A fundamental explanation for the tiny value of the cosmological constant
aa r X i v : . [ phy s i c s . g e n - ph ] N ov A fundamental explanation for the tiny value of the cosmological constant
Cl´audio Nassif
CBPF-Centro Brasileiro de Pesquisas F´ısicas, Rua Dr.Xavier Sigaud,150, CEP 22290-180, Rio de Janeiro-RJ, Brazil.
We will look for an implementation of new symmetries in the space-time structure and theircosmological implications. This search will allow us to find a unified vision for electrodynamicsand gravitation. We will attempt to develop a heuristic model of the electromagnetic nature ofthe electron, so that the influence of the gravitational field on the electrodynamics at very largedistances leads to a reformulation of our comprehension of the space-time structure at quantumlevel through the elimination of the classical idea of rest. This will lead us to a modification ofthe relativistic theory by introducing the idea about a universal minimum limit of speed in thespace-time. Such a limit, unattainable by the particles, represents a preferred frame associatedwith a universal background field (a vacuum energy), enabling a fundamental understanding of thequantum uncertainties. The structure of space-time becomes extended due to such a vacuum energydensity, which leads to a negative pressure at the cosmological scales as an anti-gravity, playing therole of the cosmological constant. The tiny values of the vacuum energy density and the cosmologicalconstant will be successfully obtained, being in agreement with current observational results.
PACS numbers: 98.80.Es, 11.30.Qc
I. INTRODUCTION
Motivated by Einstein’s ideas for searching for new fundamental symmetries in Nature, our main focus isto go back to that point of the old incompatibility between mechanics and electrodynamics,by extending hisreasoning in order to look for new symmetries that implement gravitation into electrodynamics of movingparticles. We introduce more symmetries into the space-time geometry,where gravitation and electromagnetismbecome coupled with each other,in such a way to enable us to build a new dynamics that is compatible withthe quantum indeterminations.Besides quantum gravity at the Planck length scale,our new symmetry idea appears due to the indispensablepresence of gravity at quantum level for particles with very large wavelengths (very low energies). This leadsus to postulate a universal minimum speed related to a fundamental (privileged) reference frame of backgroundfield that breaks Lorentz symmetry[1].Similarly to Einstein’s reasoning,which has solved that old incompatibility between nature of light and motionof matter (massive objects), let us now expand it by making the following heuristic assumption based on newsymmetry arguments:
If,in order to preserve the symmetry (covariance) of Maxwell’s equations, c is required to be constant basedon Einstein’s reasoning,according to which it is impossible to find the rest reference frame for the speed of light( c − c = 0 ( = c )) due to the coexistence of ~E and ~B in equal-footing,then now let us think that fields ~E and ~B may also coexist for moving charged massive particles (as electrons),which are at subluminal level ( v < c ).So,by making such an assumption,it would be also impossible to find a rest reference frame for a charged massiveparticle,by canceling its magnetic field,i.e., ~B = 0 with ~E = 0 . This would break the coexistence of these twofields,which would not be possible because it is impossible to find a reference frame where v = 0 ,in such a space-time. Thus we always must have ~E = 0 and also ~B = 0 for charged massive particles,due always to the presenceof a non-null momentum for the electron,in a similar way to the photon electromagnetic wave. The reasoning above leads to the following conclusion:-
The plane wave for free electron is an idealization impossible to conceive under physical reality. In theevent of an idealized plane wave,it would be possible to find the reference frame that cancels its momentum( p = 0 ),just the same way as we can find the reference frame of rest for classical (macroscopic) objects withuniform rectilineal motion (a state of equilibrium). In such an idealized case, we could find a reference framewhere ~B = 0 for charged particle. However, the presence of gravity in quantum world emerges in order toalways preserve the coexistence of ~E and ~B ( = 0) in electrodynamics of moving massive particles (section 3). That is the reason why we think about a lowest and unattainable speed limit V in such a space-time, in orderto avoid to think about ~B = 0 ( v = 0 ). This means that there is no state of perfect equilibrium (plane waveand Galilean inertial reference frame) for moving particles in such a space-time,except the privileged inertialreference frame of a universal background field associated with an unattainable minimum limit of speed V . Sucha reasoning allows us to think that the electromagnetic radiation (photon: “ c − c ′′ = c ) as well as the matter(electron: “ v − v ′′ > V ( = 0) ) are in equal-footing,since now it is not possible to find a reference frame inequilibrium ( v relative = 0 ) for both through any velocity transformations (section 6).The interval of velocity with two limits V < v ≤ c represents the fundamental symmetry that is inherent tosuch a space-time,where gravitation and electrodynamics become coupled. However,for classical (macroscopic)objects,the breaking of that symmetry,i.e., V →
0, occurs so as to reinstate Special Relativity (SR) as a particular(classical) case,namely no uncertainties and no vacuum energy,where the idea of rest,based on the Galileanconcept of reference frame is thus recovered.In another paper,we will study the dynamics of particles in the presence of such a universal (privileged)background reference frame associated with V , within a context of the ideas of Mach[2],Schroedinger[3] andSciama[4],where we will think about an absolute background reference frame in relation to which we havethe inertia of all moving particles. However,we must emphasize that the approach we will intend to use isnot classical as the machian ideas (the inertial reference frame of fixed stars),since the lowest limit of speed V ,related to the privileged reference frame connected to a vacuum energy,has origin essentially from the presenceof gravity at quantum level for particles with very large wavelengths.We hope that a direct relationship should exist between the minimum speed V and Planck’s minimum length l p = ( G ~ /c ) / ( ∼ − m ) treated by Double Special Relativity theory (DSR)[20-25] (4th section).In the next section,a heuristic model will be developed to describe the electromagnetic nature of the matter.It is based on the Maxwell theory used for investigating the electromagnetic nature of a photon when theamplitudes of electromagnetic wave fields are normalized for one single photon with energy ~ w . Thus,due tothe reciprocity and symmetry reasoning,we shall extend such a concept for the matter (electron) through theidea of pair materialization after γ -photon decay,so that we will attempt to develop a simple heuristic model ofthe electromagnetic nature of the electron that will experiment a background field in the presence of gravity.The structure of space-time becomes extended due to the presence of a vacuum energy density associatedwith such a universal background field (a privileged reference frame connected to a zero-point energy of back-ground field,which is associated with the minimum limit of speed V for particles moving with respect to such abackground reference frame). This leads to a negative pressure at the cosmological length scales,behaving likea cosmological anti-gravity for the cosmological constant whose tiny value will be determined (section 8). II. ELECTROMAGNETIC NATURE OF THE PHOTON AND OF THE MATTERA. Electromagnetic nature of the photon
In accordance with some laws of Quantum Electrodynamics[5],we shall take into account the electric field ofa plane electromagnetic wave whose amplitude is normalized for just one single photon[5]. To do this,considerthat the vector potential of a plane electromagnetic wave is ~A = acos ( wt − ~k.~r ) ~e, (1)where ~k.~r = kz ,admitting that the wave propagates in the direction of z,being ~e the unitary vector of polariza-tion. Since we are in vacuum,we must consider ~E = − c ∂ ~A∂t = ( wac ) sen ( wt − kz ) ~e (2)In the Gaussian system of units,we have | ~E | = | ~B | . So the average energy density of the wave shall be h ρ eletromag i = 18 π D | ~E | + | ~B | E = 14 π D | ~E | E (3)Substituting (2) into (3),we obtain h ρ eletromag i = 18 π w a c , (4)where a is an amplitude that depends upon the number of photons in such a wave. Since we wish to obtainthe plane wave of one single photon ( ~ w ),then by making this condition for (4) and by considering an unitaryvolume for the photon ( v ph = 1),we have a = r π ~ c w (5)Substituting (5) into (2),we obtain ~E ( z, t ) = wc r π ~ c w sen ( wt − kz ) ~e, (6)from where,we deduce that e = wc r π ~ c w = √ π ~ w, (7)where e could be thought of as an electric field amplitude normalized for 1 single photon,with b = e (Gaussiansystem),being the magnetic field amplitude normalized for 1 photon. So we may write ~E ( z, t ) = e sen ( wt − kz ) ~e (8)Substituting (8) into (3) and considering the unitary volume ( v ph = 1),we obtain h E eletromag i = 18 π e ≡ ~ w (9)Now,starting from the classical theory of Maxwell for the electromagnetic wave,let us consider an averagequadratic electric field normalized for one single photon,which is e m = e / √ rD | ~E | E . So doing such aconsideration,we may write (9) in the following alternative way: h E eletromag i = 14 π e m ≡ ~ w, (10)where we have e m = e √ wc r π ~ c w = √ π ~ w (11)It is important to emphasize that,although the field in (8) is normalized for only one photon,it is still a classicalfield of Maxwell because its value oscillates like a classical wave (solution (8)). The only difference is that wehave thought about a small amplitude field for one photon. Actually the amplitude of the field ( e ) cannot bemeasured directly. Only in the classical approximation (macroscopic case), when we have a very large numberof photons ( N → ∞ ),we can somehow measure the macroscopic field E of the wave. Therefore,although wecould idealize the case of just one photon as if it were an electromagnetic wave of small amplitude,the solution(8) is even a classical one,since the field ~E presents oscillation.Actually we already know that the photon wave is a quantum wave, i.e.,a de-Broglie wave,where its wavelength( λ = h/p ) is not interpreted classically as the oscillation frequency (wavelength due to oscillation) of a classicalfield. However,in a classical case,using the solution (8),we would have E eletromag = 14 π | ~E ( z, t ) | = 14 π e sen ( wt − kz ) (12)In accordance with (12),if the wave of a photon were really classical, then its energy would not be fixed,as wecan see in (12). Consequently,its energy ~ w would just be an average value [see (10)]. Hence,in order to achieveconsistency between the result (10) and the quantum wave (de-Broglie wave),we must interpret (10) to berelated to the de-Broglie wave of the photon with a fixed discrete energy value ~ w instead of an average energyvalue,since now we consider that the wave of one single photon is a non-classical wave,i.e.,it is a de-Broglie wave.Thus we rewrite (10) as follows: E eletromag = E = pc = hcλ = ~ w ≡ π e ph , (13)where we conclude that λ ≡ πhce ph , (14)where λ is the de-Broglie wavelength. Now,in this case (14),the single photon field e ph should not be assumedas a mean value for oscillating classical field,and we shall preserve it in order to interpret it as a scalar quantumelectric field (a microscopic field) of a photon. So basing on such a heuristic reasoning,let us also call it “scalarsupport of electric field” ,representing a quantum (corpuscular)-mechanical aspect of electric field for the photon.As e ph is responsible for the energy of the photon ( E ∝ e ph ),where w ∝ e ph and λ ∝ /e ph ,indeed we see that e ph presents a quantum behavior,as it provides the dual aspects (wave-particle) of the photon,where its mechanicalmomentum may be written as p = ~ k = 2 π ~ /λ = ~ e ph / hc [refer to (14)],or simply p = e ph / πc . B. Electromagnetic nature of the matter
