- with attempt to unificate particle and radiation form of matery -


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The intention of this paper is to point to an interesting (reciprocal) relationship and global link between macro (...galaxy clusters, planets...) and micro objects (...elektrons, neutrinos...) with a possible connection among corpuscular objects and their representation in wave form. All research done in the field of theoretical physics in the past century, proved that energy is quantifiable. This fact offers a possibility of energy density quantifying, and, basically some kind of space quantifying, which is expressed by the existence of "stable" objects. Observing the entire uniformity of space-time-matter from objects whose mass is almost negligible (e.g. neutrino) to the whole Universe whose mass is estimated at about 1053 [kg], one can see as the existence of "stable" objects on an entire scale of values.


-table formation-

- Stable object definition:

   An object is stable if its lifetime is sufficiently long, or much longer than its time of formation.

   The computation of the mean value density for some stable objects shows that their value is approximately p=103k [g/cm3], where k is a real number close to its entire value.

   From these results Hypothesis 1 is introduced.

- Hypothesis 1:

   The stable objects density is defined as: pk=103k..........................(1)
   where k is a whole number.

- Stable object table (Table I) formation:

   m = ordinary or virtual (L. de Broglie) mass of object [kg],
   r = classical or fictious radius of object [m],
   p = objects density (specific mass), in the sense of energy density [g/cm3],
   pk = stable object density as defined in Hypothesis 1 [g/cm3],
l = Compton wavelength, l=h/mc [m],
   Ms = Solar mass [2] : Ms=1.989*1030 [kg].

a) Electron [1],[2],[3]:

   me=9.1*10-31 [kg], l=2.43*10-12 [m], 10-19 [m] < re < 10-15 [m],
   with rmean: 10-17 [m] => p=2.1*1017 [g/cm3] and k=5.72.
   Hence, using Hypothesis 1, k=6 and p6=1018 [g/cm3].
   Then the estimated radius is: re ~ 0.6*10-17 [m].

b) Proton [1],[2], c) Earth [2], d) Sun [2]: see Table I

e) Galaxy [4],[5]:

   1.4*1011Ms [kg] < mg < 2*1011Ms [kg], mgmean=3.38*1041 [kg].
>   From the particular form of the galaxy /4/ we obtained rgmean ~ 1.6*1020 [m], where rgmean is the radius of the fictious sphere with the same volume as (usually) disc form of mean galaksy.
   => p=1.94*10-23 [g/cm3] and k=-7.57.
   From /5/: p=6*10-24 [g/cm3]. => k=-7.75.
   Hence, applying Hypothesis 1, k=-8. => p-8=10-24 [g/cm3].
   Then the estimated radius is: rg ~ 4.3*1020 [m].

f) Galaxy Cluster [4],[5]:

   5*mg < mcl < 104*mg, mclmean=3.38*1043 [kg], rclmean ~ 1.89*1022 [m].
   => k=-9.3. Hence, using Hypothesis 1, k=-9. => p-9=10-27 [g/cm3].
   The estimated radius is now: rg.cl ~ 0.93*1022 [m].

g) Universe [2],[5]:

   mu ~ 1053 [kg], 3.08*1017 [s] < Hubble time < 15.41*1017 [s].
   For Universe age of 15*109 years ( 4.73*1017 [s]), ru=1.42*1026 [m].
   => p=8*10-30 [g/cm3], and k=-9.7.
   Hence, using Hypothesis 1, k=-10. => p-10=10-30 [g/cm3].

   In view of the fact that Universe age is more precise data than Universe mass, the estimated value of (low limit) Universe mass is mu ~ 1.2*1052 [kg].
   In Figure 1 the mass of stable objects in function of k is given. The best approximation of the mass function in order to minimize the error is assumed by the relation:

    m=10-5.2k [kg].......................................................................................(2)

   The Earth and the Sun have not been used in this analysis (explanation succeed). Combining (1) and (2) we obtain:

   m=2.5r1.9 [kg]........................................................................................(3) and
   p=m-0.575 [g/cm3, for m [kg]]...............................................................(4)

   From these results Hypothesis 2 is introduced:

- Hypothesis 2:

