Motto: An Universe made of 10 ^{122} etherons
Ether and Etherons  A Possible Reappraisal of the Ether Concept
by IoanIovitz Popescu
iovitzu@gmail.com
;
http://www.iipopescu.com/
Translated from the Roumanian Academy journal of physics
Stud. Cercet. Fiz., vol. 34, 451468 (1982)
Both original and translated versions are also presented and discussed at
Ether and Etherons. A Possible Reappraisal of the Concept of Ether. 1982
Etherons as predicted by IoanIovitz Popescu in 1982
NOTE. A later support of this paper is offered by Nicholas IonescuPallas,
Reflections Concerning the Genuine Origin of Gravitation, Romanian Reports in Physics,
55, 742 (2003),
http://rrp.infim.ro/2003_55_1/d00_pallas.pdf; see also I.I. Popescu and R.E. Nistor,
Subquantum medium and fundamental particles, Romanian Reports in Physics,
57, No. 4, 659–670 (2005),
http://www.rrp.infim.ro/2005_57_4/11659670.pdf
Outline
Abstract.
A new explanation of the Newtonian law of gravitation is given, proceeding from the following statements: a) the Universe
is finite and filled with some particles of very small mass, traveling at speed of light; b) all material bodies
in the Universe are made up of such particles called �etherons�; c) the matter in the Universe is prevailingly under
the form of etherons. The uncertainty principle of quantum mechanics and some dimensionless relations of relativistic
cosmology  among which Mach�s principle  are adopted in view of establishing the intrinsic characteristics of etherons
as well as their number in the Universe. By applying statistical ratiocinations to the etheronic background, expressions
of Hubble�s and Newton�s constants are derived in terms of some kinetic entities pertaining to the ether. The emergence
of the inverse square law of force entails at the same time a very strong coupling of the etherons in a nucleon and
a saturation character of the binding forces. A wide discussion is undertaken concerning the consistency of the physical
world picture suggested by the etheronic conjecture with the already constituted frame of conventional physics, drawing
interesting and encouraging conclusions.
The idea of an universal medium filling the whole space is very old. Since Aristotle and BhagavadGita until nowadays, the philosophers and the physicists and, more recently, the cosmologists strived to understand the �most subtle� state of matter, occasionally called �ether�. The historical persistence of this concept, which escapes from the usual control by experiment � though intimately bound to the basic phenomena of the physical world, gets its motivation not only in the Latin aphorism �Natura abhorret vacuum�, but mainly in the need to explain the phenomena by a causal infrastructure, whose existence is left to be subsequently tested. A study on the internal logic and the historical roots of various evaluations of the ether concept within the framework of the modern physical theories has recently been given by Liviu Sofonea and Nicolae IonescuPallas [1].
The history of the luminiferous ether, prevailing in the European physics of the XIXth century, is well known  see, for instance, Edmund Whittaker [2]. Some new aspects regarding the irrelevant character of the ether, as well as its compatibility with the special relativity theory, have been investigated by Nicolae IonescuPallas [3]. The �irrelevance� of the ether seemed in the past stranger than today, when physicists are already used to �magnetic monopoles�, �partons�, �quarks� and others.
In the present paper we will consider such an irrelevant entity  the �etheron�  in connection with the cosmological role of the ether, so much discussed in the last decade. Fist of all we will shortly expose the major achievements in cosmology as obtained by adoption or adaptation of the ether concept just to satisfy the modern principles of �covariance�, �minimal action�, �physical field� and so on.
The first serious attempt to elaborate an etheronic scheme of the matter is due to Georg Szekeres [4]. Extensions of this trial, aiming to obtain separate conditions of conservation for the ether and the substance, have been done by Nicolae IonescuPallas [5] in his recent treatise entitled �General Relativity and Cosmology�. Retaining the hypothesis of the existence of two kinds of conservative �matter� � ether and substance � and trying at the same time to lessen the differential order of the field equations, Nicolae IonescuPallas and Liviu Sofonea [6] succeeded to build a cosmological model; here appears a sort of universal ether and Newton�s constant G, as well as the cosmological constant L, vary just to ensure an adiabatic expansion of the Universe. The latter model, called also �Cosmologia Veradiensis�, allows to get an idea of the way to reconcile the ether concept with the present theories of Big Bang and expanding Universe. Another remarkable model  also based on the ether concept and having some common features with Cosmologia Veradiensis, is due to Nathan Rosen [7]. The exceptional value of Rosen�s model consists in the fact of representing an oscillating system, thus preventing the collapse at maximum contraction.
The question of what effectively consists the physical structure of the ether remains an extremely controversial subject, in spite of valuable suggestions made by physicists of mark such as E. Sudarshan et al. (the ether as a superfluid state of particles and antiparticles [8]), J. P. Vigier et al. (the ether made up of bosons of minute mass [9]), A. Das and P. Agrawal (the ether of quanta or particles of extremely tiny mass [10]), J. R. Rao et al. (the ether of particles responsible for the �strong� gravity [11]).
Let us remind, finally, two hypotheses based on options favorable to an ether with discrete structure. The first, due to Nicolae IonescuPallas and Ioan Gottlieb [12], accredits the opinion that the Hubble�s expansion would be determined by a scalar field with quanta of a tiny rest mass, as given by the expression
m _{0} = (3/2)(h _{B}H/c ^{2}) @ 10 ^{ 69} kg (1)
where H is Hubble�s constant, c the light speed in vacuum and h _{B }= h/2 p the Planck�s reduced constant �hBar�. The second hypothesis, more recent, argues on the possibility of an universal medium structured of neutrinos [13].
In continuation will be presented some considerations regarding relation (1) which represents, in fact, the starting point of our approach. Let us first observe that this relation, basic for the following, results immediately if the Hubble�s constant, H, is interpreted as the angular frequency, w _{0 }, of an oscillatory process occurring at cosmic scale. Thus, considering the temerarious identification of the physical Universe with a threedimensional isotropic harmonic oscillator, with the proper frequency w _{0 }= H , one observes that relation (1) is a consequence of the expression of the ground state energy, namely (3/2)h _{B} w _{0 }= (3/2)h _{B}H = m _{0}c ^{2} @ 10 ^{ 33} eV. As a support may be invoked the model of oscillatory Universe of Richard Tolman [14], according to which the angular frequency of the cosmic pulsation is w _{0} @ H. We are also led to accept that the neighboring �excited� states of the Universe are energetically equally distant by h _{B} w _{0} = h _{B}H and that the minimal energy which can be exchanged between the interacting material systems is given by the quantum h _{B} w _{0} = h _{B}H.