Our objective is to extend the idea of the photon electromagnetic energy [equation (13)] for the matter. Bydoing this,we shall provide heuristic arguments that rely directly on de-Broglie reciprocity postulate, which hasextended the idea of wave (photon wave) for the matter (electron), behaving also like wave. Thus the relation (14)for the photon,which is based on de-Broglie relation ( λ = h/p ) may also be extended for the matter (electron),inaccordance with the idea of de-Broglie reciprocity. In order to strengthen such an argument,we are going toassume the phenomenon of pair formation,where the photon γ decays into two charged massive particles,namelythe electron ( e − ) and its anti-particle,the positron ( e + ). Such an example will enable us to better understandthe need of extending the idea of the photon electromagnetic mass ( m electromag = E electromag /c ) (equation 13)for the matter ( e − and e + ), by using that concept of field scalar support. .Now consider the phenomenon of pair formation,i.e., γ → e − + e + . Then,by using the conservation of energyfor γ -decay,we write the following equation: E γ = ~ w = m γ c = m − c + m +0 c + K − + K + = 2 m c + K − + K + , (15)where K − and K + represent the kinetic energy for electron and positron respectively. We have m − c = m +0 c ∼ =0 , γ electromagnetic energy is E γ = hν = m γ c = π e γ ,or else, E γ = ǫ e γ given in theInternational System of Units (IS),and also knowing that e γ = cb γ (IS),where b γ represents the magnetic fieldscalar support for the photon γ ,so we also may write E γ = cǫ ( e γ )( b γ ) (16)Photon has no charge,however,when it is materialized into the pair electron-positron,its electromagnetic con-tent given in (16) ceases to be free or purely kinetic (purely relativistic mass) to become massive through thematerialization of the pair. Since such massive particles ( v (+ , − ) < c ) also behave like waves in accordancewith de-Broglie idea,now it would also be natural to extend the relation (14) (of the photon) for representingwavelengths of the matter (electron or positron) after the photon- γ decay,namely: λ (+ , − ) ∝ hcǫ [ e (+ , − ) s ] = hǫ [ e (+ , − ) s ][ b (+ , − ) s ] , (17)where e (+ , − ) s and b (+ , − ) s play the role of the electromagnetic content for energy condensed into matter ( scalarsupport of electromagnetic field for the matter). Such fields are associated with the total energy of the movingmassive particle,whose mass has essentially an electromagnetic origin,given in the form m ≡ m electromag ∝ e s b s , (18)where E = mc ≡ m electromag c .Basing on (16) and (17),we may write (15) in the following way: E γ = cǫ e γ b γ = cǫ e − s b − s v − e + cǫ e + s b + s v + e = [ cǫ e − s b − s v e + K − ] + [ cǫ e + s b + s v e + K + ] = (19)2 cǫ e (+ , − ) s b (+ , − ) s v e + K − + K + = 2 m c + K − + K + ,where m c = m (+ , − )0 c = cǫ e (+ , − ) s b (+ , − ) s v e ∼ = 0 , M eV . e (+ , − ) s and b (+ , − ) s represent the proper electromag-netic contents of electron or positron. Later we will show that the mass m does not represent a classic restmass due to the inexistence of rest in such a space-time. This question shall be clarified in 5th section. Thevolume v e in (19) is a free variable to be considered.In accordance with equation (19),the present model provides a fundamental point that indicates electron isnot necessarily an exact punctual particle. Quantum Electrodynamics,based on Special Relativity deals withthe electron as a punctual particle. The well-known classical theory of the electron foresees for the electronradius the same order of magnitude of the radius of a proton,i.e., R e ∼ − m .The most recent experimental evidence about scattering of electrons by electrons at very high kinetic energiesindicates that the electron can be considered approximately a point particle. Actually electrons have an extentless than collision distance,which is about R e ∼ − m [6]. Actually such an extent is negligible in comparison tothe dimensions of an atom (10 − m ),or even the dimensions of a nucleus (10 − m ),but it is not exactly a point.By this reason,the present model can provide a very small non-null volume v e for the electron. But, if we justconsider v e = 0 according to (19),we would have an absurd result,i.e,divergent internal fields e s = b s → ∞ .However, for instance,if we consider R e ∼ − m ( v e ∝ R e ∼ − m ) for our model,and knowing that m c ∼ = 0 , M eV ( ∼ − J ),thus,in such a case (see (19)),we would obtain e s ∼ V /m . Such a value isextremely high and therefore we may conclude that the electron is extraordinarily compact,with a very highenergy density. So,for such an example,if we imagine over the “surface” of the electron,we would detect a field e s ∼ V /m instead of an infinite value for it. According to the present model,the field e s inside the almostpunctual non-classical electron with such a radius ( ∼ − m ) would be finite and constant ( ∼ V /m )instead of a function like 1 /r with divergent classical behavior. Indeed,for r > R e ,the field E decreases like1 /r ,i.e, E = e/r . For r = R e , E = e/R e ≡ e s . Actually,for r ≤ R e ,we have E ≡ e s = constant ( ∼ V /m ).The next section will be dedicated to the investigation about the electron coupled to a gravitational field.
III. ELECTRON COUPLED TO A GRAVITATIONAL FIELD
When a photon with energy hν is subjected to a certain gravitational potential φ ,its energy (or frequency)increases to be E ′ = hν ′ ,where E ′ = hν ′ = hν (1 + φc ) (20)By convention,as we have stipulated φ > ν ′ > ν . By considering (16) forany photon and by substituting (16) into (20),we alternatively write E ′ = cǫ e ′ ph b ′ ph = cǫ e ph b ph √ g , (21)where g is the first component of the metric tensor,where √ g = (1 + φc ) and e ph = cb ph .From (21),we can extract the following relationships,namely: e ′ ph = e ph q √ g , b ′ ph = b ph q √ g (22)In the presence of gravity,such fields e ph and b ph of the photon increase according to (22),leading to theincreasing of the photon frequency or energy,according to (20). Thus we may think about the following incre-ments,namely: ∆ e ph = e ′ ph − e ph = e ph ( q √ g − , ∆ b ph = b ′ ph − b ph = b ph ( q √ g −
1) (23)In accordance with General Relativity (GR),when a massive particle of mass m moves in the presence of agravitational potential φ ,its total energy E is given in the following way: E = mc = m c √ g + K, (24)where we can think that m (= m (+ , − )0 ) is the mass of the electron or positron,emerging from γ -decay in thepresence of a gravitational potential φ .In order to facilitate the understanding of what we are proposing,let us consider K << m c ,since we areinterested only in obtaining the influence of the potential φ . Therefore we write E = m c √ g (25)As we already know that E = m c = cǫ e (+ , − ) s b (+ , − ) s v e ,we can also write the total energy E ,as follows: E = cǫ e (+ , − ) s b (+ , − ) s v e = cǫ e (+ , − ) s b (+ , − ) s v e √ g , (26)from where we can extract e (+ , − ) s = e (+ , − ) s q √ g , b (+ , − ) s = b (+ , − ) s q √ g . (27)So we obtain ∆ e s = e (+ , − ) s ( q √ g − , ∆ b s = b (+ , − ) s ( q √ g − , (28)where we have ∆ e s = c ∆ b s .As the energy of the particle can be represented as a condensation of electromagnetic fields in scalar forms e s and b s ,this model is capable of assisting us to think that the well-known external fields ~E and ~B for the movingcharged particle,by storing an energy density ( ∝ | ~E | + | ~B | ) should also suffer some influence (shifts) in thepresence of gravitational potential. In accordance with GR,every kind of energy is also a source of gravitationalfield. This non-linearity that is inherent to the gravitational field leads us to think that,at least in a certainapproximation in the presence of gravity,the external fields E and B should experiment positive small shifts δE and δB ,which are proportional to the intrinsic increments (shifts) ∆ e s and ∆ b s of the particle,namely: δE = ( E ′ − E ) ∝ ∆ e s = ( e s − e s ) , δB = ( B ′ − B ) ∝ ∆ b s = ( b s − b s ) (29)Here we have omitted the signs (+ , − ) in order to simplify the notation. Since ∆ e s = c ∆ b s ,then δE = cδB .In accordance with (29),we may conclude that there is a constant of proportionality that couples the externalelectromagnetic fields E and B of the moving charge with gravity by means of the small shifts δE and δB . Sucha constant works like a fine-tuning,namely: δE = ξ ∆ e s , δB = ξ ∆ b s , (30)where ξ is a dimensionaless constant to be obtained. We expect that ξ << δE and δB depend only on φ over the electron.Substituting (28) into (30),we obtain δE = ξe s ( q √ g − , δB = ξb s ( q √ g − . (31)Due to the very small positive shifts δE and δB in the presence of a weak gravitational potential φ ,the totalelectromagnetic energy density in the space around the charged particle is slightly increased,as follows: ρ totalelectromag = 12 ǫ ( E + δE ) + 12 µ ( B + δB ) (32)Substituting (31) into (32) and performing the calculations,we will finally obtain ρ totalelectromag = 12 [ ǫ E + 1 µ B ] + ξ [ ǫ Ee s + 1 µ Bb s ]( q √ g −
1) + 12 ξ [ ǫ ( e s ) + 1 µ ( b s ) ]( q √ g − (33)We may assume that ρ totalelectromag = ρ (0) electromag + ρ (1) electromag + ρ (2) electromag for representing (33), where the firstterm ρ (0) electromag is the free electromagnetic energy density (zero order) for the ideal case of a charged particleuncoupled from gravity ( ξ = 0),i.e,the ideal case of a free particle (a perfect plane wave,which does not exist inreality due always to the presence of gravity). We have ρ (0) ∝ /r ( coulombian term ).The coupling term ρ (1) (second term) represents an electromagnetic energy density of first order,that is,itcontains an influence of 1st order for δE and δB ,as it is proportional to δE and δB due to a certain influenceof gravity. Therefore it is a mixture term that behaves essentially like a radiation term . Thus we have ρ (1) ∝ /r ,since e s (or b s ) ∼ constant and E (or B ) ∝ /r . It is very interesting to notice that such a radiationterm of a charge in a true gravitational field corresponds effectively to a certain radiation field due to an slightlyaccelerated charge in free space,however such an equivalence is weak due to the very small value of ξ .The last coupling term ( ρ (2) ) is purely interactive due to the presence of gravity only. This means that itis a 2nd order interactive electromagnetic energy density term,since it is proportional to ( δE ) and to ( δB ) .Hence we have ρ (2) ∝ /r ∼ constant ,being ρ (2) = ǫ ( δE ) + µ ( δB ) = ǫ ( δE ) = µ ( δB ) ,which variesonly with the gravitational potential ( φ ). Since we have ρ (2) ∝ /r ,it has a non-locality behavior. This meansthat ρ (2) behaves like a kind of non-local field,that is inherent to the space ( a constant term for representing abackground field ). It does not depend on the distance r from the charged particle. So it is a constant energydensity for a fixed potential φ ,and fills the whole space. ρ (2) always exists due to the inevitable presence ofgravity and therefore it cannot be cancelled. Due to this fact,the increment δB that contributes for the densityof interactive energy ρ (2) cannot vanish since the electron is not free ( ρ (2) = 0). This always assures a non-zerovalue of magnetic field ( δB = 0) for any transformation,and so this is the fundamental reason why the fields E and B should coexist in the presence of gravity,where the charge experiments a background field ( ρ (2) ∝ ( δB ) )connected to a privileged reference frame of an unattainable minimum speed that justifies in a kinematic pointof view the impossibility of finding δB = 0. This minimum speed ( V ) is a universal constant that should berelated directly to gravity ( G ), since V is also responsible for the coexistence of E and B . We will see such aconnection in the next section.Usually we have ρ (0) >> ρ (1) >> ρ (2) . For a very weak gravitational field,we can consider a good practicalapproximation as ρ totaleletromag ≈ ρ (0) . However,from a fundamental point of view,we cannot neglect the couplingterms,specially the last one for large distances,as it has a vital importance in this work,permiting us to under-stand a non-local vacuum energy that is inherent to the space,i.e. ρ (2) ∝ /r . Such a background field withenergy density ρ (2) has deep implications for our understanding of the space-time structure at very large scalesof length (cosmological scales),since ρ (2) does not have r -dependence,i.e,it remains for r → ∞ .In the next section,we will estimate the constant ξ and consequently the idea of a universal minimum velocityin the space-time. Its cosmological implications will be treated in section 8. IV. THE FINE ADJUSTMENT CONSTANT ξ AND ITS IMPLICATIONS
Let us begin this section by considering the well-known problem that deals with the electron at the boundstate of a coulombian potential of a proton (Hydrogen atom). We start from this subject because it poses acertain similarity with the present model for the electron coupled to a gravitational field. We know that the finestructure constant ( α F = 1 / ξ plays an even more fundamental role than the fine structure α F ,by considering that ξ couples gravity to the electromagnetic field of the electron charge.Let’s initially consider the energy that bonds the electron to the proton at the fundamental state of theHydrogen atom,as follows: ∆ E = 12 α F m c , (34)where ∆ E is assumed as module. We have ∆ E << m c , where m is the electron mass,which is practicallythe reduced mass of the system ( µ ≈ m ).We have α F = e / ~ c = q e / πǫ ~ c ≈ /
137 (fine structure constant). Since m c ∼ = 0 .
