mplc · This shows that the SBH-particle is in thermody- evolution of the gravitational power can be ap- namical equilibrium[12] with the Hawking- temper- proached with this as follows: ature of the stimulated-black-hole. t 5 ∞ 0 α c P (t)dt = dt = M c2 (36) ∼ r5(t) G U 4.3 The baryogenesis Z0 Ztpl The present model takes the energy to create ele- Inserting for r(t) the expansion of the EdS model mentary particles from the energy of gravitational as referred in (1) the constant α can be determined waves. As a source for this gravitational energy one and one gets after some elementary calculations: may bring in the Universe of the big bang or big bounce. The gravitational radiation power [21] of 7 1/6 24/3 7 Gh¯ 1 two critical objects with dimensions compareable P (t)=( · ) (37) 3 · H6c5 ·t10/3 to their Schwarzschild radii is:  0  5 3 2 c R R From this the time tB of baryogenesis through grav- P 1 2 (33) ≈ G · r · r itational waves can be calculated. This is the time    

8 in which the energy density of εn (35) for a virtual over the whole Universe, like shown by Heisenbergs Schwarzschild area was available. uncertainity relation. The relation (24) shows re- 2 semblance to the Dirac-mass- term[20], which fol- En P (tB ) tn P(tB) 3c ε = = · = · (38) lows from the socalled see-saw-mechanism in GUT- n Vol. ∼ 4 3 8 ¯ 3 πρn πnGh theory: 2 in which t is the time for traversing ρ (m ). In- mD n SS n mν ∼= (41) serting (37) and solving for tB gives: MR In this case, the Dirac-mass MD could be identified 1/3 7 1 28 2 Gh¯ 1 20 3/10 with the Planckmass mpl and the mass of the right- tB =( · ) 5 6 3 c · H0 handed neutrino MR with the mass of the Universe. h  i t0.7 =1.643 pl = 2 10 25 sec (39) 0.3 ∼ − · H0 · 4.6 The photon and the mass of the Universe The time tB is independent on the eigenvalue n and only has a small dependence on the value of From the theory of nucleosynthesis, which uses the hubble constant because the 3/10-power of 2 3 4 7 − the relation of the occurence of D, He, He, Li, H0 doesn’t matter so much as the 7/10-power of a fraction of photons to baryons was derived[3]: tpl. Great standardization theories [3] (GUT) pre- 35 dict the baryogenesis for the time between 10− sec nB 10 10 η = =(4 1) 10− (42) and 10− sec after the big bang. The time value nγ ± · for a baryogenesis through primordial gravitational waves (39) is compatible with this prediction. From the measured photon-density, which is about 3 nγ = 400/cm , the mass density of the Universe, 4.4 The mass of the proton mainly , should be about 10 to 12 percent of the critical mass density[3] 0.1 ρc ρB 0.12 ρc. · ≤ ≤ · As (39) points out the gravitational power of the As most of these photons are cold ones, hav- evolving Universe is greater than the power needed ing the temperature of the cosmic-background- to create virtual black holes for the first primordial radiation 2.7 K, photons give not a worth men- 25 tB 2 10− sec. This time may be interpreted as a tioning amount of energy to the mass density of kind≈ of· phase change as the Universe stops boiling. the Universe. But these photons are ’late’ photons, On the other hand the uncertainity equation for which means that these photons origin somewhere the Universe at this time gives a mass-equivalent else in the Universe and are much red-shifted due to h¯ of m = 2 . Inserting (39) gives: the expansion. In the SBH-model, the photon can l 2c tB be identified with a bosonic excitation of the Uni- 13 6 h¯ H0 1 verse (i.e. space-time). As the Universe is isotropic m =0.239 ( ) 20 = 1.8 m (40) l · c5G7 ∼ · p and homogenous at every point, one has to esti- mate the energy of the photon as a typical photon for a =87 km , which relates close to the H0 Mpc sec like it is emitted by the photospheres of stars in our · proton mass as a limiting upper value for SBH- neighbourhood, which have much higher tempera- masses. tures or typical wavelenghts around 500 nm. With this one gets a fraction of Eγ = hc/λ as 4.5 The mass of the neutrino Eγ /η h In this model, the neutrino is the less weighted = =6.607 (43) m c2 cληm fermion, resulting from the fermionic excitation p p of the Universe as the underlying virtual-mass- 2 which is more than 6 times the energy-density of 1 mpl particle. The mass relation (24) ∆mn = the baryonic mass density. For this example, with 2 · mn gives a minimum mass for the lightest neutrino of a baryonic mass density of ρB =0.131ρc and a hH¯ 0 33 mν 4c2 =0.46 10− eV, which can also be inter- averge photon wavelength of λγ = 500nm at its preted≥ as the minimum· mass for a particle, blurred origin, the photon would bring up the missing 86.9

