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Formation of Complex Matter Structures and Mutual Relations Between the Mass of Elementary

arXiv:gr-qc/0111113v1 30 Nov 2001

ˇ Jozef Sima and Miroslav S?ken? u ?k Slovak Technical University, FCHPT, Radlinsk?ho 9, e 812 37 Bratislava, Slovakia e-mail: sima@chtf.stuba.sk, sukenik@minv.sk
Abstract The model of Expansive Nondecelerative Universe leads to a conclusion stating that at the end of radiation era the Jeans mass was equal to the upper mass limit of a black hole and, at the same time, the e?ective gravitational range of nucleons was identical to their Compton wavelength. At that time nucleons started to exert gravitational impact on their environment which enabled to large scale structures become formed. Moreover, it is shown that there is a deep relationships between the inertial mass of various leptons and bosons and that such relations can be extended also into the realm of other kinds of elementary particles.



Particle physics comprises some hundreds of so-called elementary particles. Their list is regularly updated, reviewed and published [1]. The particles are characterized by their mass (energy), charge, magnetic moment, decay mode and mean life, etc. In spite of general believe that there must be very deep fundamental interrelationships between the particles parameters, these are usually presented and taken into account as independent characteristics. In our previous contributions we manifested a coupling of inertial mass for some kind of particles, namely for electron, muon, and tau neutrinos [2], and for electron, proton, and Planckton [3]. 1

This contribution brings the results relating to two aspects of elementary particles properties. The ?rst part deals with their recombination leading to the formation of large scale structures. The second section represents a continuation of our research devoted to unveiling relations between the properties of fundamental particles, particularly to the relations between the mass of various leptons or bosons.


Recombination and Formation of Large Scale Structures

Jeans mass mJ is the mass at which gravity is balanced by pressure forces, i.e. Gm2 4π 3 J = pr (1) r 3 Expressing the mass through the average matter densityρ mJ = it is obtained for the Jeans mass mJ = 3 4π

4πr 3 ρ 3



p3/2 G3/2 ρ2


where p is the radiation pressure. Except of the relations where numerical factors are given by de?nitions, such factors (e.g. (3/4π)1/2 in the above equation) will be further omitted. At the end of radiation era (the quantities related to this time are denoted by the subscript r) the Universe was in the state of thermodynamic equilibrium, i.e. pr = ρr c2 3 (4)

The model of Expansive Nondecelerative Universe (further ENU) [4-6] has answered the problem of matter density at the end of the radiation era giving ρr = 3c2 8πGa2 r (5)

where the gauge factor ar had at the end of the radiation era the value of [7] ar ? 1022 m = 2 (6)

Based on (1) to (6) is follows ar c2 ? 49 mJ,r ? = = 10 kg 2G


Immediately after the recombination, the radiation pressure dropped by S times, where S means the speci?c entropy de?ned as the mean number of photons per one nucleon. It holds for speci?c entropy [7,8] S ? 109 = (8)

and thus after the recombination (gauge factor did not signi?cantly changed during recombination), mJ approached to the value mJ = ar c2 2G (S)3/2 ? 1035 kg = (9)

Stemming from the ENU model background and entropy considerations it was possible to estimate an upper mass limit of black holes m(BH) max and its time evolution [6]. Their gravitational radius r(BH) max is generally expressed as 1/4 r(BH) max = a3 lP c (10) At the beginning of the matter era it had to hold m(BH) max = (a3 lP c ) r 2G
1/4 2

c ? 35 = 10 kg


which is identical value to that provided by relation (9). Putting (9) and (11) equal, relation ar (12) S6 ? = lP c is obtained. Speci?c entropy can also be expressed [9] as S= mp c2 hνr (13)

where mp is the proton mass (1.67262158 × 10?27 kg) and hνr is the mean photon energy at the end of the radiation era. From the beginning of the Universe expansion up to the end of the radiation era the photon energy gradually decreased in time as documented by relation hνr ≈ a?1/2 3 (14)

In accordance with (13) and (14), the photon energy at the end of the radiation era was 1/2 2 lP c hνr = mP c c (15) ar where mP c and lP c are Planck mass and length [1], respectively mP c = lP c = hc ? G G? h 3 c

= 2.176716 × 10?8 kg

(16) (17)

= 1.616051 × 10?35 m

Then, based on (13) to (17) it follows for speci?c entropy S= mp mP c ar lP c


and stemming from (12) and (18), at the end of the radiation era h2 ? ar ? = Gm3 p (19)

The above relation is of key signi?cance [5,6]. The ?nal relation for speci?c entropy, based on (18) and (19), adopts the form S? = mP c mp



Relationships Between the Elementary Particles Masses

It seems to be obvious that completing the recombination, the Compton wavelength of nucleons equals to their e?ective gravitational range. This is the time when gravitational in?uence of nucleons on their environment started to be e?ective that, in turn, enabled to more complex matter structures be formed. Before the recombination nucleons could not exert gravitational impact. The mass in equation (7) represents in the ENU the total Universe mass at the end of radiation era, i.e. a limit mass. The above conclusions suggest the existence of an important relationship between the mass of nucleons, ionization energy of the hydrogen atom E(H) , and the mass of the electron me . It should be pointed out that after the 4

recombination, the mass mJ is identical to the maximum mass of a black hole at the given time. The recombination started at the temperature Tr . Using relation (20) it may be written mp 1/2 ? ?E S ?1 ? (21) = = exp ? mP c kTr where ?E = E(H) ? kTr (22) Providing that hνr ? kTr = it follows from (21), (22), and (23) that hνr ? = E(H) 1 ? ln
mp 1/2 mP c



Since the ionization energy of the hydrogen atom can be expressed as α2 me c2 E(H) ? e = 2 (25)

where αe is the constant of hyper?ne structure (? 7.3 × 10?3 ). When (15) = and (24) are put equal, using (19) and (25) the relation (26) relating the electron, proton and Planckton masses is obtained 2 1 ? ln
mp 1/2 mP c 2 αe m3 p mP c 1/2

me ? =

? 4.0 × 10?31 kg =


Taking into account the simpli?cation of some relations (e.g. the omissions of numerical coe?cients), the calculated electron mass is in good agreement with its known value (? 9.1 × 10?31 kg). = We believe that relationships analogous to (26) should exist also for other couples of particles. In this part we manifest such connections for some couples of stable leptons and bosons. Near the energy of 100 GeV the uni?cation of electromagnetic and weak interaction occurs. The energy corresponds to the mass of vector bosons Z and W. When the proton mass is substituted in (26) for the vector boson W mass, and adjusted value [10] of the splitting constant is taken into calculation, relation (26) leads to the lepton ? mass (? 230 me ) which is very close to its actual mass (206.7me ). Another relation can be found for the heavy lepton τ mass (which is about 3.03 × 10?27 kg [1]) and the mass of one of the 5

Higgs bosons substituting the proton mass in (26) for the Higgs boson mass 7 × 10?25 kg [1]. It may be demonstrated that (26) can be taken as a bridge between the macro-world (the Universe) and the micro-world (particles). Substitution of the proton mass in (26) by the value of 5.35 × 10?12 kg (the mass of bosons X,Y) leads directly to Planck mass.

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