51 Mev,we have∆ E ≈ . E = m c = cǫ e s b s v e ,so we may write (34) in the following alternative way:∆ E = 12 α F cǫ e s b s v e = 12 cǫ ( α F e s )( α F b s ) v e ≡ cǫ (∆ e s )(∆ b s ) v e , (35)from where we extract ∆ e s ≡ α F e s , ∆ b s ≡ α F b s . (36)It is interesting to observe that (36) maintains a certain similarity with (30),however,first of all,we mustemphasize that the variations ∆ e s and ∆ b s for the electron energy have a purely coulombian origin,since thefine structure constant α F depends solely on the electron charge. Thus we can write the electric force betweentwo electronic charges in the following way: F e = e r = q e πǫ r = α F ~ cr , (37)where e = q e / √ πǫ .If we just consider a gravitational interaction between two electrons,we would have F g = Gm e r = β F ~ cr , (38)from where we obtain β F = Gm e ~ c . (39)We have β F << α F due to the fact that the gravitational interaction is much weaker than the electric one,sothat F e /F g = α F /β F ∼ ,where β F ∼ = 1 . × − . Therefore we shall call β F the superfine structure con-stant ,since gravitational interaction creates a bonding energy extremely smaller than the coulombian bondingenergy considered for the fundamental state (∆ E ) in the Hydrogen atom.To sum up,whereas α F ( e ) provides the adjustment for the coulombian bonding energies between two elec-tronic charges, β F ( m e ) gives the adjustment for the gravitational bonding energies between two electronicmasses. Such bonding energies of electrical or gravitational origin increment the particle energy through ∆ e s and ∆ b s .Now,following the above reasoning,we notice that the present model enables us to introduce the very fine-tuning (coupling) ξ between gravity (a gravitational potential generated by the mass m e ) and electrical field(electrical energy density generated by the charge q e (refer to (30))). Thus for such more fundamental case,wehave a kind of bond of the type m e q e (mass-charge) through the adjustment (coupling) ξ . So the subtlenesshere is that the bonding energy density due to ξ ,by means of the increments δE and δB (see (30), (31),(32) or(33)) occurs on the electric and magnetic fields generated in the space by the own charge q e .Although we could show a laborious and step by step problem for obtaining the constant ξ ,the way we followhere is shorter because it starts from important analogies by using the ideas of fine structure α F = α F ( e ),i.e.,aneletric interaction ( charge-charge ) and also superfine structure β F = β F ( m e ),i.e.,a gravitational interaction( mass-mass ). Hence,now it is easy to conclude that the kind of mixing coupling we are proposing,of the type“ m e q e ” ( mass-charge ) represents a gravi-electrical coupling constant,which leads us naturally to think that sucha constant ξ is of the form ξ = ξ ( m e q e ),and therefore meaning that ξ = p α F β F , (40)which represents a geometrical average between electrical and gravitational couplings,and so we finally obtainfrom (40) ξ = r G πǫ m e q e ~ c , (41)where indeed we have ξ = ξ ( m e q e ) ∝ m e q e . From (41) we obtain ξ ∼ = 3 . × − . Let us call ξ fine adjustmentconstant. The quantity √ Gm e in (41) can be thought of as a gravitational charge e g ,so that ξ = e g e/ ~ c . we must not mistake superfine structure β F with hyperfine structure (spin interaction), as they are completely different. α F = e / ~ c = v B /c , where v B = e / ~ = c/ ξ ,we may also write (41) as the ratio of two velocities,as follows: ξ = Vc , (42)from where we have V = ξc = e g e ~ = r G πǫ m e q e ~ , (43)where V ∼ = 1 . × − m/s . In the newtonian (classical) universe,where c → ∞ and V → ξ → V → ξ → ξ ∼ − ,gravitation is coupled to electrodynamics of moving particles. The quantum uncertainties should naturallyarise from such a symmetric space-time structure ( V < v ≤ c ),which will be denominated Symmetrical SpecialRelativity (SSR) due to the existence of two limits of speed.Similarly to the Bohr velocity ( v B ) for fundamental bound state,the speed V is also a universal fundamentalconstant,however the crucial difference between them is that V is associated with a more fundamental boundstate in the Universe as a whole,since gravity ( G ),which is the weakest interaction plays now an important rolefor the dynamics of the electron (electrodynamics) in such a space-time. This may be observed in (43) because,ifwe make G → V → V as an unattainable universal (constant) minimum speed associated with a privilegedframe of background field,but before this,we must provide a better justification of why we consider the electronmass and charge to calculate V ( V ∝ m e q e ),instead of masses and charges of other particles. Although thereare fractionary electric charges as the case of quarks,such charges are not free in Nature for bonding onlywith gravity. They are strongly connected by the strong force (gluons). Actually the charge of the electronis the smallest free charge in Nature. Besides this, the electron is the elementary charged particle with thesmallest mass. Therefore the product m e q e assumes a minimum value. And in addition to that, the electronis completely stable. Other charged particles such as for instance π + and π − have masses that are greaterthan the electron mass, and they are unstable,decaying very quickly. Such a subject may be dealt with moreextensively in another article.We could think about a velocity Gm e / ~ ( << V ) that has origin from a purely gravitational interac-tion,however such a much lower bound state does not exist because the presence of electromagnetic interactionsis essential at subatomic level. And since neutrino does not interact with electromagnetic field,it cannot beconsidered to estimate V .Now we can verify that the minimum speed ( V ) given in (43) is directly related to the minimum length ofquantum gravity (Planck length),as follows: V = √ Gm e e ~ = ( m e e r c ~ ) l p , (44)where l p = p G ~ /c .In (44),as l p is directly related to V , if we make l p → G → V → V in (44),associated with very low en-ergies (very large wavelengths) is directly related to the universal constant of minimum length l p (very highenergies),whose invariance has been studied in DSR by Magueijo,Smolin,Camelia et al [20-25].1The natural consequence of the presence of a more fundamental level associated with V in the space-timeis the existence of a privileged reference frame of background field in the Universe. Such a frame should beconnected to a kind of vacuum energy,that is inherent to the space-time (refer to ρ (2) in equation (33)). Thisidea reminds us of the conceptions of Mach[2], Schroedinger[3] and Assis[7],although such conceptions are stillwithin the classical context.Since we are assuming an absolute and privileged reference frame ( V ), which is underlying and also inherentto the whole space-time geometry,we shall call it ultra-referential- S V . By drawing inspiration from some of thenon-conventional ideas of Einstein in relation to the “ether”[8],let us assume that such an ultra-referential ofbackground field S V ,which in a way redeems his ideas,introduces a kind of relativistic “ether” of the space-time.Such a new concept has nothing to do with the so-called luminiferous ether (classical ether) established beforeRelativity theory.The present idea about a relativistic “ether” for the ultra-referential S V aims at the implementation of thequantum principles (uncertainties) in the space-time. This line of investigation resumes those non-conventionalEinstein’s ideas [8][9],who attempted to bring back the idea of a new “ether” that cannot be conceived ascomposed of punctual particles and having a world line followed in the time.Actually such an idea of “ether” as conceived by Einstein should be understood as a non-classical concept ofether due essentially to its non-locality feature. In this sense, such a new “ether” has a certain correspondencewith the ultra-referential S V due to its totality as a physical space, not showing any movement. In fact,as S V would be absolutely unattainable for all particles (at local level), V would prohibits to think about a perfect planewave (∆ x = ∞ ),since it is an idealized case associated with the perfect equilibrium of a free particle (∆ p = 0).So the ultra-referential S V would really be non-local (∆ x = ∞ ),which is in agreement with that Einstein’sconception about an “ether” that could not be split into isolated parts and which,due to its totality in thespace,would give us the impression that it is actually stationary. In order to understand better its non-localityfeature by using a symmetry reasoning,we must perceive that such a minimum limit V works in a reciprocal waywhen compared with the maximum limit c ,so that particles supposed in such a limit V ,in contrast of what wouldhappen in the limit c ,would become completely “ defrosted ” in the space ( ∆ x → ∞ ) and time ( ∆ τ → ∞ ),beingin anywhere in the space-time and therefore having a non-local behavior. This super ideal condition correspondsto the ultra-referential S V ,at which the particle would have an infinite de-Broglie wavelength,being completelyspread out in the whole space. This state coincides with the background field for S V , however S V is unattainablefor all the particles.In vain,Einstein attempted to satisfactorily redeem the idea of a new “ether” under Relativity in variousmanners[9, 10, 11, 12, 13, 14] because,in effect,his theory wasn’t still able to adequately implement the quantumuncertainties as he also tried to do[15, 16, 17],and in this respect,Relativity is still a classical theory,although thenew conception of “ether” presented a few non-classical characteristics. Actually it was Einstein who coined theterm ultra-referential as the fundamental aspect of Reality. To him,the existence of an ultra-referential cannotbe identified with none of the reference frames in view of the fact that it is a privileged one in respect of theothers. This seems to contradict the principle of Relativity,but,in vain,Einstein attempted to find a relativistic“ether” (physical-space),that is inherent to the geometry of the space-time,which does not contradict such aprinciple. That was the problem because such a new “ether” does not behave like a Galilean reference frameand, consequently,it has nothing to do with that absolute space filled by the luminiferous ether,although itbehaves like a privileged background field in the Universe.The present work seeks to naturally implement the quantum principles into the space-time. Thanks to thecurrent investigation,we shall notice that Einstein’s non-conventional ideas about the relativistic “ether” andalso his vision[18] of making quantum principles to emerge naturally from a unified field theory become closelyrelated between themselves.2 V. A NEW CONCEPTION OF REFERENCE FRAMES AND SPACE-TIME INTERVAL: AFUNDAMENTAL EXPLANATION FOR THE UNCERTAINTY PRINCIPLEA. Reference frames and space-time interval
The conception of background privileged reference frame (ultra-referential S V ) has deep new implications forour understanding of reference systems. That classical notion we have about the inertial (Galilean) referenceframes,where the idea of rest exists, is eliminated at quantum level, where gravity plays a fundamental role forsuch a space-time with a vacuum energy associated with S V ( V ∝ G / / ~ ).Before we deal with the implications due to the implementation of such a ultra-referential S V in the space-time at quantum level,let us make a brief presentation of the meaning of the Galilean reference frame (referencespace),well-known in Special Relativity. In accordance with that theory,when an observer assumes an infinitenumber of points at rest in relation to himself,he introduces his own reference space S . Thus,for another observer S ′ who is moving with a speed v in relation to S , there should also exist an infinite number of points at rest athis own reference frame. Therefore, for the observer S ′ ,the reference space S is not standing still and it has itspoints moving at a speed − v . For this reason, in accordance with the principle of relativity,there is no privilegedGalilean reference frame at absolute rest,since the reference space of a given observer becomes movement foranother one.The absolute space of pre-einsteinian physics,connected to the ether in the old sense,also constitutes by itselfa reference space. Such a space was assumed as the privileged reference space of the absolute rest. However,asit was also essentially a Galilean reference space like any other, comprised of a set of points at rest,actually itwas also subjected to the notion of movement. The idea of movement could be applied to the “absolute space”when,for instance,we assume an observer on Earth, which is moving with a speed v in relation to such a space.In this case,for an observer at rest on Earth,the points that would constitute the absolute space of referencewould be moving at a speed of − v . Since such an absolute space was connected to the old ether,the Earth-boundobserver should detect a flow of ether − v ,however the Michelson-Morley experiment has not detected such anether.Einstein has denied the existence of the ether associated with a privileged reference frame because it hascontradicted the principle of relativity. Therefore this idea of a Galilean ether is superfluous,as it would alsomerely be a reference space constituted by points at rest,as well as any other. In this respect,there is nothingspecial in such a classical (luminiferous) ether.However,motivated by the provocation from H. Lorentz and Ph. Lenard Lorentz[8],Einstein attempted tointroduce several new conceptions of a new “ether”,which did not contradict the principle of relativity. After1925,he started using the word “ether” less and less frequently,although he still wrote in 1938: “This word‘ether’ has changed its meaning many times,in the development of Science... Its history,by no means finished,iscontinued by Relativity theory[10]... ” .In 1916,after the final formulation of GR,Einstein proposed a completely new concept of ether. Such a new“ether” was a relativistic “ether”,which described space-time as a sui generis material medium,which in noway could constitute a reference space subjected to the relative notion of movement. Basically,the essentialcharacteristics of the new “ether” as interpreted by Einstein can be summarized as follow:- It constitutes a fundamental ultra-referential of Reality,which is identified with the physical space,being arelativistic ether,i.e., it is covariant because the notion of movement cannot be applied to it,which represents akind of absolute background field that is inherent to the metric g µν of the space-time .- It is not composed of points or particles,therefore it cannot be understood as a Galilean reference space forthe hypothetical absolute space. For this reason,it does not contradict the well-known principle of Relativity .- It is not composed of parts,thus its indivisibility reminds the idea of non-locality .- It constitutes a medium which is really incomparable with any ponderable medium constituted of parti-cles,atoms or molecules. Not even the background cosmic radiation of the Universe can represent exactly sucha medium as an absolute reference system (ultra-referential)[19] .- It plays an active role on the physical phenomena[11] [12].