9 percent of the critical Universe-mass-density 17.In This ratio lies just between the eigenvalues for this interpretation, the photon energy could give SBH- and SBH-electrons. If the virtual mass a considerable contribution to the mass density of of a SBH-particle would be present as a higher order the Universe: the average-photon-energy and also effect in any unknown way, the gravitational force a not vanishing mass of the neutrino could close the between a SBH-particle and the virtual-SBH-mass Universe without the need of exotic particles. of its vis-a-vis would be:

∆mn mn ch¯ FG virtual = G · = (45) 4.7 The electromagnetic force − r2 4r2 As this force is independent on n and therefore in- Moreover a black hole has not to gravitate because dependent on the mass of the particle one may re- as an effect of its event-horizon no gravitational late it to an electromagnetic charge: waves or gravitons may leave it. An ordinary black hole gravitates because it leaves behind the gravi- 1 Q2 ch¯ FQ = 2 = FG virtual = 2 (46) tational field of a collapsing object. If e.g. a Dae- 4πε0 · r − 4r mon would cut out a exactly at which gives the Schwarzschild-border and place it somewhere else in the Universe, such a black hole would not Q = πε0ch¯ = 5.853 e (47) gravitate except by the small mass equivalent of ± ± · its . This mechanism defines and is about 6 timesp the elementary charge. As this elementary particles as the Hawking radiation of assumption is just a plain one (as it must be an ef- virtual-miniature-black-holes. fect of higher order), this charge is not so far away The main difference to classical quantumdynam- from unity as it seems at first sight. Maxwells equa- ics is the assumption that the Compton-wavelength tions of electrodynamics should be an outcome of of a particle is assoziated with a virtual black hole a theory of quantum gravitation and for this rea- or a black brane, maybe a D-brane like suggested son should give some hints to the formulation of in string-theory, which means a quantum space- . time curvature of this diameter. Such a virtual- miniature-black-hole is charged with the constant 4.8 Interactions of SBH-particles (30). This charge is independent on the mass Cm±n of the virtual black hole and could give rise to A crucial requirement for SBH-particles is that this an electrical charge by quantum gravity effects: particles should behave like known particles. So The gravitational force between a non-gravitating what happens if two SBH-particles collide ? In virtual-black-hole and a gravitating stimulated- this plain model, SBH-particles have at least two mass would be 2n-times stronger than the gravi- quantum-numbers: The spin sz = h/¯ 2andthe virtual-mass-charge C = ch/¯ 4G±, which corre- tational force between two stimulated masses. m±n sponds to the electrical charge± e and a virtual- The classical ratio between the gravitational ± 2 mass of mn. force F = me and the electromagnetic force ± G G r2 A SBH- has the opposite values of 1 e2 FQ = 2 for an electron in any given distance this quantum-numbers, so SBH-electron and SBH- 4πε0 · r r is: positron annihilate to a radiation of 2-times the 2 FQ e 42 energy of the stimulated mass ∆mn, just like elec- = 2 =4.167 10 (44) FG 4πε0Gme · tron and positron do. When two SBH-electrons meet, they will not merge to a double-massive SBH- 17 At the beginning the Universe was indeed radiation electron, as the two ¯h/2-spins would add to ¯h or 0, dominated: the photon energy-density of the early Universe was a multiple of the baryonic density and this situation con- but the double-massive SBH-electron would have a tinued for about 1 Million years. As the Universe expanded, h/¯ 2-spin. the wavelength of the photon expanded as well, and the en- ± Recent experiments[5] in a quantum-Hall envi- ergy of the photon seems to vanish. In the SBH-picture, only ronment show the possibility of the creation of frac- the energy-density of the electromagnetic waves decreases, but not the overall energy. See also page 128 ff. of the book tional charged pseudo-electrons. This splitted elec- of R.u.H. Sexl[21] trons, with e.g. e/3ore/5 charges, are created