   The stable objects mass has the approximate value
   of its radius (raised to) power two: m ~ r2

   Density analysis of the Earth and the Sun shows that their masses are (approximately) equal to the radius power four. For the neutron stars, power is eight. Therefore, hypothesis 2 could be generalized as m=r2n, where n is an even number.
   All previous results and considerations lead to the conclusion that all stable objects parameters (mass, radius, energy density) remain in well determined and rather than random relations. On the other hand, for "objects" which appear in wave form, Compton wavelength
lc is used instead of the mass m, and the relation (5) becomes:

   lc = 10-41.9476-1.9 log r, or: lc ~ 10-42-2log r ..................................(6)

- Further considerations:

h) Neutrino [6]:

   mn ~ 5 - 9 [eV] => mn ~ 1.3*10-35 [kg].
   As there is no information about the neutrino radius, relation (4) is used.
   => p=1.14*1020 [g/cm3], and k=6.68
   Hence, using Hypothesis 1, k=7. => p7=1021 [g/cm3]. With value of mn:
   => rn ~ 1.45*10-20 [m], and
lc ~ 1.7*10-7 [m].

   The radius and wavelength values from Table I, corresponding to stable objects, are now used to establish relations among them in the global space (Figure 2). It is interesting to note that all these stable objects are placed on function (6), which is linear in the logarithmic scale. This line (linear function) crosses the 2nd, the 3rd and the 4th quadrant. It is obvious that the 2nd quadrant contains the macro - space objects. In the 3rd quadrant are situated the micro - space objects. Man and his environment are positioned in the area between them. By analogy in the 4th quadrant would be placed objects with qualitative and quantitative differences in comparison with objects from the 3rd quadrant in a similar way as these objects are different from objects in the 2nd quadrant. Regarding scales these 4th quadrant objects would have a very small radius (r < 10-22 [m]) and at the same time a very large wavelength (l > 1 [m])
   As a curiosity one can note that the electromagnetic component of ultralow-frequency "brain-waves", measured by electroencephalogram (EEG), are classified in five types (
a, b, g, d, q waves) of frequencies. For instance a waves (8 - 13 Hz) corresponds to l = 0.3*108 [m]. If we use this value as Compton wavelength then we obtain:

i) a wave - corpuscle [5]:

   lc= la =0.3*108 m. => mo,a=2.38*10-50 [kg].
   As there is no information about the
a wave "radius", relation (4) is used.
   => p=1.7*1028 [g/cm3], and k=9.41.
   Hence, using Hypothesis 1, k=9. => p9=1027 [g/cm3]. With value of mo
   => r
a ~ 0.26*10-26 [m]. (Associations...universe radius is ru=1.42*1026 [m].)

j) Planck length [5]:

   To complete Table I, some imaginary object by the name "Planck length" has been introduced. Planck length (10-35 [m]) is assigned to this "object" as its radius.

   Using rel.(3) => mo,pl=0.79*10-66 [kg], p=0.18*1039 [g/cm3], k=12.75.
   Hence, using Hypothesis 1 => k=13 and p13=1039 [g/cm3].
   Now estimated values are: mo,pl ~ 0.15*10-67 [kg], rpl ~ 0.15*10-36 [m].

   From the previous consideration it is obvious that the stable object mass and radius cannot have random values. It points out the quantification of stable objects in the sense of energy density and (essentially) the quantification of its radius. However, this radius quantification has to be conceived of as the impossibility of the existence of a random relation between the energy and the space occupied by the stable object rather than as of the exclusive discontinuity of space.
   This could be explained by the fact that unstable objects are characterized by the relation energy/space which is different from the relation proper to the primary object in the same basic group. We can note that the mean lifetime of unstable objects is as shorter as their masses are more different from the mass of the primary object in the same basic group of objects (e.g. Barions: proton/~endless, lambda/2.51*10-10 sec...).
   Macro space circumstances are similar. The established mass of the smallest observed star is 3*1029 [kg] and that of the largest one 2*1032 [kg]. Almost all other star masses (Sun is in this group) range between values of 1030 [kg] and 1031 [kg].