In accordance with Einstein,it is impossible toformulate a complete physical theory without the assumption of an “ether”(a kind of non-local vacuum field),3because a complete physical theory must take into consideration real properties of the space-time.The present work attempts to follow this line of reasoning that Einstein did not finish,providing a new modelwith respect to the fundamental idea of unification,namely the electrodynamics of a charged particle (electron)moving in a gravitational field.As we have interpreted the lowest limit V (formulas (43) and (44)) as unattainable and constant (invari-ant),such a limit should be associated with a privileged non-Galilean reference system,since V must remaininvariant for any frame with v > V . As a consequence of such a covariance of the relativistic “ether” S V ,newspeed transformations will show that it is impossible to cancel the speed of a particle over its own referenceframe,in such a way to always preserve the existence of a magnetic field ~B for such a charged particle. Thuswe should have a speed transformation that will show us that “ v − v ′′ > V for v > V (see section 6), where theconstancy of c remains,i.e.,“ c − c ′′ = c for v = c .Since it is impossible to find with certainty the rest for a given non-Galilean reference system S ′ with a speed v with respect to the ultra-referential S V ,i.e., “ v − v ′′ = 0( > V ) (section 6),consequently it is also impossibleto find by symmetry a speed − v for the relativistic “ether” when an “observer” finds himself at the referencesystem S ′ assumed with v . Hence,due to such an asymmetry,the flow − v of the “ether” S V does not existand therefore,in this sense,it mantains covariant ( V ). This asymmetry breaks that equivalence by exchangeof reference frame S for S ′ through an inverse transformation. Such a breakdown of symmetry by an inversetransformation breaks Lorentz symmetry due to the presence of the background field for S V (section 6).There is no Galilean reference system in such a space-time, where the ultra-referential S V is a non-Galileanreference system and in addition a privileged one (covariant),exactly as is the speed of light c . Thus the newtransformations of speed shall also show that “ v ± V ′′ = v (section 6) and “ V ± V ′′ = V (section 6).Actually,if we make V → c remains. In this classical case (SR),we have reference systemsconstituted by a set of points at rest or essentially by macroscopic objects. Now,it is interesting to notice thatSR contains two postulates which conceptually exclude each other in a certain sense,namely:1) - the equivalence of the inertial reference frames (with v < c ) is essentially due to the fact that we haveGalilean reference frames,where v rel = v − v = 0 , since it is always possible to introduce a set of points at relativerest and,consequently,for this reason,we can exchange v for − v by symmetry through inverse transformations.
2) - the constancy of c ,which is unattainable by massive particles and therefore it could never be related to aset of infinite points at relative rest. In this sense, such “referential”( c ),contrary to the 1st. one,is not Galileanbecause we have “ c − c ′′ = 0 (= c ) and,for this reason,we can never exchange c for − c . However,the covariance of a relativistic “ether” S V places the photon ( c ) in a certain condition of equalitywith the motion of other particles ( v < c ),just in the sense that we have completely eliminated the classicalidea of rest for reference space (Galilean reference frame) in such a space-time. Since we cannot think abouta reference system constituted by a set of infinite points at rest in such a space-time,we should define a non-Galilean reference system essentially as a set of all those particles which have the same state of motion ( v ) inrelation to the ultra-referential- S V of the relativistic “ether”. Thus SSR should contain 3 postulates as follow:1) - the constancy of the speed of light ( c ) .2) - the non-equivalence (asymmetry) of the non-Galilean reference frames, i.e.,we cannot exchange v for − v bythe inverse transformations, since “ v − v ′′ > V ( ∝ √ G/ ~ ), which breaks Lorentz symmetry due to the universalbackground field associated with S V .3) - the covariance of a relativistic “ether” (ultra-referential S V ) associated with the unattainable minimumlimit of speed V . The three postulates described above are compatible among themselves, in the sense that we completelyeliminate any kind of Galilean reference system for the space-time of SSR.Figure 1 illustrates a new conception of reference systems in SSR.Under SR,there is no ultra-referential S V ,i.e., V →
0. Hence,the starting point for observing S ′ is the referenceframe S ,at which the classic observer thinks he is at rest (Galilean reference frame S ).Under SSR,the starting point for obtaining the actual motion of all particles of S ′ is the ultra-referential S V (see Fig.1). However,due to the non-locality of S V ,that is unattainable by the particles,the existence of anobserver (local level) at it ( S V ) becomes inconceivable. Hence,let us think about a non-Galilean frame S for4 FIG. 1: S V is the covariant ultra-referential of background field (relativistic “ether”). S represents the non-Galileanreference frame for a massive particle with speed v in relation to S V ,where V < v < c . S ′ represents the non-Galileanreference frame for a massive particle with speed v ′ in relation to S V . In this instance,we consider V < v ≤ v ′ ≤ c .FIG. 2: As S is fixed (universal),being v ( >> V ) given with respect to S V ,we should also consider the new interval V ( S V ) < v ( S ′ ) ≤ v ( S ) . This non-classical regime for v introduces a new symmetry in the space-time,leading toSSR. Thus we expect that new and interesting results take place. In such an interval ( V < v ≤ v ),we will see that < Ψ( v ) ≤ (see equations (60),(72) and Fig.7). a certain intermediate speed mode ( V << v << c ) in order to represent the starting point at local level for“observing” the motion of S ′ across the ultra-referential S V . Such a frame S (for v with respect to S V ) playsthe similar role of a “rest”,in the sense that we restore all the newtonian parameters of the particles,such asthe proper time interval ∆ τ ,i.e.,∆ t ( v = v )=∆ τ ,the mass m , i.e., m ( v = v ) = m ,among others. Therefore S plays a role that is similar to the frame S under SR,where ∆ t ( v = 0) = ∆ τ , m ( v = 0) = m , etc. However,herein SSR,the classical relative rest ( v = 0) of S should be replaced by a universal “quantum rest” v ( = 0) ofthe non-Galilean frame S . We will show that v is also a universal constant. In short, S is a universal non-Galilean reference frame with speed v given with respect to S V . At S ,the well-known proper mass ( m ) orproper energy E = m c of a particle is restored . This means that,at such a frame S , we have the properenergy E = E = m c = m c Ψ( v ),such that Ψ( v ) = 1,as well as γ ( v = 0) = 1 for the particular case ofLorentz transformations,where V →
0. So we will look for the general function Ψ( v ) of SSR,where we have E = m c Ψ( v ). In the limit V →
0, indeed we expect that the function Ψ( v ) → γ ( v ) = (1 − v /c ) − / (seeFig.7).By making the non-Galilean reference frame S (Fig.1) coincide with S ,we get Figure 2.In general,we should have the total interval V < v < c for S ′ (Fig.2). In short we say that both of the frames S V and S are already fixed or universal,whereas S ′ is a rolling frame to describe the variations of the movingstate of the particle within such a total interval. Since the rolling frame S ′ is not a Galilean one due to theimpossibility to find a set of points at rest on it,we cannot place the particle exactly on the origin O ′ ,sincethere would be no exact location on x ′ = 0 ( O ′ ) (an uncertainty ∆ x ′ = O ′ C : see Figure 3). Actually we wantto show that ∆ x ′ (Fig.3) is a function which should depend on speed v of S ′ with respect to S V ,namely,forexample,if S ′ → S V ( v → V ),then we should have ∆ x ′ → ∞ (infinite uncertainty),which is due to the non-local5 FIG. 3:
We have four imaginary clocks associated with non-Galilean reference frames S , S ′ , the ultra-referential S V (for V ) and also S c (for c ). We observe a new result,namely the proper time (interval ∆ τ ) elapses much faster closer toinfinite ( ∆ τ → ∞ ) when one approximates to S V . On the other hand,it tends to stop ( ∆ τ → ) when v → c ,providingthe strong symmetry for SSR. Here we are fixing ∆ t (∆( t )) and letting ∆ τ vary. aspect of the ultra-referential S V . On the other hand,if S ′ → S c ( v → c ),then we should have ∆ x ′ → O ′ ). Thus let us search for a function ∆ x ′ = ∆ x ′ ( v ) = ∆ x ′ v ,starting from Figure 3.At the frame S ′ in Fig.3,let us consider that a photon is emitted from a point A at y ′ ,in the direction AO ′ .This occurs only if S ′ were Galilean (at rest over itself). However,since the electron cannot be thought of asa point at rest on its proper non-Galilean frame S ′ ,and cannot be located exactly on O ′ ,its non-location O ′ C (= ∆ x ′ v )(see Fig.3) causes the photon to deviate from the direction AO ′ to AC . Hence,instead of just thesegment AO ′ ,a rectangular triangle AO ′ C is formed at the proper non-Galilean reference frame S ′ ,where it isnot possible to find a set of points at rest.From the non-Galilean frame S (“quantum rest”),which plays the role of S ,from where one “observes” themotion of S ′ across S V ,one can see the trajectory AB for the photon. Thus the rectangular triangle AO ′ B isformed. Since the vertical leg AO ′ is common to the triangles AO ′ C (for S ′ ) and AO ′ B (for S ≡ S ),we have( AO ′ ) = ( AC ) − ( O ′ C ) = ( AB ) − ( O ′ B ) , (45)or else ( c ∆ τ ) − (∆ x ′ v ) = ( c ∆ t ) − ( v ∆ t ) . (46)If ∆ x ′ ( v ) = ∆ x ′ v = 0 ( V → ⇒ S V ≡ S ( ≡ S )), we go back to the classical case (SR),where we consider forinstance a train wagon ( S ′ ),which is moving in relation to a fixed rail ( S ). At a point A on the ceiling of thewagon,there is a laser that releases photons toward