10 between groups of close circulating Hall-electrons. [3] Coughlan, G. and Dodd, J.: 1991, The ideas of They seem to behave like one would assume for particle physics, Vol. 2nd edition, Cambridge SBH-electrons: in their close vicinity space-time University Press can be bend enough to form a virtual fractional- electron. [4] Davies, P. and Brown, J.R.: 1986, The Ghost In the SBH-picture, the photon is the most ele- in the Atom, Cambridge University Press mentary particle: Massive elementary particles are [5] Daviss, B.:1998, New Scientist 31 Jan. 1998, photons ’captured’ by virtual-black-holes. And in- Splitting the Electron deed, every elementary particle can be transformed 18 to photons by annihilation . [6] Durrer, R.: 1997, Physik in unserer Zeit 1/97, 16

4.9 Resume [7] Frenk, C. and Couchman, H.: 1996, Astron- A simple model of the Universe and its elementary omy 8/96,22 parts was derived by assuming that a gravitational [8] Gawiser, E. and Silk, J.: Science, ,1405 wave should not overrun its own event-horizon. 280 (1998), Extracting Primordial Density Fluctu- From this assumption follows the relation between ations, astro-ph/9806197 masses, Compton-size, and spins of fermions and a vivid interpretation of the particle/wave-dualism is [9] Hawking, S.: 1988, A Brief History of Time, given. The origin of electromagnetic charge as an Vol. 4th edition 1995, Bantam Books, New effect of the virtual mass of a particle was proposed. York Also follows the quantization of black holes, as was also shown e.g. by Khriplovich[13] by dimensional [10] Horowitz, G.T.: 1997, The origin of Black arguments. Hole entropy in string theory, UCSBTH-96-07 All matter should be build up by the event- gr-qc/9604051 horizons of gravitational waves. Every natural wave, like waves running with the speed of light [11] Kaku, M.: 1993, Quantum Field Theory: A or sound, are kinds of black-branes: No informa- modern Introduction, Oxford University Press tion will leave the wavefront to the outer regions. [12] Kiefer, C.: 1997, Physik in unserer Zeit 1/97, As gravitation gravitates, these waves build up 22 closed and stable regions, defining fermionic ele- mentary particles and even the Universe itself. The [13] Khriplovich, I.B.: 1998, On Quantization of Hawking-radiation of the black-branes of the de- Black Holes, BINP 98-17 gr-qc/9804004 rived particles resemble the masses of the known fermions. [14] Linde, A.: 1995, Spektrum der Wissenschaft 1/95,32 References [15] Luminet, J.-P.: 1992, Black Holes, Cambridge University Press [1] B¨orner, G.: 1997, Physik in unserer Zeit 1/97, 6 [16] Olavo, L.S.F.: 1998, Quantum Mechanics as a Classical Theory IV: The negative mass con- [2] Concar, D.: 1997, New Scientist 24 May jecture, quant-ph/9503024 1997,38 [17] Oppenheimer and Snyder: 1939, Phys. Rev. 18Another process to do this, is the capture of a particle 56, 455 (1939) by an ordinary black hole. The captured particle adds its mass to the black hole and after a while it is radiated away [18] Priester, W.: 1995, Uber¨ den Ursprung by Hawking-radiation and the black hole restores its mass as des Universums: Das Problem der Singu- it was before the capture. By this mechanism, also known as the information-paradoxon, the particle is transformed to larit¨at, Nordrh.-Westf¨alische Akademie der a photon loosing its quantum numbers. Wissenschaften, Westdeutscher Verlag 1995

11 [19] Schulz, H.: 1997, Sterne und Weltraum 2/97, 124 [20] Senjanovic, G.: 1998, Neutrino Mass,hep- ph/9402359 [21] Sexl, R.: 1979, Weisse Zwerge - Schwarze L¨ocher, Vieweg, Braunschweig/Wiesbaden [22] Smolin, L.: 1997, The Life of the Cosmos,Ox- ford University Press, New York [23] t’Hooft, G.: 1997, In search of the ultimate building blocks, Cambridge University Press [24] Vaas, R.: 1994, Naturwissenschaftliche Rund- schau 2/94,43

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