In the following we will call �etheron� the quantum of energy of h _{B} w _{0} = h _{B}H = m _{E}c ^{2}. Because the energy of this quantum is extremely small (of the order 10 ^{ 33} eV) and, on the other hand, since the gravity is the most feeble of known interactions, there arises the plausible supposition that the etherons represent the exchange particles associated to the gravitational interaction. Moreover, as we will further argue, we are led to postulate the existence of an interaction associated to any form of energy of the type �Energy = Energy + Etherons�, where Energy means any substructure of the Universe, including elementary particles. Generally, the existence of an interaction of this type leads to a stationary potential of the Yukawa type, F ~ (1/r)exp(r/ l), where l is the Compton wavelength associated to the particle mediating the interaction. For gravitational interactions, presumably mediated by etherons, l _{E} = h _{B}/m _{E}c @ c/H @ R @ 10 ^{26} m, that is of the order of magnitude of the Universe radius. For strong interactions, mediated by pions (a presumably �multietheronic� process, m _{p }@ nm _{E}), l _{p} = h _{B}/m _{ p }c ( @ l _{E}/n) @ r _{n} @ 10 ^{ 15} m, that is of the order of magnitude of nucleon radius. The mass quantification introduces in this way a finite range for all interactions, which cannot exceed the dimension of the Universe. By calling �etheron� this quantum of mass there should be no envision about the properties of absolute reference frame of the ether. The ether concept would only reflect the occurrence of some entities having particle properties, by the �condensation� of which (under the form of inertial mass and of �interaction� mass of �transit� etherons) we have to explain the extremely complex structure of microobjects challenging us nowadays.
Another reflection inspired by relation (1) is connected to the observance of the process of emission, respectively of absorption of the quantum of energy h _{B} w _{0} = h _{B}H. Thus, according to the uncertainty principle of Werner Heisenberg, the lapse of time during which such a process occurs with certainty is given by t @ (1/2) h _{B}/h _{B} w _{0} = 1/2 w _{0} = 1/2H, that is of the order of magnitude of the cosmic epoch (of the Universe �age�). Due to their tiny mass and extreme rarity of the events (collisions, processes) in which they are involved, the etherons travel (almost) at light speed, revealing rather quantum than particle properties. Arguments in favor of this seemingly strange situation (but essential for what follows) are brought within the theory of Louis de Broglie regarding the photons with nonzero rest mass and with velocity close to the light speed in vacuum [15]. In this context, the mass of the order of magnitude given by expression (1), m @ h _{B}H/c ^{2 } @ 10 ^{ 69} kg , is also presently mentioned as the �photon rest mass� or the �boson mass� [16]. Note added on January 6, 2003: An expression of the quantified mass in a �spacetime cavity� [29], indicates again the etherons as the ultimate building blocks of matter [see Addendum 4].
Another interesting argument in favor of quanta of energy h _{B} w _{0} = h _{B}H is the following. Thus, due to the fact that, according to a �gedanken experiment�, the detection time of an etheron is of the order of 1/H, one can not avoid an uncertainty of the order h _{B} w _{0} = h _{B}H in the measurement of energy, respectively a mass uncertainty of the order h _{B} w _{0}/c ^{2} = h _{B}H/c ^{2}. Adopting the Einstein�s static model with cosmological constant, any fluctuation of the Universe mass, M, induces, via the relation GM/c ^{2}R = p/2, a fluctuation of the curvature radius, R, of the Universe (where G is Newton�s constant). From dM = h _{B}H/c ^{2} in association with the last relation it results dR = (2/ p)(h _{B}G/c ^{3})(H/c) or dR ^{2} = (4/ p)(h _{B}G/c ^{5})(HR/c). Since HR/c @ 1 and L _{P} = (h _{B}G/c ^{3}) ^{1/2} is the Planck�s gravitational length, it results that the quadratic fluctuation of the Universe radius of curvature is of the order of magnitude of Planck�s gravitational radius, namely
( dR ^{2}) ^{1/2} = (2/ p ^{1/2})(HR/c) ^{1/2}(h _{B}G/c ^{3}) @ L _{P} @ 10 ^{ 35} m (2)
This conclusion agrees with the opinion of Arthur Eddington regarding the fluctuations of the curvature radius of the Universe [17].
The energy quantum h _{B} w _{0} = h _{B}H denominated here as �etheron� is assumed to be, by definition, the constitutive particle of the cosmic ether. As far as the etheron has the smallest mass compatible with the uncertainty principle of quantum mechanics, it follows that the ether represents the most �fine� fluid, yet having a discrete (corpuscular) structure [see Footnote 1]. For sure, the ether is a form of existence of the matter but qualitatively different from the common (atomic and molecular) substance or radiation (photons). Moreover, we will assume that the ether is governed by the principle of inertia and produces by its presence a modification of the spacetime geometry. According to the static model of Einstein, the mass of the Universe (conceived as finite but unbounded) is given by the expression M = ( p/2)c ^{2}R/G; the magnitude of the radius of curvature, R, is of the order c/H. Thus, the mass of the whole Universe, predicted theoretically, is exclusively expressed in terms of universal constants, namely M @ c ^{3}/GH @ 10 ^{53 }kg. A second way of estimation of this mass is based on the formula M = 2 p ^{2}R ^{3} r where R @ c/H and r is the mass density in the Universe, an observational quantity deduced from the mass and distribution of the galaxies. As it is known, the theoretical estimation M @ 10 ^{53} kg is about two orders of magnitude greater than the �observational� mass, as if the Universe mass would be stored in the space under a form which escapes to the conventional observation (the problem of the so called �hidden mass�). We take this opportunity to suggest that the �hidden mass� could be under the form of ether.
In order to explain the universal law of gravitation by means of the ether concept, as argued above, we need still two essential hypotheses, namely: a) all material bodies are build up of etherons; b) the universal attraction is, actually, the result of the decompensation of the hydrodynamic pressure, exerted upon the bodies by the universal ether, as a result of mutual screening. The aim of this article is to present the way of acting of these hypotheses and the manner in which one can obtain the global consistency of the model, both in itself and in comparison with the already established frame of general relativity and modern cosmology. We mention that the explanation of the gravitation, as will be presented in this article, has some common traits with the theory of Iosif Adamut, a theory based on the Lesage�s hypothesis and on a medium made up of quanta [18].
But before proceeding to the demonstration of the gravity law, let us present an additional argument regarding the speed of the etherons, as well as the consequences which follow from their ultra relativistic character. For this purpose, we will appeal again to the uncertainty principle  this time with reference to the relationship coordinatemomentum. Thus, the smallest possible error in the determination of the momentum of a physical system is given by the momentum p _{E} of an etheron (randomly emitted or absorbed), that is dp = p _{E} = m _{E}v _{E} = (h _{B}H/c ^{2})v _{E}. This quantity should be corroborated with the greatest possible error of the position coordinate dx in conformity with Heisenberg�s relation dp dx @ h _{B}/2. Since the �characteristic dimension� of the Universe is c/H it results that dx @ (1/2)(c/H) and, consequently, v _{E } @ c. By developing this argument we considered the quantity h _{B}H/c ^{2} as the dynamic mass rather than the rest mass of the etheron. Actually, we can assume that the speed of the etheron is not just, but a little less than c  so that the rest mass should be of the same order as the dynamic mass (for instance, if v _{E}/c = (1/2)3 ^{1/2 } @ 0.866 , then m _{0E} = (1/2)m _{E} = (1/2)h _{B}H/c ^{2}. On the other side, in conformity with the representations of statistical mechanics, one can assume that the velocities of the etherons are distributed around a mean value a little smaller than c and in a narrow band which, practically, can be neglected. A similar situation, in which �particles having quantum properties�, of given energy, move at speed c, occurs in the theory of gravitation of J. L. Synge [19].