   In the second part of Table I are placed objects with the wave nature (radiation form). For this class of objects the (virtual) mass is computed from Compton wavelength which is obtained from the relation:

   lc=l / (1-v2/c2) 1/2 .................................................................................(7)

   where l is the mean value of the appropriate wavelengths for this "bject". This technique makes the class of the wave "objects" equivalent to the objects from the first part of Table I, where the Compton wavelength is computed for the remaining mass and now we compute the remaining mass from the Compton wavelength. At any rate, the value of (1-v2/c2)1/2 cannot be complex. This is the reason for introducing Hypothesis 3.

- Hypothesis 3:

   The largest possible object velocity is defined as vmax= (1/m0e0)1/2 .....(8)

Lorentz`s relation

   Hence, light propagation velocity assuredly is fotons moving speed but, regarding ours supposition (Hip. 3), this is not value implied in T.R. as largest velocity in nature. By ours opinion, this largest velocity value (vmax) is established by Maxwell (James Clerk, 1831-1879). Its quantity is (really slightly) larger than the experimentally measured light speed (c=2.99792458*108 [m/s]). This vmax velocity value originates from vacuum characteristics and is: vmax= (36p109/4p10-7)1/2 =3*108 [m/s]. This really negligible (quantitative) difference is, probably, fundamental reason why Maxwell (and all other scientists) was equalize and, in essence, identify this two velocity values. By ours opinion, although Lorentz`s relations stay (almost) same, qualittative dissimilarity is very large, so, in accordance with that, using hipothesis 3, we can write the expression (7) as:

   lc=l / (1-v2/vmax2)1/2 .............................................................................. (9)

and continue with further considerations:

k) Photon [5]:

   For the optical spectre lmean ~ 5*10-6 [m].
   Hence, for v=c (now this is less than vmax), from the expression (9) =>
lc=1.35*10-4 [m].
   Virtual mass (De Broglie) is mo=1.6*10-38 [kg]. Using (2) => k=7.24.
   Hence, using Hypothesis1 => p7=1021 [g/cm3] and rph=1.5*10-21 [m].

l) Iks rays [5]:

   lmean=10-9 [m]. Then from (9) => lc=2.69*10-8 [m].
   Virtual mass is mox=8.2*10-35 [kg]. Using (2) => k=6.9.
   Hence using Hypothesis 1 => p7=1021 [g/cm3] and rx=2.7*10-20.

   One can note that all parameters defining X rays are almost similar to the parameters defining neutrino.

m) Gama rays [5]:

   lmean=5*10-11 [m]. From (9) => lc=1.34*10-9 [m] and mo =1.65*10-33 [kg]
   For k=6.3 using Hypothesis 1 => p6=1018 [g/cm3] and r =7.3*10-19 [m].

n) Cosmic rays [5]:

   lmean=10-13 [m]. => lc=2.69*10-12 and mo,cos=8.2*10-31 [kg].
   For k=5.7 using Hypothesis 1 => p6=1018 [g/cm3] and rcos=0.58*10-17 [m].

   As we can see cosmic ray parameters are similar to electron parameters.

   In considering Hypothesis 3 the speed of light is reachable for objects with a real remaining mass (mo). Therefore this mass can be expressed by the De Broglie relation for the velocities in the range which includes the speed of light (c). Regarding that fact, the Planck energy relation E=hv becomes equivalent to Einsteins total energy relation E=mc2 because the mass now has a real and finite value:

   m=mo / (1-v2/vmax2)pow-1 ....................................................................(10)