y ′ ,reaching the point O ′ assumed in the origin of S ′ (on thefloor of the train wagon). For Galilean- S ′ ,the trajectory of the photon is AO ′ . For Galilean- S ,its trajectory is AB .6Since ∆ x ′ v is a function of v ,assumed as a kind of “displacement” (uncertainty) given on the proper non-Galilean reference frame S ′ ,we may write it in the following way:(∆ x ′ v ) = f ( v )∆ τ, (47)where f ( v ) is a function of v , which also presents dimension of velocity,i.e.,it is a certain velocity in SSR,whichcould be thought of as a kind of internal motion v int of the particle,being responsible for the increasing ordilation (stretch) of an internal dimension of the particle on its own non-Galilean frame S ′ . Such an internaldilation is given by the non-classical “displacement” ∆ x ′ v = O ′ C (see Fig.3). This leads us to think that thereis an uncertainty of position for the particle,as we will see later. Hence,substituting (47) into (46),we obtain∆ τ [1 − ( f ( v )) c ] = ∆ t (1 − v c ) , (48)where we use the notation ∆ t or ∆ t ( S ≡ S ),and where we have f ( v ) = v int to be duly interpreted.Thus,since we have v ≤ c ,we should have f ( v ) ≤ c in order to avoid an imaginary number in the 1st. memberof (48).The domain of f ( v ) is such that V ≤ v ≤ c . Thus,let us also think that its image is V ≤ f ( v ) ≤ c ,since f ( v )has dimension of velocity and also represents a speed v int (internal motion), which also must be limited for theextremities V and c .Let us make [ f ( v )] /c = f /c = v int /c = α , whereas we already know that v /c = β . v is thewell-known external motion (spatial velocity). Thus we have the following cases originated from (48),namely:- (i) When v → c ( β → β max = 1),the relativistic correction in its 2nd. member (right-hand side) pre-vails,whereas the correction on the left-hand side becomes practically neglected,i.e.,we should have v int = f ( v ) << c ,where lim v → c f ( v ) = f min = ( v int ) min = V ( α → α min = V /c = ξ ). ξ ∼ = 3 . × − (refer to (41)).-(ii) On the other hand,due to idea of symmetry,if v → V ( β → β min = V /c = ξ ),there is no substantialrelativistic correction on the right-hand side of (48),whereas the correction on the left-hand side becomes nowconsiderable,namely we should have lim v → V f ( v ) = f max = ( v int ) max = c ( α → α max = 1).In short,from (i) and (ii),we observe that,if v → v max = c , then f → f min = ( v int ) min = V ,and if v → v min = V , then f → f max = ( v int ) max = c . So now we perceive that the internal motion v int (= f ( v )) works like areciprocal speed ( v Rec ) in such a symmetrical structure of space-time in SSR. In other words,we notice that the(external or spatial) velocity v increases to c whereas the internal (reciprocal) one ( v int = v Rec ) decreases to V .On the other hand,when v tends to V ( S V ), v int tends to c ,leading to a large internal stretch (uncertainty ∆ x ′ v )due to a non-locality behavior much closer to the ultra-referential S V . Due to this fact,we reason that f ( v ) = v int = v Rec = av , (49)where a is a constant that has dimension of square speed. Such a reciprocal velocity v Rec will be betterunderstood later. It is interesting to know that a similar idea of considering an internal motion for microparticleswas also thought by Natarajan[26].In addition to (48) and (49),we already know that,at the referential S (see Fig.2 and Fig.3),we should havethe condition of equality of the time intervals,namely ∆ t = ∆ τ for v = v ,which,in accordance with (48),occursonly if [ f ( v )] c = v c ⇔ f ( v ) = v (50)By comparing (50) with (49) for the case v = v ,we obtain a = v (51)7Substituting (51) into (49),we obtain f ( v ) = v int = v Rec = v o v (52)According to (52) and also considering (i) and (ii),indeed we observe respectively that f ( c ) = V = v /c ( V isthe reciprocal velocity of c ) and f ( V ) = c = v /V ( c is the reciprocal velocity of V ),from where we immediatelyobtain v = √ cV (53)As we already know the value of V (refer to (43)) and c ,we obtain the velocity of “quantum rest” v ∼ =5 . × − m/s ,which is also universal just because it depends on the universal constants c and V . However,wemust stress that only c and V remain invariant under speed transformations in such a space-time of SSR (section6).Finally,by substituting (53) into (52) and after into (48),we finally obtain∆ τ r − V v = ∆ t r − v c , (54)where α = f ( v ) /c = v int /c = V /v and β = v/c inside (54). In fact,if v = v = √ cV in (54),so we have ∆ τ = ∆ t .Therefore we conclude that S ( v ) is the intermediate (non-Galilean) reference frame such that,if:a) v >> v ( v → c ) ⇒ ∆ t >> ∆ τ : It is the well-known time dilation .b) v << v ( v → V ) ⇒ ∆ t << ∆ τ : Let us call this new result contraction of time . This shows us the noveltythat the proper time interval (∆ τ ) is variable,so that it may expand in relation to the improper one (∆ t in S ).∆ τ is an intrinsic variable for the particle on its proper non-Galilean frame S ′ . Such an effect of dilation of ∆ τ with respect to ∆ t would become more evident only for v → V ( S V ),since we would have ∆ τ → ∞ in such alimit S V . In other words,this means that the proper time ( S ′ ) would elapse much faster than the improper oneat S .In SSR,it is interesting to notice that we restore the newtonian regime when V << v << c ,which representsan intermediate regime of speeds,where we can make the approximation ∆ τ ≈ ∆ t .Substituting (52) into (47) and also considering (53),we obtain O ′ C = ∆ x ′ v = v int ∆ τ = v Rec ∆ τ = v v ∆ τ = Vc v ∆ τ = αc ∆ τ (55)Actually we can verify that,if V → v → O ′ C = ∆ x ′ v = 0, restoring the classical case(SR),where there is no such an internal motion. And also,if v >> v ,this implies ∆ x ′ v ≈ v → c ,we have ∆ x ′ ( c ) = V ∆ τ and,if v → V ( S V ), we have∆ x ′ ( V ) = c ∆ τ . This means that,when the particle momentum with respect to S V increases ( v → c ),it becomesmuch more localized upon itself over O ′ ( V ∆ τ →
0) and,when its momentum decreases ( v → V ),it becomes muchless localized over O ′ ,because it gets much closer to the non-local ultra-referential S V ,where ∆ x ′ v = ∆ x ′ max =0 ′ C max = c ∆ τ → ∞ . Thus,now we begin to perceive that the velocity v (momentum) and the position (non-localization ∆ x ′ v = v Rec ∆ τ ) operate like mutually reciprocal quantities in such a space-time of SSR,since thenon-localization is ∆ x ′ v ∝ v Rec ∝ v − (see (49) or (52)). This really provides a basis for the fundamentalcomprehension of the quantum uncertainties in a context of objective reality of the space-time,according toEinstein’s vision[18].It is very interesting to observe that we may write ∆ x ′ v in the following way:∆ x ′ v = ( V ∆ τ )( c ∆ τ ) v ∆ τ ≡ ∆ x ′ ∆ x ′ ∆ x ′ , (56)8where V ∆ τ = ∆ x ′ , c ∆ τ = ∆ x ′ and v ∆ τ = ∆ x ′ . We also know that c ∆ t ≡ c ∆ t = ∆ x and v ∆ t ≡ v ∆ t = ∆ x for the frame S ( ≡ S ). So we write (46) in the following way:∆ x ′ − ∆ x ′ ∆ x ′ ∆ x ′ = ∆ x − ∆ x , (57)where ∆ x ′ corresponds to a fifth dimension of temporal nature. Therefore we may already conclude that the newgeometry of space-time has three spatial dimensions ( x , x , x ) plus two temporal dimensions ( c ∆ t , V ∆ τ ),being V ∆ τ normally hidden. However,we will perceive elsewhere that we can also describe such a space-time in acompact form as effectively a 4-dimensional structure,because V ∆ τ and c ∆ t represents two complementaryaspects of the same temporal nature, and also mainly because V ∆ τ appears as an implicit variable for thespace-time interval c ∆ τ (see (61), (62) or (63)).If ∆ x ′ → V → S = ∆ x − ∆ x = ∆ S ′ = ∆ x ′ .As we have ∆ x ′ v > S ′ = ∆ x ′ > ∆ S = ∆ x − ∆ x . Hence,we may write (57),as follows:∆ S ′ = ∆ S + ∆ x ′ v , (58)where ∆ S ′ = AC , ∆ x ′ v = O ′ C and ∆ S = AO ′ (refer to Fig.3).For v >> V or also v → c ,we have ∆ S ′ ≈ ∆ S , hence θ ≈ π (see Fig.3). In macroscopic world (or very largemasses),we have ∆ x ′ v = ∆ x ′ = 0 (hidden dimension),hence θ = π ⇒ ∆ S ′ = ∆ S . The quantum uncertaintiescan be neglected in such a particular regime (Galilean reference frames of SR).For v → V ,we would have ∆ S ′ >> ∆ S , where ∆ S ′ ≈ c ∆ τ ,with ∆ τ → ∞ and θ → π . In this new relativisticlimit (relativistic “ether” S V ),due to the maximum non-localization ∆ x ′ v → ∞ ,the 4-dimensional interval ∆ S ′ loses completely its equivalence in respect to ∆ S ,because 5th dimension ( V ∆ τ ) increases drastically much closerto such a limit,i.e.,∆ x ′ → ∞ . So it ceases to be hidden for such very special case.Equation (58) or (57) shows us a break of the 4-interval invariance (∆ S ′ = ∆ S ),which becomes noticeableonly at the limit v → V ( S V ). However,a new invariance is restored when we implement a 5th.dimension ( x ′ )to be intrinsic to the particle (frame S ′ ) through the definition of a new (effective) general interval,where theinterval V ∆ τ appears as an implicit variable,namely:∆ S = p ∆ S ′ − ∆ x ′ v = ∆ x ′ s − ∆ x ∆ x ′ = c ∆ τ r − V v , (59)such that ∆ S ≡ ∆ S (see (58)).We have omitted the index ′ for ∆ x ,as such an interval is given only at the non-Galilean proper referenceframe ( S ′ ),that is intrinsic to the particle. Actually such a 5-interval or simply an effective 4-interval c ∆ τ ∗ = c ∆ τ √ − α guarantees the existence of a certain effective internal dimension for the electron. However,from apractical viewpoint,for experiments of higher energies,the electron approximates more and more to a punctualparticle,since ∆ x becomes hidden. So in order to detect its internal dimension, it should be at very lowenergies,namely very close to S V .Comparing (59) with the left side of equation (54),we may alternatively write∆ t = Ψ∆ τ = ∆ S c q − v c = ∆ τ q − V v q − v c , (60)where ∆ S is the invariant effective interval given at the frame S ′ . We have Ψ = √ − α √ − β = q − V v q − v c and,alternatively,we can also write Ψ = √ − β int √ − α int = r − v intc r − V v int ,since α = V /v = β int = v int /c and β = v/c = α int =9 V /v int ,from where we get v int = v Rec = cV /v = v /v (see (52)). Only for v = v ,we obtain v int = v = v .Although we cannot obtain directly v int by any experiment (just the uncertainty ∆ x is obtained),we could alsouse Ψ in its alternative form Ψ( v int ). However,let us use Ψ( v ).For v >> V ,we get ∆ t ≈ γ ∆ τ ,where Ψ ≈ γ = (1 − β ) − / .Substituting (55) into (46) and using the notation ∆ t ≡ ∆ t ,we obtain c ∆ τ = 1(1 − V v ) [ c ∆ t − v ∆ t ] , (61)from where,we also obtain the equation (54).By placing (61) in a differential form and manipulating it,we will obtain c (1 − V v ) dτ dt + v = c (62)We may write (62) in the following alternative way: dS dt + v = c , (63)where dS = c q − V v dτ .Equation (62) shows us that the speed related to the marching of time (“temporal-speed”),which is v t = c q − V v dτdt , and the spatial speed,which is v in relation to the background field for S V form respectively thevertical and horizontal legs of a rectangular triangle.We have c = ( v t + v ) / ,which represents the space-temporal velocity of any particle (hypothenuse ofthe triangle= c ). The novelty here is that such a space-time implements the ultra-referential S V . Such animplementation arises at the vertical leg v t of such a rectangular triangle.We should consider 3 importants cases as follow:a) If v ≈ c ,then v t ≈ >> t >> ∆ τ ( dilation of time ).b) If v = v = √ cV ,then v t = p c − v ,where Ψ = Ψ = Ψ( v ) = 1, since ∆ t = ∆ τ ( “quantum rest” S ).c) If v ≈ V ,then v t ≈ √ c − V = c p − ξ ,where Ψ << t << ∆ τ ( contraction of time ).In SR,when v = 0,we have v t = v tmax = c . However,in accordance with SSR, due to the existence of aminimum limit V of spatial speed for the horizontal leg of the triangle,we see that the maximum temporal-speed is v tmax < c . This means that we have v tmax = c p − ξ . Such a result introduces a strong symmetryin such a space-time of SSR,in the sense that both of spatial and temporal speeds c become unattainable for allmassive particles.The speed v = c is represented by the photon (massless particle), whereas v = V is definitely inaccessible forany particle. Actually we have V < v ≤ c ,but,in this sense,we have a certain asymmetry,as there is no particleat the ultra-referential S V where there should be a kind of sui generis vacuum energy density ( ρ (2) ) to bestudied elsewhere.In order to produce a geometric representation for that problem ( V < v ≤ c ), let us assume the world line ofa particle limited by the surfaces of two cones,as shown in Figure 4.A spatial speed v = v P in the representation of light cone shown in Figure 4 (horizontal leg of the rectangulartriangle) is associated with a temporal speed v t = v tP = p c − v P (vertical leg of the same triangle) given inanother cone representation,which could be denominated temporal cone (Figure 5).We must observe that a particle moving just at one spatial dimension always goes only to left or to right,sincethe unattainable non-null minimum limit of speed V forbids it to stop its spatial velocity ( v = 0) in order toreturn at this same spatial dimension. On the other hand,in a complementary way to V , the limit c is temporalin the sense that it forbids to stop the time (temporal velocity v t = 0) and also to come back to the past.However,if we consider more than one spatial dimension,at least 2 spatial dimensions( xy ), the particle can nowreturn by moving at the additional dimension(s). So SSR provides the reason why we must have more than one0 FIG. 4:
The external and internal conical surfaces represent respectively c and V ,where V is represented by the dashed line,that is a definitely prohibited boundary. For a point P in the interior of the two conical surfaces,there is a correspondinginternal conical surface,such that V < v P ≤ c . (1) spatial dimension ( d >
1) for representing movement in reality,although we could consider 1 d just as a goodapproximation for some cases in classical space-time of SR (classical objects). Such a minimum limit V hasdeep implications for understanding the irreversible aspect of time connected to movement,since we can nowdistinguish the motions to left and to right in the time. Such an asymmetry generated by SSR really deservesa deeper treatment elsewhere.Based on the relation (61) or also by substituting (55) into (46),we obtain c ∆ t − v ∆ t = c ∆ τ − v v ∆ τ (64)In (64),when we transpose the 2nd.term from the left side to the right side and divide the equation by ∆ t ,weobtain (62) in differential form. Now,it is important to observe that,upon transposing the 2nd.term from theright side to the left one and dividing the equation by ∆ τ ,we obtain the following equation in the differentialform,namely: c (1 − v c ) dt dτ + v v = c (65)From (59) and (54),we obtain dS = cdτ √ − α = cdt p − β . Hence we can write (65) in the followingalternative way: dS dτ + v v = c (66)We see that equation (65) or (66) reveals a complementary way of viewing equation (62) or (63). This leadsus to that idea of reciprocal space for conjugate quantities. Thus let us write (65) or (66) in the following way: v tRec + v Rec = c , (67)1 FIG. 5:
Comparing this Figure 5 with Figure 4,we notice that the dashed line on the internal cone of Figure 4 ( v = V )corresponds to the dashed line on the surface of the external cone of this Figure 5,where v t = √ c − V ,which representsa definitely forbidden boundary in this cone representation of temporal speed v t . On the other hand, v = c (photon)is represented by the solid line of Figure 4,which corresponds to the temporal speed v t = 0 in this Figure 5,coincidingwith the vertical axis t . In short,we always have v + v t = c ,being v for spatial (light) cone (Figure 4) and v t fortemporal cone represented in this Figure 5,such that an internal point P x is related to a temporal velocity v tP ,where photon ) ≤ v tP (= p c − v P ) < √ c − V . The horizontal axis is S = c p − V /v τ ,so that v t = dS /dt = c p − V /v dτ /dt = √ c − v (see equation (54)). where v tRec = ( v t ) int = dS /dτ = c q − v c dtdτ ,which represents an internal (reciprocal) temporal velocity . Theinternal (reciprocal) spatial velocity is v int = v Rec = f ( v ) = v v . Therefore we can also represent a rectangulartriangle,but now displayed in a reciprocal space. For example, if we assume v → c (equation (62)),we obtain v Rec = lim v → c f ( v ) → v c = V (equation (65)). In this same case,we have v t → v tRec = dS dτ → √ c − V (equation (65) or (66)). On the other hand,if v → V (eq.(62)),we have v Rec → v V = c (eq.(65)),where v t → √ c − V (eq.(62)) and ( v t ) int = v tRec → two spatial representations : a ) v = dxdt , in equation (62) ,represented in F ig. b ) v Rec = dx ′ v dτ = v v , in equation (65) . (68) two temporal representations : a ) v t = dS dt = c q − V v dτdt = c q − v c ,in equation (62) , represented in F ig. b ) v tRec = dS dτ = c q − v c dtdτ = c q − V v ,in equation (65) . (69)The chart given in Figure 6 shows us those four representations.2 FIG. 6:
The spatial representations in a (also shown in Figure 4) and b are related respectively to velocity v (momentum)and position (non-localization ∆ x ′ v = f ( v )∆ τ = v int ∆ τ = v Rec ∆ τ = ( v /v )∆ τ ),which represent conjugate (reciprocal)quantities in space. On the other hand,the temporal representations in a (also shown in Figure 5) and b are relatedrespectively to time ( ∝ v t ) and energy ( ∝ v tRec = ( v t ) int ∝ v − t ),which represent conjugate (reciprocal) quantities in thetime. Hence we can perceive that such four cone representations of SSR provide a basis for the fundamental understandingof the two uncertainty relations. Now,by considering (54),(60),(69) and also looking at a and b in Fig.6,we may observe thatΨ − = ∆ τ ∆ t = q − v c q − V v = v t c q − V v = v t v tRec ∝ ( time ) (70)and Ψ = ∆ t ∆ τ = q − V v q − v c = v tRec c q − v c = v tRec v t ∝ E ( Energy ∝ ( time ) − ) (71)From (71),since we have energy E ∝ Ψ,we write E = E Ψ,where E is a constant of proportionality. Hence,ifwe consider E = m c ,we obtain3 FIG. 7: v represents the velocity of “quantum rest” in SSR, from where we get E = E = m c ,being Ψ = Ψ( v ) = 1 . E = m c q − V v q − v c , (72)where E is the total energy of the particle in relation to the absolute inertial frame of universal backgroundfield S V . Such a result shall be explored in a coming article about the dynamics of the particles in SSR. In (71)and (72),we observe that,if v → c ⇒ E → ∞ and ∆ τ → t fixed. If v → V ⇒ E → τ → ∞ , alsofor ∆ t fixed. If v = v = √ cV ⇒ E = E = m c (energy of “quantum “rest””). Figure 7 shows us the graphfor the energy E in (72). B. The Uncertainty Principle