One of the most important consequences resulting from the ultra relativistic character of the etherons resides in the fact that the �primary aggregates� buildup of etherons should reveal themselves as exceptionally stable, due to the major contribution of the part of speed dependent binding energy. In spite of the fact that this assumption cannot be directly proven, we can, however, illustrate it in the sole rigorous case of the twobody problem within the frame of special relativity. Specifically, let us refer to a potential inversely proportional with the distance between the particles, a case independently elaborated by Alfred Schild [20] (starting from the symmetric electrodynamics) and by Nicolae IonescuPallas and Liviu Sofonea [21] (starting from the �invariantive mechanics� of Octav Onicescu). Schild�s formula reads
E = m _{01}c ^{2}(1 v _{1} ^{2}/c ^{2}) ^{1/2 }+ m _{02}c ^{2}(1 v _{2} ^{2}/c ^{2}) ^{1/2 } (3)
where the energy E of the system vanishes as (v
_{1
}, v
_{2}) come closer to c. As it will be shown in the following, the �etheronic
model� appears particularly encouraging, inasmuch as it allows the deduction of
Newton�s law for gravitation, as well as the fact that primary aggregates, directly
made up of etherons, have a mass defect comparable to the sum of the etheronic
constituents. Actually, as it is known, an almost unity ratio between the binding
energy and rest energy is characteristic for nucleons [22]. Is there an indication
that the �partons� or the �quarks� might be modes of etheronic collective motion?
Until here we prepared the following
remarkable hypothesis: �The Universe is filled almost exclusively with particles
of tiny mass, m
_{E} , moving at random at light speed, c . The aggregated mass, stored in
stars and galaxies, can be formally considered as constructed of such particles
of mass m
_{E}  called here etherons  whose number is proportional to the ratio between
the inertial mass of the body and the mass of etherons. In order to exploit this
supposition for the clarification of the �mechanism� of gravitation, we need
a corpus of quantitative relationships already established and allowing a conciliation
of the etheronic theoretical approach with relativistic cosmology. This will
be achieved by adopting the following set of six simple relationships
m _{E}c ^{2}/h _{B}H = k _{1}  GM/c ^{2}R = k _{2}  m _{E}cR/h _{B} = k _{3}  (46) 
m _{E}c ^{2}/(h _{B} ^{2}/m _{E}R ^{2}) = k _{4}  r _{E}N _{E} ^{1/2}/R = k _{5 }  V/2 pR ^{3} = k _{6 }  (79) 
where k _{1} , k _{2} , �, k _{6 }are nondimensional constants of the order of magnitude of unity; (c, h _{B }= h/2 p) are the speed of light in vacuum and the Planck�s reduced constant; (G, H) are Newton�s constant, respectively Hubble�s constant; (m _{E} , r _{E} , N _{E}) are the mass, dimension, and total number of etherons in the finite Universe; finally, (M, R, V) are the mass, dimension (that is the curvature radius), and the volume of the finite (but unbounded) Universe. The fact that we adopted simultaneously the static model of Einstein and the Hubble�s constant does not necessarily constitutes a contradiction by virtue of two reasons: 1) the expansion is not the sole explanation for the constant of Hubble; 2) even the static model provides the right order of magnitude of the characteristics of the Universe. Let us comment upon the origin and opportunity of the relationships (49).
Relation (4) simply affirms that the etherons exist; this is our axiomatic point which we accept together with its sustaining arguments.
Relation (5) is an expression of the Mach principle, independent of adopted cosmological model. For the static model of Einstein with positive curvature k _{2} = p/2 ; for the expanding Universe k _{2} = p [6].
Relation (6) represents an adaptation to the etheron of the relation of Feza Gürsey [23] and Fred Hoyle [24] and requires a scalar particle of an extremely small mass. This is compatible with relation (4), showing that the curvature radius, R, and the ratio c/H have the same order of magnitude [5].
Relation (7) is, formally, a consequence of relation (6) and introduces a restriction for the unknown constants (k _{3} , k _{4}), namely k _{4} = k _{3} ^{2}. However, this relation has a relevant physical meaning, allowing us to consider it as an independent relationship. Thus, this affirms that the rotation quantum h _{B} ^{2}/m _{E}R ^{2} has the same order of magnitude as the oscillation quantum h _{B} w _{0} = h _{B}H @ m _{E}c ^{2}. In other words, the uncertainty relations discussed above can be rewritten in a form replacing the oscillation quantum with the rotation quantum. This fact can be interpreted as a proof of the stability of the Universe not only against oscillations (when an energy of the order m _{E}c ^{2} is by chance emitted or absorbed), but likewise against rotations (when an energy of the order h _{B} ^{2}/m _{E}R ^{2} is involved in a similar manner).
Relation (8) represents an ad litteram transposition for etherons of the famous relation established by Arthur Eddington for protons [17]. A simplified version of Eddington�s reasoning, given by Nicolae IonescuPallas [5], is: �If in the finite and unbounded Universe of Einstein would exist a single particle (proton), this would be described by a wave which, due to the space curvature, would prescribe an incertitude of the center of inertia equal to R. Assuming that in the Universe there exist a finite number N _{p} of protons, the uncertainty is reduced according to the laws of statistical mathematics to R/N _{p} ^{1/2}. This quantity is identified by Eddington with the spatial extension of the particle (which becomes, in this way, non punctual)�. Obviously, if the free particles filling predominantly the Universe are not protons, but etherons, the above reasoning is equally valid also for our model of etheronic Universe, whence it results relation (8).
Relation (9) has a pure geometric content and affirms that the Universe volume and the third power of its characteristic dimension (of the curvature radius) are in a constant ratio. Thus, the constant k _{6} has the value 2/3 in an Euclidean geometry and the value p in a Riemannian geometry (topological closure).
The most plausible values which will be adopted here for the set of constants (k _{1 }, �, k _{6}) are the following:
k _{1} = 1 , k _{2} = p/2 , k _{3} = 1 , k _{4} = 1 , k _{5 }= 1/2 , k _{6} = p (10)
The value k _{1 }= 1 results from the manner in which we concretized the etheron concept. The values k _{2} = p/2 and k _{6} = p arise from the static cosmological model of Einstein. The special value k _{5} = 1/2 was chosen to give correctly the proton dimension (r _{p} = 1.4 10 ^{ 15 }m) when formula (8) is used in the original interpretation of Eddington. The value k _{3} = 1 results as a consequence of the relationship R = (k _{3}/ k _{1})c/H, of the choice already done for k _{1} = 1 and of the accepted fact of contemporary cosmology that, in the present epoch, R @ c/H [5, 6, 25]. Once the value k _{1} = 1 is admitted, it then results k _{4} = k _{3} ^{2} = 1.
Further we shall
see that the set of constants (10) leads to a very strong
coupling for etherons, assumed to be the constituents of
the nucleon. It is interesting to notice how a macroscopic
condition at cosmic scale such as, for instance, the topological
closure of the Universe, leads to an energetic consequence
at infranucleonic level.