   The fact that both Energy relations are now identical, shows that objects energy contents are completely equal, indicating that it is (can be) the same object ( e.g. neutrino - x rays, electron - cosmic rays...) seen on two different ways. With speed increase, the corpuscular object energy increases also. This means that its mass value grows, but when velocity reaches the speed of light, the mass of this object is not infinite. If the corpuscular object speed exceeds the speed of light (c<v<vmax), the energy and the mass increase or its wave length decreases. In both cases object energy grows, but its value is finite too.
   The light dualism can be (illustration) explained by the fact that the SAME OBJECT can be seen (or expressed) in two (different) ways (fig 3). Object whose (relativ) velocity (relate to observer) is less than light (propagation) speed, observer perceive as corpuscular object with real remaining mass. Identical object whose velocitu (v) is larger than light (propagation) speed (c), but (regarding hyphotezis 3) less than her theoretical vallue (vmax) same observer will perceive as wave (without measurable remaining mass). Explanation is related to space-time-mater nature, and is exposed in Eternal oscilating universe model (To find something more about this presentation immediately, click  here . Objects whose velocity is very close to light (propagation) speed express dualism.
   Moreover, to every minimal energy (maximal wavelength) in the separate radiation ranges on the scale of electromagnetic waves, corresponds maximal energy (Compton wavelength) of some corpuscular objects. It is the reason why we suppose that there is an analogy between the (number of) different radiation ranges and (number of) basic stable corpuscles.

- Examples:

   In the Barion group the only stable corpuscle is proton. In the lepton group the electron and its neutrino, apart from other corpuscles, have a sufficiently long lifetime. In this group, there are the mion neutrino and the neutrino which corresponds to the heavy Tau corpuscle.

1) Hypothetical corpuscle 1 [5],[6]:

   We suppose that gama ray maximal wavelength corresponds to some hypothetical corpuscle 1 which Compton wavelength have the same value.

   From [5] lc=10-9 [m]. This is equivalent to the mass mo,hip1=2.2*10-33 [kg]

   or mo,hip1=1.2 keV. According to [6, page 1166] the upper limit for the mion neutrino mass is 500 keV, which enables the association link between the hypothetical corpuscle 1 and the mion neutrino.

2) Hypothetical corpuscle 2 [5],[6]:

   One can reverse this proposition: From [6, page 1166] the mass upper limit for Tau neutrino is 31 MeV (or 5.5*10-29 [kg]).

   Equivalent Compton wavelength is lc=4*10-14 [m].

   This is the same value as the limit wavelength for cosmic rays.

   Approximately, assuming that individual radiation ranges overlap (fig3) the association link between the Tau neutrino and hypothetical corpuscle 2 is very possible...


   The perceived regularity and harmony which link the mass (energy), radius and corresponding wavelength for some object, do not allow precise and highly reliable results. However, knowing only one of these values enables us to estimate the other values and reach fairly correct results.

   The greatest value of the results obtained in the present paper is in that it offers us a glimpse into the world of matter, energy and radiation in the unfathomable deepness of micro and macro space where official physics axioms are powerless.


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1. Oznake, jedinice, nazivi i fundamentalne konstante u fizici, Naucna knjiga, Beograd, 1990.god.
2. Astrophysical Data: Planets and Stars, Springer - Verlag, New York, 1992.god.
3. kvarkovi, Harald Fritzsch, R.Piper Co Verlag, Munchen, 1981.g./ skolska knjiga, zagreb 1988.god.
4. The Hamlyn guide to astronomy, David baker, Hong kong, 1989.god.
5. Opsta enciklopedija Larousse, Tom 2, Vuk karadzic, Beograd, 1972.god.
6. Proceedings of the xxVI International Conference on HIGH ENERGY PHYSICS, Vol 1, American institute of Physics, New York, 1993.god.
7. Od atoma do nebeskih tela, P. Savic, Radnicki univerzitet "R.Ciripanov", N.Sad, 1978.god.
8. Enciklopedijski leksikon, mozaik znanja, fizika, Interpres, Beograd, 1972.god.

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m (kg)

r (m)

p (g/cm3)

lc (m)

Planck length





a wave-corpuscule



































Galaxy Clasters















X waves*





g waves





Cosmic rays**






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Mass of stable objects


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objects relations in global space


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Dualism vawe-corpuscle


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Instead of the real THE END
and without final conclusions...

   Considering suppositions exposed in this work, it is possible to create very interesting, multidimesional model of Eternal oscilating universe (see fig. 4), where EPR effect (e.g.) is very obvious, blue shift is totally normal event, corpuscular object velocities are possible to exceed experimenaly measured vallue off light speed - wihout discredit and/or ignore quantum theory or theory of relativity....

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copyright Goran Marjanovic
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