The particle actual momentum (in relation to S V ) is P = Ψ m v , whose conjugate value is ∆ x ′ v = v v ∆ τ = v v ∆ t Ψ − ,where ∆ τ = Ψ − ∆ t (refer to (54)). From S V it would be possible to know exactly the actualmomentum P and the total energy E of the particle,however,since S V represents an ultra-referential which isunattainable (non-local) and also inaccessible for us,so one becomes impossible to measure such quantities withaccuracy. And for this reason,as a classical observer (local and macroscopic) is always at rest ( v = 0) in hisproper reference frame S ,he measures and interprets E without accuracy because his frame is Galilean, beingrelated essentially to macroscopic systems (a set of points at rest) ,whereas on the other hand,non-Galileanreference frames for representing subatomic world in SSR are really always moving for any transformation insuch a space-time and therefore cannot be related to a set of points at rest . Due to this conceptual discrepancybetween the nature of non-Galilean reference frames in SSR (no rest) and the nature of Galilean reference framesin SR for classical observes (with rest) ,the total energy E in SSR (eq.(72)) behaves as an uncertainty ∆ E forsuch classical observers at rest, i.e., E (for S V ) ≡ ∆ E (for any Galilean- S at rest). Similarly P also behaves as anuncertainty ∆ p ( P ( S V ) ≡ ∆ p (Galilean-S)) and, in addition,the non-localization ∆ x ′ v as simply an uncertainty∆ x . Hence we have ∆ x ′ v P ≡ (∆ x ∆ p ) classical observer S = v v ∆ t Ψ − Ψ m v = ( m v )( v ∆ t ) (73)and4∆ τ E ≡ (∆ τ ∆ E ) classical observer S = ∆ t Ψ − Ψ m c = ( m c )( c ∆ t ) , (74)where we consider again ∆ t fixed and let ∆ τ vary for each case. In obtaining (73) and (74),we also haveconsidered the relations ∆ x ′ v = v v ∆ τ , ∆ τ = ∆ t Ψ − , P = Ψ m v and E = Ψ m c .Since we know the actual momentum P of the particle moving across the relativistic “ether”- S V ,its de-Brogliewavelength is λ = hP = h Ψ m v = hm v q − v c q − V v (75)If v → c ⇒ λ → spatial contraction or temporal dilation ),and if v → V ⇒ λ → ∞ ( spatial dilation or temporal contraction ). In such a space-time of SSR,actually we should interpret the spatial scales as wavelengths λ given at the background frame S V ,in accordance with (75).The relationship (75) shows us a strong symmetry that enables us to understand the space as an elasticstructure,which is capable of contracting ( λ → v → c ) and also expanding ( λ → ∞ for v → V ( S V )).The wavelength λ in (75) may be thought of as being related to the non-localization ∆ x ′ v ,namely λ ∝ ∆ x ′ v .Such a proportionality is verified by comparing (55) with (75) and also by considering ∆ τ = Ψ − ∆ t . Hence wehave λ ∝ ∆ x ′ v = v v ∆ τ = v v ∆ t q − v c q − V v , (76)where λ ∝ ∆ x ′ v ( ≡ ∆ x )= v int ∆ τ = v Rec ∆ τ ∝ ( v Ψ) − . We also make ∆ t fixed and let ∆ τ vary,such that 0 < ∆ τ < ∞ . Now,we can perceive that the quantum nature of the wave is derived from the internal motion v int = v Rec of the proper particle,since its wavelength for S V is λ ∝ v Rec . This leads to a fundamental explanation forthe wave-particle duality in such a space-time of SSR. Natarajan[26] also used a kind of internal motion v in [26]of the microparticle to explain in alternative way such a dual aspect of the matter. In approximation for SR,wehave V → v → v Rec = 0 ⇒ λ = 0. Indeed this means that the wave nature of the matteris not included in SR.Now let us observe that,if we make v = v in (76) and (75),and then compare these two results,we obtain v ∆ t ≡ v T ∼ λ = hm v ∼ m, (77)where we fix ∆ t ≡ T ∼ hm v , m being the electron mass. T represents the period of the wave with length λ ,such that T ∼ s . λ is a special standard intermediate scale for the frame S . Since λ ∼ m , indeed itrepresents a typical scale of a classical observer (human scale).Finally,by substituting (77) into (73),we obtain∆ x ′ v P ≡ ∆ x ∆ p ∼ m v λ = h (78)Now,it is easy to conclude that ∆ τ E ≡ ∆ τ ∆ E ∼ m cλ c = h, (79)where c ∆ t ≡ cT c ∼ λ c = hm c (refer to (74)). λ c ∼ − m (Compton wavelength for the photon,whoseenergy mc ( ∝ e s b s ) must be equivalent to the electron energy m c ( ∝ e s b s ),that is, m ≡ m . In thisinstance,∆ t ≡ T c ∼ hm c ).5It is interesting to notice that λ c λ = v c , where λ ∼ m . It is also very curious to observe that λ c = v c λ = Vv λ ⇐⇒ v = √ cV ,which in fact represents a special intermediate point (a kind of aurum point ), namelyit represents a geometric average between c and V ,where the human scale ( λ ∼ m ) is really found as anintermediate scale. Thus we may write λ c = β λ = α λ ,such that λ c = ξλ ∼ − m ,where we have β = α = v /c = V /v = V /c = ξ ∼ − .As we already know the total energy E = m c Ψ and the momentum ~P = m ~v Ψ at S V , we can demonstratethat E = c ~P + m c (1 − V /v ),where Ψ is shown in (71). VI. TRANSFORMATIONS OF SPACE-TIME AND VELOCITY IN THE PRESENCE OF THEULTRA-REFERENTIAL S V Let us assume the reference frame S ′ with a speed v in relation to the ultra-referential S V . To simplify,considerthe motion only at one spatial dimension,namely (1 + 1) D -space-time with background field S V . So we writethe following transformations: dx ′ = Ψ( dX − β ∗ cdt ) = Ψ( dX − vdt + V dt ) , (80)where β ∗ = βǫ = β (1 − α ),being β = v/c and α = V /v ,so that β ∗ → v → V or α → dt ′ = Ψ( dt − β ∗ dXc ) = Ψ( dt − vdXc + V dXc ) , (81)being ~v = v x x . We have Ψ = √ − α √ − β . If we make V → α → S V is eliminated and simply replaced by the Galilean frame S at rest for the observer.The transformations shown in (80) and (81) are the direct transformations from S V [ X µ = ( X, ict )] to S ′ [ x ′ ν = ( x ′ , ict ′ )],where we have x ′ ν = Ω νµ X µ ( x ′ = Ω X ), so that we obtain the following matrix of transformation:Ω = (cid:18) Ψ iβ (1 − α )Ψ − iβ (1 − α )Ψ Ψ (cid:19) , (82)such that Ω → L (Lorentz matrix of rotation) for α → → γ ). Let us assume the following more general transformations: x ′ = θγ ( X − ǫ vt ) and t ′ = θγ ( t − ǫ vXc ), where θ , ǫ and ǫ arefactors (functions) to be determined. We hope all these factors depend on α ,such that,for α → V → θ = 1, ǫ = 1 and ǫ = 1). By using those transformations to perform [ c t ′ − x ′ ],we find the identity:[ c t ′ − x ′ ] = θ γ [ c t − ǫ vtX + 2 ǫ vtX − ǫ v t + ǫ v X c − X ]. Since the metric tensor is diagonal,the crossed terms mustvanish and so we assure that ǫ = ǫ = ǫ . Due to this fact,the crossed terms (2 ǫvtX ) are cancelled between themselves and finallywe obtain [ c t ′ − x ′ ] = θ γ (1 − ǫ v c )[ c t − X ]. For α → ǫ = 1 and θ = 1),we reinstate [ c t ′ − x ′ ] = [ c t − x ] of SR.Now we write the following transformations: x ′ = θγ ( X − ǫvt ) ≡ θγ ( X − vt + δ ) and t ′ = θγ ( t − ǫvXc ) ≡ θγ ( t − vXc + ∆),wherewe assume δ = δ ( V ) and ∆ = ∆( V ),such that δ = ∆ = 0 for V → ǫ = 1. So from such transformationswe extract: − vt + δ ( V ) ≡ − ǫvt and − vXc + ∆( V ) ≡ − ǫvXc ,from where we obtain ǫ = (1 − δ ( V ) vt ) = (1 − c ∆( V ) vX ). As ǫ is adimensionaless factor, we immediately conclude that δ ( V ) = V t and ∆( V ) = V Xc ,such that we find ǫ = (1 − Vv ) = (1 − α ). Onthe other hand,we can determine θ as follows: θ is a function of α ( θ ( α )),such that θ = 1 for α = 0,which also leads to ǫ = 1 inorder to recover Lorentz transformations. So,as ǫ depends on α ,we conclude that θ can also be expressed in terms of ǫ ,namely θ = θ ( ǫ ) = θ [(1 − α )],where ǫ = (1 − α ). Therefore we can write θ = θ [(1 − α )] = [ f ( α )(1 − α )] k ,where the exponent k >
0. Thefunction f ( α ) and k will be estimated by satisfying the following conditions: i) as θ = 1 for α = 0 ( V = 0),this implies f (0) = 1.ii) the function θγ = [ f ( α )(1 − α )] k (1 − β ) = [ f ( α )(1 − α )] k [(1+ β )(1 − β )] should have a symmetry behavior,that is to say it goes to zero closer to V ( α →
1) in the same way it goes to infinite closer to c ( β → θγ ,which dependson α should have the same shape of its denumerator,which depends on β . Due to such conditions,we naturally conclude that k = 1 / f ( α ) = (1 + α ),so that θγ = [(1+ α )(1 − α )] [(1+ β )(1 − β )] = (1 − α ) (1 − β ) = √ − V /v √ − v /c = Ψ,where θ = √ − α = p − V /v . det Ω = (1 − α )(1 − β ) [1 − β (1 − α ) ],where 0 < det Ω <
1. Since V ( S V ) is unattainable ( v > V ),thisassures that α = V /v < det Ω = 0 ( > det Ω = ±
1) and so it does not represent a rotation matrix ( det Ω = 1) in such aspace-time due to the presence of the privileged frame of background field S V that breaks the invariance ofthe norm of 4-vector. Such a break occurs strongly closer to S V because the particle experiments an enormousdislocation (uncertainty) from the origin O ′ of the frame S ′ (see Fig.3). This leads to the strong inequality∆ S ′ >> ∆ S ,where ∆ x ′ v → ∞ for v → V (see (55),(56),(57) and (58)). Actually such an effect ( det Ω ≈ α ≈
1) emerges from such a new relativistic physics for treating much lower energies at infrared regime (verylarge wavelengths),where a new implicit dimension (∆ x ) ceases to be hidden and then stretches drasticallyto the infinite closer to S V (see (56)). In the limit S V ,the “particle” would loose its identity,by dissolvingcompletely in the background field (∆ x ′ v = ∞ ). So the matrix Ω would become singular ( det Ω = 0), however,assuch a limit V is unattainable,this really assures the existence of an inverse matrix for Ω.We notice that det Ω is a function of the speed v with respect to S V . In the approximation for v >> V ( α ≈ det Ω ≈ x ′ v ≈ S ′ ≈ ∆ S . Alternatively,if we make V → α → det Ω = 1 (∆ S ′ = ∆ S ,∆ x ′ v = 0).The inverse transformations (from S ′ to S V ) are dX = Ψ ′ ( dx ′ + β ∗ cdt ′ ) = Ψ ′ ( dx ′ + vdt ′ − V dt ′ ) , (83) dt = Ψ ′ ( dt ′ + β ∗ dx ′ c ) = Ψ ′ ( dt ′ + vdx ′ c − V dx ′ c ) . (84)In matrix form,we have the inverse transformation X µ = Ω µν x ′ ν ( X = Ω − x ′ ),so that the inverse matrix isΩ − = (cid:18) Ψ ′ − iβ (1 − α )Ψ ′ iβ (1 − α )Ψ ′ Ψ ′ (cid:19) , (85)where we can show that Ψ ′ =Ψ − / [1 − β (1 − α ) ],so that Ω − Ω = I .Indeed we have Ψ ′ = Ψ and therefore Ω − = Ω T . This non-orthogonal aspect of Ω has an important physicalimplication. In order to understand such an implication,let us consider firstly the orthogonal (e.g: rotation)aspect of Lorentz matrix in SR. Under SR,we have α = 0,so that Ψ ′ → γ ′ = γ = (1 − β ) − / . This symmetry( γ ′ = γ , L − = L T ) happens because the Galilean reference frames allow us to exchange the speed v (of S ′ ) for − v (of S ) when we are at rest at S ′ . However,under SSR,since there is no rest at S ′ (non-Galilean frame),wecannot exchange v (of S ′ ) for − v (of S V ) due to that asymmetry (Ψ ′ = Ψ, Ω − = Ω T ). Due to this fact, S V must be covariant,namely V remains invariant for any change of non-Galilean frame. Thus we can notice thatthe paradox of twins,which appears due to that symmetry by exchange of v for − v in SR should be naturallyeliminated in SSR,because only the non-Galilean reference frame S ′ can move with respect to S V that remainscovariant (invariable for any change of reference frame).We have det Ω = Ψ [1 − β (1 − α ) ] ⇒ [( det Ω)Ψ − ] = [1 − β (1 − α ) ]. So we can alternatively writeΨ ′ =Ψ − / [1 − β (1 − α ) ] = Ψ − / [( det Ω)Ψ − ] = Ψ /det Ω. By inserting this result in (85) to replace Ψ ′ ,weobtain the relationship between the inverse matrix and the transposed matrix of Ω,namely Ω − = Ω T /det Ω.Indeed Ω is a non-orthogonal matrix,since we have det Ω = ± v Rel = v ′ − v + V − v ′ vc + v ′ Vc , (86)where we have considered v Rel = v Relative ≡ dx ′ /dt ′ and v ′ ≡ dX/dt . v ′ and v are given with respect to S V ,with v Rel being related between them. Let us consider v ′ > v . If V → v ′ and v are given in relation to a certain Galilean frame S at rest. Since (86)implements the ultra-referential S V ,the speeds v ′ and v are now given with respect to S V , which is covariant(absolute). Such a covariance is verified if we assume that v ′ = v = V in (86). Thus,for this case,we obtain v Rel = “ V − V ′′ = V . Let us also consider the following cases: a) v ′ = c and v ≤ c ⇒ v Rel = c . This just verifies the well-known invariance of c . b) if v ′ > v (= V ) ⇒ v Rel = “ v ′ − V ” = v ′ . For example,if v ′ = 2 V and v = V ⇒ v Rel = “2 V − V ” = 2 V .This means that V really has no influence on the speed of the particles. So V works as if it were an “ absolutezero of movement ”,being invariant. c) if v ′ = v ⇒ v Rel = “ v − v ′′ ( = 0) = V − v c (1 − Vv ) . From ( c ) let us consider two specific cases,namely:- c ) assuming v = V ⇒ v Rel = “ V − V ” = V as mentioned before.