Many physical properties of the ether can now be derived from the statements presented above, expressed by the fundamental cosmological relationships (49), from the (presupposed) quantum properties of the etheron, and from the conventional methods of statistical mechanics.
Let us start
with the intrinsic characteristics of the etheron, whose
similarity with those of the photon is transparent. Thus,
the energy E
_{E}
, the mass m
_{E }, the momentum p
_{E} , and the associated de Broglie wavelength
l
_{E} = h
_{B}/p
_{E}
are given by the relations
E _{E }= m _{E}c ^{2} = k _{1}h _{B}H @ 10 ^{ 33} eV  
m _{E} = E _{E}/c ^{2} = k _{1}h _{B}H/c ^{2} @ 10 ^{ 69} kg  (11) 
p _{E} = m _{E}c = k _{1}h _{B}H/c 
respectively
l _{E} = h _{B}/p _{E} = c/k _{1}H = R/k _{3} @ 10 ^{26} m (12)
The last relation represents the mathematical equality of two rather different entities, thus binding the quantum properties of the etheron with the geometrical properties of the Universe.
Further, from equation (5) and the equality R = (k _{3}/k _{1})c/H we can express the mass of the Universe in the form
M = (k _{2}k _{3}/k _{1})(c ^{3}/GH) @ 10 ^{53} kg (13)
Since the ether represents the dominant component of matter in the Universe, we can suppose that the entire mass of the Universe is practically constituted of free etherons. This allows to write M = N _{E}m _{E }, where N _{E} is the total number of free etherons in the Einstein Universe,
N _{E} = M/m _{E} = (k _{2}k _{3}/k _{1} ^{2})(c ^{5}/h _{B}GH ^{2}) @10 ^{122} (14)
The dimension of the etheron can be derived from equations (8), (14) and R = (k _{3}/k _{1})c/H, so that
r _{E} = k _{5}(k _{3}/k _{2}) ^{1/2}(h _{B}G/c ^{3}) ^{1/2} = k _{5}(k _{3}/k _{2}) ^{1/2}L _{P } @ 10 ^{ 35} m (15)
As expected, the dimension of the etheron is of the order of magnitude of the Planck�s length, that is of the quantum fluctuation of the space (according to formula (2)).
Let us proceed to the statistical properties of the ether by defining, firstly, a �classical� cross section for the etheron  etheron collision with the help of the formula s _{E} = p(2r _{E}) ^{2}, that is
s _{E} = 4 pk _{5} ^{2}(k _{3}/k _{2})h _{B}G/c ^{3} = 4 pk _{5} ^{2}(k _{3}/k _{2})L _{P} ^{2 } @ 10 ^{ 70} m ^{2} (16)
A particular meaning of the last formula consists in the fact that it allows to express Newton�s constant of the universal attraction in terms of the cross section s _{E} , a quantity of statistical nature, that is
G = (1/4 p)(k _{2}/k _{3}k _{5} ^{2})c ^{3} s _{E}/h _{B} (17)
This unexpected result can be an evidence that gravitation itself might be of statistical origin (in terms of the hydrodynamic model of Lesage). We shall mention in this context that Edward Milne, in his �Kinematic Relativity� [26], deduced for the first time the Newtonian law of the attraction force within a theory which is compatible with Mach�s principle.
Another interesting relationship, connecting inframicroscopic and ultramacroscopic entities, is L _{P} ^{2} = k _{2}k _{3} l _{E} l _{U} , where l _{U} = h _{B}/Mc = (k _{1}/k _{2}k _{3})h _{B}GH/c ^{4} is the Compton length associated to the Universe [see Footnote 2].
In spite of the their tiny mass and dimension, the density of etherons in the Universe is impressing. Indeed, from V = 2 pk _{6}R ^{3} = 2 pk _{6}(k _{3}c/k _{1}H) ^{3} and from the assumed homogeneity and isotropy of etheron distribution, we get
n _{E} = N _{E}/V = (k _{1}k _{2}/2 pk _{3} ^{2}k _{6})Hc ^{2}/h _{B}G @ 10 ^{43} m ^{ 3} (18)
so that the mean distance between etherons is r _{EE} = 0.554n _{E} ^{�1/3} @ 10 ^{�15} m and characterizes the �radius� of statistical fluctuations (within which the punctual elementary particles set up).
The quantities s _{E} and n _{E} define the �classical� mean free path for etheron � etheron collision, namely
l _{E} = (1/2 ^{1/2})n _{E} s _{E} = (1/8 ^{1/2})(k _{3}k _{6}/k _{1}k _{5} ^{2})c/H = (1/8 ^{1/2})(k _{6}/k _{5} ^{2})R @ 10 ^{26} m (19)
amounting to the order of the curvature radius of the Universe.
We also can define the mean collision frequency of etherons, i.e.
n _{E} = c/ l _{E} = 8 ^{1/2}(k _{1}k _{5} ^{2}/k _{3}k _{6})H @ 10 ^{ 18} s ^{ 1} (20)
In this way the Hubble�s constant (the second of cosmological interest, besides Newton�s constant) gets a statistical explanation too.
Finally, another three
statistical characteristics
of the etheronic gas
complete the table
of the properties of
this strange fluid,
namely the collision
rate R
_{E}, the pressure
P
_{E} of the ultra
relativistic etheron
gas (analogous to the
Planck�s radiations
pressure), and the
temperature T
_{E} of the etheronic
gas, as given respectively
by
R _{E} = (1/2)n _{E} ^{2} s _{E}c = (1/2 p)(k _{1} ^{2}k _{2}k _{5} ^{2}/k _{3} ^{3}k _{6} ^{2})H ^{2}c ^{2}/h _{B}G @ 10 ^{25} m ^{ 3}s ^{ 1}  (21) 
P _{E} = (1/3)n _{E}m _{E}c ^{2} = (1/6 p)(k _{1} ^{2}k _{2}/ k _{3} ^{2}k _{6})H ^{2}c ^{2}/G @ 10 ^{ 13} atm  (22) 
T _{E} = (3P _{E}/a) ^{1/4 } @ 30 K [where a = (8 p ^{5}/15)k ^{4}/c ^{3}h ^{3}]  (23) 
Adopting for the Hubble�s
constant the value
H = 1/(6.53 10
^{17} s) and for
the constants k
_{i} the probable
values given by the
set (10), it results
a temperature of
the etherons of about
30 K, a value which
is only one order
of magnitude higher
than that observed
for the cosmic Planck
radiation. This estimation
of the ether temperature
accounts for the
fact that the partial
pressure of the free
etherons is considerably
higher than that
of the complex etheronic
aggregates (such
as presumably are
the elementary particles
and the photons).
Now we will proceed to the deduction of the famous law of the Newtonian force. The demonstration will start firstly with two nucleons and, then, we will examine the circumstances in which the result can be extended to macroscopic bodies. Thus, let us consider two spherical and homogeneous bodies (nucleons), A and B, containing N _{A} and N _{B} etherons respectively, placed in the universal ether (the etheronic gas) at a distance r _{AB} greater than any radius of the considered material spheres. In addition, we will assume that r _{AB} << l _{E} @ l _{E} @ R so that the potential of the Yukawa type becomes practically Newtonian and the scattering of etherons is negligible.
Each of the bodies would be in thermodynamic equilibrium if it were alone in the Universe, as a result of the compensation of the ether pressure exerted in all the directions of the space, supposed isotropic and homogeneous. The total hydrodynamic force acting on an etheron is just the Pascal force
F _{E} = P _{E} s _{E} = (2/3)(k _{1} ^{2}k _{5} ^{2}/k _{3}k _{6})h _{B}H ^{2}/c @ 10 ^{ 78} N (24)
ensuring the equilibrium of the considered etheron against the surrounding etheronic background. However, in the presence of another body there appears a decompensation produced by the latter. Let us suppose that the considered etheron belongs to the body A and evaluate the decompensation produced by another etheron belonging to the body B. Because we consider r _{AB} << l _{E} @ R , the mutual screening of the considered etheron pair results geometrically
dF _{E} =  F _{E}(d W/4 p) =  F _{E}[ p(2r _{E}) ^{2}/4 pr _{AB} ^{2}] =  F _{E} s _{E}/4 pr _{AB} ^{2} (25)
The Newton�s force between the two bodies (A, B) will be the resultant of all screenings of the etherons of the body A by the etherons of the body B (and conversely), that is
F _{AB} = N _{A}N _{B} dF _{E} =  GM _{A}M _{B}/r _{AB} ^{2} (26)
where Newton�s constant has the expression
G = (1/4 p)(k _{2}/k _{3}k _{5} ^{2})c ^{3} s _{E}/h _{B} (17)
and the mass of the bodies (A, B) is
M _{A,B} = (2/3) ^{1/2}[k _{1}k _{5} ^{2}/(k _{2}k _{6}) ^{1/2}](h _{B}H/c ^{2})N _{A,B} (27)
Let us analyze this expression of the mass by replacing the values of the adopted constants (10); we get
M _{A,B} = (1/2 p3 ^{1/2})m _{E}N _{A,B} = m _{E}N _{A,B}  [1  (1/2 p3 ^{1/2})]m _{E}N _{A,B } (28)
It results from this that the ratio between the binding energy per etheron, E _{bE} , and the energy of the free etheron is extremely high, namely
E _{bE}/m _{E}c ^{2} = 1  (1/2 p3 ^{1/2}) = 0.908 (29)
a fact which is qualitatively confirmed by the exceptional stability of some elementary particles [see Footnote 3]. On the other side, the binding energy is proportional with the number N _{A,B} of constituents, revealing a saturation character, a fact either in accordance with the known properties of infranuclear forces [22]. For sure, we not expect to be able to systematically deduce the structure and the properties of matter at infranucleonic level from the sole cosmological hypothesis (the existence of the etheron) of interest for gravitation. However, if the microscopic consequences of this assumption prove to be consonant with the principal features of the infranucleonic interactions, this very fact is heartening in some respect.
We shall further investigate the gravitational interaction of two nuclei. Proceeding exactly as above, we get
F _{A,B} =  GM _{A}M _{B}/r _{AB} ^{2}
where
M _{A,B} = Km _{E}N _{A,B} = Km _{E}(N _{A,B} ^{(p)}n _{p} + N _{A,B} ^{(n)}n _{n}) = m _{p}N _{A,B} ^{(p)} + m _{n}N _{A,B} ^{(n)}
Here K = (2/3) ^{1/2}k _{5} ^{2}/(k _{2}k _{6}) ^{1/2} and the new notations represent: N _{A,B} ^{(p)}, the number of protons in the nucleus A, respectively B; N _{A,B} ^{(n)} , the number of neutrons in the same nuclei; n _{p} , n _{n} , the number of etherons constituting a proton, respectively a neutron. N _{A,B} still represents the total number of etherons of the body (here nucleus) A, respectively B, but M _{A,B} no longer represents the masses of the nuclei  because no longer include their binding masses. This difficulty can be avoided by considering the saturation character of the nuclear forces, so that the binding masses are proportional with the number of nucleons. Actually, in the presence of nuclear matter the mass of a nucleon is not m _{p,n} but m _{p,n}[1  (8/939)] so that, consequently, the mass of a nucleus is not M _{A,B} but M ^{*} _{A,B} = M _{A,B}[1  (8/939)]. Introducing a new constant G* = G[1  (8/939)] ^{ 2} we are able now to write the macroscopic law of the Newtonian force as
F _{AB} @  G*M* _{A}M* _{B}/r _{AB} ^{2} (26�)
where, this time, M* _{A,B} are the masses of the bodies and the new constant G* has to be identified with Newton�s constant proper. Yet better approximations for the masses can be done with the help of the well known expression of Weizsäcker. At the precision level of the latter, the determination of the gravity constant from the force law of Newton leads to values slightly dependent on the nature of the material used in experiments. The present status of the experimental technique does not allow, however, to test in this way the etheronic hypothesis. If we identify the Newton�s constant with G*, and not with G , then, according to the etheronic model, it follows that the gravitational interaction between two nucleons is weaker by the factor [1  (8/939)] ^{ 2} than the value of the field theory, which involves an universal coupling for gravitation. Neither this possibility is suitable for the experimental proof with the presently available equipment.
Further
advance
from
nuclei
to
macroscopic
bodies
(with
atomic
and
molecular
structure)
do
not
present
any
difficulty,
the
errors
being
nevertheless
smaller
than
those
already
introduced
when
estimating
the
nuclear
masses.
In the preceding paragraphs we conceived the ether as an universal fluid, predominantly spread in the Universe and being, in many ways, similar to common fluids. Consequently, we performed some statistical ratiocinations and gave a statistical interpretation of Newton�s constant, G, and of Hubble�s constant, H. On the other hand, the peculiar properties of the ether as compared with common gases have been concretized in the ultra relativistic character of the etheronic gas and in the minute values of the mass and dimension of the etheron. Besides this, we based our reasoning from known cosmological formulae, left formally unchanged but with their meaning so adapted as to promote the etheron concept. Proceeding in this way, we implicitly assumed that there is no contradiction between the adapted cosmological framework and the presupposed hypothesis of the ether. This actually means that the geometric properties of the spacetime are practically defined only by the ether and not by the common matter. Since no real movement of the cosmic ether is observed, it results a comobile metric and, consequently, we can write
R _{ mn }  (1/2)g _{ mn }R + Lg _{ mn } =  (8 pG/c ^{2})(h _{B}H/c ^{2})n _{E } d _{0 m } d _{0 n } (30)
This represents a modified version of Einstein�s equation [27], compatible with the formulae (49), with the constants (10), and with the condition L = 1/R ^{2}. In this way the constant of Hubble gets the statute of an actual constant.
The transition from the static to the dynamic model (of an expanding Universe), if necessary, should be accomplished by preserving this character of veritable constant for H. More specific, this means that the model leading to an expansion law of the form R(t) = R(t _{0})exp[H(t  t _{0})] is preferable versus the model for which H ~ 1/t . To this aim, for the future there remains to further investigate the collective properties of the ether in order to obtain a set of relativistic hydrodynamic equations capable to explain such fundamental phenomena as the expansion of the Universe, propagation at light speed of small transverse perturbations, stability, spin, and charge of the particles.
In the absence of such a theory, we will tentatively assume the validity of the following simple hydrodynamic equation of the Navier type
m _{E}n _{E}( ¶/ ¶t + v _{g} .Ñ ) v _{g} =  ÑP _{E} + f (31)
where the etheron pressure is given by P _{E} = (1/3)n _{E}m _{E}c ^{2} and the friction force, f = Cm _{E} n _{E}m _{E} v _{E} , has the most simple form. Introducing in equation (31) the expressions of the pressure and of the friction force, expressing n _{E} through H and considering v _{E} = c, one gets the simple equation
¶n _{E}/ ¶r + (H/c)n _{E} = 0 (31�)
where the value of the constant C = (1/3) p2 ^{1/2} was chosen to fit the relativistic law of the cosmological redshift. Thus, considering also the photon as made up of (transit) etherons, the photon energy is E _{f }= h _{B} w ~ n _{E}m _{E}c ^{2}, so that from (31�) there results the well known Hubble�s law of redshift
d w/ w =  (H/c)dr = Hdt (32)
The etheronic model allows to conceive a generalization of this law in the form
(1/E)dE/dt £  H
for any etheronic aggregate of total energy E = h _{B} w = mc ^{2} [see Addenda 5 and 6 for applications]. The mode of explanation of this law, sketched above, is similar to that of the model of De Sitter�s Universe, where the spacetime geometric properties are likewise determined by ether (introduced with cosmological constant) [5].
Another interesting connection of the etheronic model can be achieved with the theory of gravitation of J. L. Synge [19]. In conformity with this theory, the Newton�s law of the gravitation force is deduced by considering that the two bodies exchange mutually quanta propagating with the light speed. It results from this that the potential energy of the system of bodies equals the energy of transiting quanta. For attraction it is necessary to assume a negative mass of the quanta. By logical transposition, the quanta with negative mass can be interpreted, within the etheronic model, as a deficit of etherons caused by the mutual screening of the bodies. We notice that Synge�s approach gives only the proportionality F ~ 1/r ^{2}. In addition, for the complete deduction of the Newton�s force law the following statements are necessary: 1) the capacity of etheronic emission, C ^{Em}, of a body is equal to its capacity of absorption, C ^{Abs}; 2) the capacity of emission is proportional to the number of etherons contained in the body; 3) the number of emitted quanta (etherons) is proportional to the capacity of emission of the emitting body and to the capacity of absorption of the absorbing body. Therefore, the potential energy of the twobody system (A, B) reads
U _{A,B}(r) = S _{transit}E _{E} ~ (C _{A} ^{Em}C _{B} ^{Abs} + C _{B} ^{Em}C _{A} ^{Abs}) ~ (C _{A} ^{Em}C _{B} ^{Em} + C _{B} ^{Abs}C _{A} ^{Abs}) ~ C _{A} ^{Em}C _{B} ^{Em} ~ N _{A}N _{B} ~ M _{A}M _{B}
In this way, the etheronic hypothesis can complete the demonstration of Synge, leading eventually to Newton�s law of gravitational force with the requirement that any material body should be constructed of etherons.
A temerarious conjecture such as the etheronic hypothesis can rise many and difficult problems regarding, for instance, the motion of a large number of etherons in a nucleon. Of course, when we speak of �partons� instead of etherons, the problems by no means become simpler and there is no satisfactory solution so far. A suitable model should explain the charge and the spin as hydrodynamicstatistical effects of the collective motion of particle constituents. Perhaps the relativity theory itself has to be reformulated in this respect on statistical bases, as recently sketched in a recent paper by J. C. Aron [28].
In
spite
of
the
serious
problems
raised
by
the
etheronic
hypothesis,
the
possibilities
of
partial
explanation
discussed
above,
as
well
as
the
suggested
connections
between
the
physical
phenomena
occurring
at
cosmic
and
infranuclear
levels,
are
tempting
and
even
encouraging
for
this
model,
as
a
possible
way
towards
a
more
unitary
picture
of
the
physical
world.
If
this
way
will
be
proven,
then
the
gravity

this
yet
so
poorly
known
interaction

will
play
a
more
important
role
then
it
is
considered
nowadays.
The
rise
of
the
interest
in
the
last
decade
for
the
concept
of
ether
could
be
an
indication
in
this
respect.
A new explanation of the Newtonian law of gravitation is given, proceeding from the following statements: a) the Universe is finite and filled with some particles of exceedingly small mass, traveling chaotically at the speed of light; b) all the material bodies in the Universe are made up of such particles called �etherons�; c) the matter in the Universe is prevailingly under the form of etherons; d) the hydrodynamic mechanism of Lesage for the gravitational interaction is valid, the cosmic background being the ether made up of etherons. The uncertainty principle of quantum mechanics and some dimensionless relations of relativistic cosmology  among which Mach�s principle  are adopted in view of establishing the intrinsic characteristics of etherons as well their number in the Universe. By applying statistical ratiocinations to the etheronic background (fluid), expressions of Hubble�s and Newton�s constants are derived in terms of some kinetic entities pertaining to the ether. The emergence of the inverse square law of force entails at the same time a very strong coupling of the etherons in a nucleon and a saturation character for the binding forces. A wide discussion is undertaken concerning the consistency of the physical world picture suggested by the etheronic conjecture with the already constituted frame of conventional physics, drawing interesting and encouraging conclusions.
Acknowledgements.
The
author
is
indebted
to
his
colleague
Dr.
Nicolae
IonescuPallas
for
the
kindness
of
favoring
with
a
critical
discussion
of
the
whole
problem
and
for
his
help
in
clarifying
many
special
aspects.
Thanks
are
also
due
to
Professors
Ioan
Gottlieb
and
Liviu
Sofonea,
as
well
as
to
Andrei
Dorobantu
for
appreciation
and
moral
assistance,
to
the
young
physicist
Silviu
Olariu
for
stimulating
discussions,
and
to
all
who,
in
one
way
or
other,
manifested
interest
for
this
work.
1. We remind here the conception about ether of the Roumanian philosopher Prince Grigorie Sturdza at the end of the 19th century; he has had at the time a correct intuition of the order of magnitude of the implied quantities, in spite of the incipient stage of the cosmology in that epoch.
2. It is interesting to accomplish, in this context, a comparison between gravitational and strong interactions. As argued above, it is plausible that the gravitational static potential is of the Yukawa type:
F(r) =  (Gm/r)exp( r/ l _{E}) =  (mc ^{2}/M)(R/r)exp( r/R)
where m is the mass of the body and the �coupling constant�, G, is Newton�s constant. A similar expression results for the strong interactions if we introduce the pion mass, m _{ p } , nucleon mass, m _{n} , nucleon radius, r _{n} , Compton length of the pion, l _{p } = h _{B}/m _{ p }c @ r _{n} , Compton length of the nucleon, l _{n} , cross section of the pion, s _{p} = l _{p}l _{n} , and the nucleonic coupling constant, G _{n} = c ^{3} s _{p} /h _{B } @ r _{n}c ^{2}/m _{n }. Let us remark some ratios between quantities at cosmic and infranucleonic scales, namely l _{E}/ l _{p }@ 10 ^{41} and G _{n}/G @ 10 ^{39} [Kretschet, Caldirola and others (16)].
3.
From
(29)
it
would
result
that
about
90.8
%
of
the
mass
of
the
constituents
of
a
nucleon
is
annihilated,
leading
in
this
way
to
a
very
strong
coupling.
Note
added
on
January
6,
2003:
It
is
worthwhile
to
notice
that
some
slight
modifications
of
the
constants
k
_{i
}allow
to
approximate
the
mass
expression
(28)
by
M
_{A,B}
=
(1/4
p)m
_{E}N
_{A,B}
, thus suggesting that the �ultimate� particle, the etheron, might result by the fusion of 4
p
@
12
¸
13
etherons,
just
as
needed
to
ensure
the
most
compact,
icosahedral
symmetry
[29].
Further
considerations
on
the
connection
between
the
etheron
conjecture
and
the
modern
string
theory
are
given
in
[30].
1. In the recent monograph of J. Heidmann, devoted to the relativistic cosmology (Springer Verlag, 1980), the applicability of the uncertainty relation dE dt @ h _{B} is confirmed at the scale of the whole Universe if dt @ H ^{ 1}; obviously, this implies the existence of an energy quantum with mass m = h _{B}H/c ^{2}.
2. In the work of L. S. Mayants, � On the existence of zero mass particles� [Found. Phys., 11, 577 (1981)], a concept is argued according to which the electromagnetic field is replaced by a gas of particles, called �emons�, having a tiny but nonzero rest mass (m < 10 ^{ 50 } kg). It is shown that the existence of emons do not contradict the special theory of relativity and confirms earlier hypotheses of Louis de Broglie regarding the massive photons [5, 15]. The theoretical considerations of Mayants are, in some way, similar to the ideas presented in this work  excepting the fact that these refer to electromagnetism and not to gravity.
3. Criticizing a few months ago the cosmological theory of Big Bang, Fred Hoyle claims that the magnitude of the cosmologic epoch t @ H ^{ 1} is too small to justify the huge information stored into highly organized beings (about 10 ^{40,000} specific modes of which about 2000 genes can be made up from ~ 1020 nucleotide chains). According to the opinion of Hoyle, the evolutional process leading to the apparition of intelligent life would necessitate several cosmological Hubble�s epochs. If this critique will be proven as realistic, then the interpretation of Hubble�s constant as a pure constant, and not as �1/Universe Age�, will acquire an unexpected support.
4. Note added on January 6, 2003 from [29]: The free particle in a spacetime cavity. Let us consider a rectangular spacetime cavity (L, L, L, T) containing a free particle which is described by a KleinGordon steady state wave function of the form
Y (x, y, z, t) = sin(n _{1} p x/L) sin(n _{2} p y/L) sin(n _{3} p z/L) sin(n _{4} p t/L)
where n _{1} , n _{2} , n _{3} , n _{4} are positive integers. The momentum components and the energy of the particle are thus subjected to the quantum conditions
p
_{x}L
=
n
_{1}
p
h
_{B}
p
_{y}L
=
n
_{2}
p
h
_{B}
p
_{z}L
=
n
_{3}
p
h
_{B}
ET
=
n
_{4}
p
h
_{B}
Let us further consider the following quadratic form of positive integers, as suggested by the discrete Minkowskian metric
n _{4} ^{2} � (n _{1} ^{2} + n _{2} ^{2} + n _{3} ^{2}) = (ET/ p h _{B}) ^{2} � (pL/ p h _{B}) ^{2} = (T/ p h _{B})[E ^{2} � (L/T)p ^{2}]
In order to ensure the largest conceivable freedom of the particle, the cavity will be extended to the observable Universe, thus obeying the cosmological relation L = cT = c/H between the Universe size L and age T. Finally, we get in this way the quantization of the rest mass m _{0} and of the rest energy E _{0} = m _{0}c ^{2} of the free particle within the Universe in the form
(E _{0}/ p h _{B}H) ^{2} = n _{4} ^{2} � (n _{1} ^{2} + n _{2} ^{2} + n _{3} ^{2})
where p h _{B}H @ 10 ^{�33} eV, according to the uncertainty principle extended to the whole Universe, represents the smallest energy that can be measured in the age of the Universe. The integers n _{i }have an upper limit imposed by the following two reasons. Thus, a first condition restricts the temporal quantum number according to n _{4} = E/ p h _{B}H £ Mc ^{2}/ p h _{B}H @ 10 ^{122}, where M @ Lc ^{2}/G @ 10 ^{53} kg is the mass of the Universe. A second condition confines the spatial quantum numbers according to n _{1} ^{2} + n _{2} ^{2} + n _{3} ^{2} = (pL/ p h _{B}) ^{2} = L ^{2}/( l/2) ^{2} £ (L/L _{P}) ^{2 } @ 10 ^{122} , where L _{P} = (h _{B}G/c ^{3}) ^{1/2} @ 10 ^{�35} m is the Planck�s length (the quantum fluctuation of the space).
The above quadratic form of the four spacetime quantum numbers, n _{i }, can be further split by analogy with the Dirac�s method and gives E _{0}/ p h _{B}H = ±[ a _{4}n _{4} � ( a _{1} n _{1} + a _{2}n _{2} + a _{3}n _{3})], where the operators a _{i} have the following properties: a _{4} ^{2} = +1; a _{1} ^{2} = a _{2} ^{2} = a _{3} ^{2} = �1; a _{ i } a _{j} + a _{ j } a _{i} = 0 (i, j = 1, 2, 3, 4; i # j).
Similar conclusions can be drawn by changing the cubic Universe into a spherical one. Indeed, in the latter case we only have to introduce the corresponding quantum conditions and will eventually get the quantified mass in terms of the temporal, n _{4} , principal, n, and orbital, l, quantum numbers, as
(E _{0}/ p h _{B}H) ^{2} = n _{4} ^{2} � (n + l/2) ^{2}
subjected
to
the
corresponding
limitation
to
(n
_{4})
_{max}
= (n +
l/2)
^{2}
_{max}
@
10
^{122}.
5. Note added on January 6, 2003 from [30]: A consequence of Hubble�s law (1/E)dE/dt £ � H , as extended from (32), would be that the orbits of motion in a central field of mass M will expand at a rate of the order of
dr/r ³ (4 p ^{3}/3)Hr ^{3/2}/(GM) ^{1/2}
per period, where r is the average dimension of the orbit (see the deduction of this formula in Addendum 6 below). Consequently, for instance, the orbit of the Moon in the field of the Earth would expand by dr/r ³ 3 10 ^{�10} per period, while the orbit of the Earth in the field of the Sun would expand by dr/r ³ 6 10 ^{�9} per period. However, this expansion might become significant at the galactic scale; thus, for a typical galaxy of mass M @ 10 ^{40} kg and a radius of 10 ^{4} light years, the expansion becomes dr/r ³ 0.1 per period and might contribute to the formation of arms of the spiral galaxies [30].
6.
Note
added
on
January
6,
2003
from
[30]:
Let
us
consider
a
mass
m
orbiting
in
the
central
field
of
mass
M
>>
m
which
decays
according
to
the
extended
Hubble�s
law
as
considered
above,
i.e.
M
=
M
_{0}exp(�Ht)
@
M
_{0}(1
�
Ht).
Denoting
by
r
the
average
orbit
radius,
the
outward
acceleration
induced
by
the
net
mass
decrease
dM
=
M
_{0}Ht
is
da
=
�
dF/m
=
�G
dM/r
^{2}
=
�GM
_{0}Ht/r
^{2}
or, integrating twice for t, the increase of the orbit radius is
dr
=
GM
_{0}Ht
^{3}/6r
^{2}.
The
relative
radius
increase
per
period
is
thus
dr/r
=
(1/6)GM
_{0}H(t/r)
^{3}
= (8
p
^{3}/6)GM
_{0}H/v
^{3},
where
we
introduced
the
tangential
velocity
v
=
2
pr/t.
On
the
other
hand,
from
mv
^{2}/r
=
GmM
_{0}/r
^{2}
we have v = (GM
_{0}/r)
^{1/2},
so
that
we
finally
get
the
expression
dr/r
³
(4
p
^{3}/3)Hr
^{3/2}/(GM)
^{1/2},
as
used
in
Addendum
5
above.
Generally,
as
it
is
well
known
for
an
adiabatic
Kepler
orbit
around
a
slowly
varying
mass
M
[L.
D.
Landau
and
E.
M.
Lifshitz,
Mechanics],
the
eccentricity
of
the
orbit
remains
unchanged,
while
the
orbit
radius
vary
as
r
~
1/m
or
dr/r
=
�
dm/m.
On
the
other
hand,
according
to
the
extended
Hubble�s
law,
dm/m
=
�
Hdt.
Finally,
we
obtain
dr/r
=
Hdt,
i.e.
the
planetary
systems
expand
with
the
recession
velocity
v
_{r}
=
dr/dt
=
Hr,
the
spiral
trajectories
getting
progressively
away
from
the
force
center.
After this work has been sent for publication (in the journal of physics Studii si Cercetari de Fizica of the Roumanian Academy, subsequently published in the issue Stud. Cercet. Fiz., 34, 451468 (1982)), the author continued the discussions initiated at Timisoara (at the Annual National Conference on � Progresses in Physics�, October 1981, where this work has been delivered as a plenary lecture). These have been carried out, among others, with Aretin Corciovei (at that time acting as head of the Theoretical Physics Department of the Institute of Physics and Nuclear Engineering at BucharestMagurele). It was found appropriate to present shortly the critiques formulated by him in a postscript. Aretin Corciovei agreed with this procedure and sent to the author some of his objections. These are presented in the text (in italic fonts) to follow.
In the present work is introduced the concept of etheron as being the smallest particle that can exist and which mediates the gravitational interactions. For the computation of the mass of this universal particle three ways of approach are suggested. For some aspects of the problem is considered that the universe ought to be static, but actually models of dynamic universe will be needed. Three ways of approach of the etheron mass are discussed below.
1. It is considered that the uncertainty relations of Heisenberg are applicable to the scale of the whole universe and the time incertitude is identified to the universe age. It is also considered that the energy incertitude represents the minimal quantum which can be exchanged between parts of the universe. The mass associated with this minimal quantum is considered to be the mass of the etheron. In order to obtain the value m _{E} = h _{B}H/c ^{2}, the author has to take the universe age equal to 1/H, H being Hubble�s constant, that is to return to the hypothesis of an universe with a linear expansion in time. It should be noted that the hypothesis of an universe linearly expanding in time means to consider the velocity of a given galaxy (for instance relative to the Sun) as constant; but because the distance between this galaxy and other galaxy (in particular relative to the Sun) increases linearly in time, the �constant� H decreases linearly in time. Therefore, the mass of the etheron should diminish also linearly and the etheron in A.D. 2000 would have a mass slightly smaller than in Democritus times. However, all known particles have fixed mass. Thus, the hypothesis of a variable mass of the etheron is equivalent to a continuous creation of etherons in the electron in order to keep the electron mass constant.
2. It is considered the whole universe as having an oscillatory motion. The pulsation w of the universe is identified to the Hubble�s constant. It is considered that the states of the universe are characterized by the quantified energies of the harmonic oscillator with the pulsation w. The spectrum is practically continuous, the difference h _{B} w between levels providing the energy of the smallest allowed quantum, the etheron. It is arrived again at m _{E} = h _{B}H/c ^{2}. Obviously, the hypothesis that the universe is oscillating in time contradicts the first hypothesis of an universe linear in time. This contradicts the hypothesis of a static universe as well.
Let us comment a little the hypothesis that the universe is oscillating in time. Let us write, for instance, R(t) = R _{0} sin wt  for the time dependence of the distance (to the Sun) of a galaxy. At the present epoch T of the universe R(T) = R _{0} sin wT . The Hubble�s constant is ( wcos wT)/ sin wt  and we notice that, in order to get w = H , we should be at an extremely particular moment, T, given by HT = p/4 . Does the universe age satisfies such a particular relation? Finally, if we take R(t) = R _{0}(1 + sin wt), a possible way to obtain w = H would be precisely T = 0. In other words, the hypothesis of the identification w = H is extremely particular.
3. Finally, it is considered the radius of the universe as the maximal radius of gravitational interaction. Analogously to the potential used for nuclear forces, it is possible to introduce a potential of the Yukawa type for the gravitational potential, namely (1/r)exp( r/R _{U}) , where R _{U} is the radius of the universe. The radius of the universe is equalized to the Compton length associated to the gravity quantum, the etheron, i.e. l = h _{B}/m _{E}c. It is considered R _{U} equal to c/H though nobody has observed Doppler shifts of some galaxies having velocities exactly equal to c. It results again m _{E} = h _{B}H/c ^{2}. In any way, the hypothesis that the galaxies found at the edge of the universe move at light speed contradicts the hypothesis of a static universe.
It is to be noticed that all three ways of approach of the problem suppose contradictory models of universe evolution, including the static model adopted in order to use the relation GM/c ^{2}R _{U} = p/2 (yet in the static model H is meaningless).
Finally, it is still to be asked about the urgent experimental facts which led to the need of a new particle, the etheron, and which are its other characteristics (spin, charge, other internal quantum numbers).
It is possible to formulate also observations of detail. Here is given only an example. Thus, in the expression for the field equations of Einstein (formula 30) it is assumed that the common pressure vanish (remaining only the cosmological pressure), but in the next formula it is assumed that the etheron travels at light speed, a case in which the pressure is maximum.
The author (I. I. P.) of the present work hopes that the presentation of such critiques as Aretin Corciovei�s above allows the perception of the