- c ) if v = c ⇒ v Rel = c , where we have the interval V ≤ v Rel ≤ c for V ≤ v ≤ c .This last case ( c ) shows us in fact that it is impossible to find the rest for the particle on its own non-Galilean frame S ′ ,where v Rel ( v ) ( ≡ ∆ v ( v )) is an increasing function. However, if we make V → v Rel ≡ ∆ v = 0 and therefore it would be possible to find the rest for S ′ ,which becomes a Galilean referenceframe ( v < c ) of SR.By dividing (83) by (84),we obtain v Rel = v ′ + v − V v ′ vc − v ′ Vc (87)In (87),if v ′ = v = V ⇒ “ V + V ′′ = V . Indeed V is invariant,working like an absolute zero point in SSR. If v ′ = c and v ≤ c ,this implies v Rel = c . For v ′ > V and considering v = V , this leads to v Rel = v ′ . As a specificexample,if v ′ = 2 V and assuming v = V ,we would have v Rel = “2 V + V ′′ = 2 V . And if v ′ = v ⇒ v Rel = “ v + v ” = v − V v c (1 − Vv ) . In newtonian regime ( V << v << c ),we recover v Rel = “ v + v ” = 2 v . In relativistic (einsteinian)regime ( v → c ),we reinstate Lorentz transformation for this case ( v ′ = v ), i.e., v Rel = “ v + v ” = 2 v/ (1 + v /c ).By joining both transformations (86) and (87) into just one,we write the following compact form: v Rel = v ′ ∓ ǫv ∓ v ′ ǫvc = v ′ ∓ v (1 − α )1 ∓ v ′ v (1 − α ) c = v ′ ∓ v ± V ∓ v ′ vc ± v ′ Vc , (88)being α = V /v and ǫ = (1 − α ). For α = 0 ( V = 0) or ǫ = 1,we recover Lorentz speed transformations.In a more realistic case for motion of the electron in SSR,due to the non-zero minimum limit of speed V forall directions in the space,actually we should also consider the existence of non-null transverse components v y and v z ,such that ~v T = v y j + v z k . So,if we also assume that such a transverse motion in 2d ( yz ) oscillates inthe time ( ~v T ( t ) = v y ( t ) j + v z ( t ) k ) around x ,where the particle has a constant longitudinal motion v = v x ,weobtain an oscillatory (jittery) motion for the electron. This so-called zitterbewegung (zbw) of the electronwas introduced by Schroedinger[27] who proposed the electron spin to be a consequence of a local circulatorymotion,constituting zbw and resulting from the interference between positive and negative energy solutions ofthe Dirac equation. Such an issue turned out to be of renewed interest[28] [29]. The present work providesnaturally a more fundamental vision for zbw ,whose origin is connected to the vacuum energy from the ultra-referential S V ,where now gravity also plays an essential role ( V ∝ √ G ). We intend to go deeper into such asubject about more general transformations elsewhere. VII. COVARIANCE OF THE MAXWELL WAVE EQUATION IN PRESENCE OF THEULTRA-REFERENTIAL S V Let us assume a light ray emitted from the frame S ′ . Its equation of electrical wave in this reference frame is ∂ ~E ( x ′ , t ′ ) ∂x ′ − c ∂ ~E ( x ′ , t ′ ) ∂t ′ = 0 (89)8As it is already known,when we make the exchange by conjugation on the spatial and temporal coordinates,weobtain respectively the following operators: X → ∂/∂t and t → ∂/∂X ; also x ′ → ∂/∂t ′ and t ′ → ∂/∂x ′ . Thusthe transformations (80) and (81) for such differential operators are ∂∂t ′ = Ψ( ∂∂t − βc ∂∂X + ξc ∂∂X ) = Ψ[ ∂∂t − βc (1 − α ) ∂∂X )] , (90) ∂∂x ′ = Ψ( ∂∂X − βc ∂∂t + ξc ∂∂t ) = Ψ[ ∂∂X − βc (1 − α ) ∂∂t )] , (91)where v = βc , V = ξc and ξ = αβ ,being α = V /v .By squaring (90) and (91),inserting into (89) and after performing the calculations, we will finally obtainΨ [1 − β (1 − α ) ] ∂ ~E∂X − c ∂ ~E∂t ! = det Ω ∂ ~E∂X − c ∂ ~E∂t ! = 0 (92)As the ultra-referential S V is definitely inaccessible for any particle,we always have α < v > V ),whichalways implies det Ω = Ψ [1 − β (1 − α ) ] >
0. And as we already have shown in section 6,such a result is inagreement with the fact that we must have det Ω >
0. Therefore this will always assure ∂ ~E∂X − c ∂ ~E∂t = 0 (93)By comparing (93) with (89),we verify the covariance of the equation of the electromagnetic wave propagatingin the relativistic “ether” (background field) S V . VIII. COSMOLOGICAL IMPLICATIONSA. Energy-momentum tensor in the presence of the ultra-referential- S V Let us write the 4-velocity in the presence of S V ,as follows: U µ = q − V v q − v c , v α q − V v c q − v c , (94)where µ = 0 , , , α = 1 , ,
3. If V → T µν = ( p + ǫ ) U µ U ν − pg µν , (95)where now U µ is given in (94). p represents a pressure and ǫ an energy density.From (94) and (95),by calculating the new component T ,we obtain T = ǫ (1 − V v ) + p ( v c − V v )(1 − v c ) (96)If V → T of the Relativity theory.9Now,in order to obtain T in (96) for vacuum limit in the ultra-referential- S V ,we perform lim v → V T = T vacuum = p ( ξ − − ξ ) = − p, (97)where ξ = V /c (see (42)).As we always must have T > p < S V . So we verify that a negative pressure emerges naturally from such new tensorin the limit of S V .We can obtain T µνvacuum by calculating the following limit: T µνvacuum = lim v → V T µν = − pg µν , (98)where we naturally conclude that ǫ = − p . T µνvac. is in fact a diagonalized tensor as we hope to be. So thevacuum- S V ,which is inherent to such a space-time works like a sui generis fluid at equilibrium and with negativepressure,leading to a cosmological anti-gravity connected to the cosmological constant. B. Cosmological constant Λ Let us begin by writing the Einstein equation in the presence of the cosmological constant Λ,namely: R µν − Rg µν = 8 πGc T µν + Λ g µν , (99)where we think that the anti-gravitational effect due to the vacuum energy has origin from the last term Λ g µν .In the absence of matter ( T µν = 0),we have R µν − Rg µν − Λ g µν = 0 (100)For very large scales of space-time,the presence of the term Λ g µν is considerable and the accelerated expansionof the universe is governed by vacuum energy density. So we can relate Λ to the vacuum energy density. To dothat,we just use the energy-momentum tensor (95) (from (94)) given in vacuum limit of the ultra-referential S V (see (98)). Thus we can rewrite equation (100) in its equivalent form for the energy-momentum tensor given inthe limit of vacuum- S V ,as follows: R µν − Rg µν − πGc T vac.µν = 0 , (101)where T vac.µν = lim v → V T µν = − pg µν (see (98)). And as p = − ǫ = − ǫ vac. = − ρ (Λ) ( p = wǫ with w = − R µν − Rg µν − πGc ρ (Λ) g µν = 0 (102)Finally,by comparing (102) with (100),we obtain ρ (Λ) = Λ c πG , (103)which gives the direct relationship between cosmological constant Λ and vacuum energy density ρ (Λ) .Our aim is to estimate Λ and ρ (Λ) by using the idea of such a universal minimum speed V and its influenceon gravitation at very large scales of length. In order to study such an influence,let us firstly start from thewell-known simple model of a massive particle that escapes from a classical gravitational potential φ ,whereits total relativistic energy for an escape velocity v is due to the presence of such a potential φ ,namely E =0 mc (1 − v /c ) − / ≡ mc (1 + φ/c ). Here the interval of velocity 0 ≤ v < c is associated with the intervalof potential 0 ≤ φ < ∞ ,where we stipulate φ > S V of background field has origin in a non-classical(non-local) aspect of gravitation that leads to a repulsive gravitational potential ( φ <
0) for very large distances(cosmological anti-gravity). In order to see such a modified aspect of gravitation[30],let us consider the totalenergy of the particle with respect to S V ,shown in (72),namely: E = m c q − V v q − v c ≡ m c (1 + φ/c ) , (104)from where we obtain φ ≡ c q − V v q − v c − (105)From (105),we observe two regimes of gravitational potential,namely: φ = φ R : − c < φ ≤ V (= ξc ) < v ≤ v ,φ A : 0 ≤ φ < ∞ for v (= √ ξc ) ≤ v < c. (106) φ A and φ R are respectively the attractive (classical) and repulsive (non-classical) potentials. We observe thatthe strongest repulsive potential is φ = − c ,which is associated with a vacuum energy for the ultra-referential S V of the universe as a whole (consider v = V in (105)). Therefore such most negative potential is related tothe cosmological constant (see (97)),and so we write: φ Λ = φ ( V ) = − c (107)The negative potential above depends directly on Λ,namely φ Λ = φ (Λ) = φ ( V ) = − c . To show that,let usconsider a simple model of spherical universe with a radius R u ,being filled by a uniform vacuum energy density ρ (Λ) ,so that the total vacuum energy inside the sphere is E Λ = ρ (Λ) V u = − pV u = M Λ c . V u is its volume and M Λ is the total dark mass associated with the dark energy for Λ ( w = − φ Λ = − GM Λ R u = − Gρ (Λ) V u R u c = 4 πGpR u c (108)By introducing (103) into (108),we find φ Λ = φ (Λ) = − Λ R u c R u , (110)where Λ S u = 24 πc ,being S u = 4 πR u .And also by comparing (108) with (107),we have ρ (Λ) = − p = 3 c πGR u , (111)1where ρ (Λ) S u = 3 c /G . (111) and (110) satisfy (103).Λ (eq. 110) is a kind of cosmological scalar field , extending the old concept of Einstein about the cosmologicalconstant for stationary universe. From (110),by considering the Hubble radius,with R u = R H ∼ m ,weobtain Λ = Λ ∼ (10 m s − / m ) ∼ − s − . To be more accurate,we know the age of the universe T = 13 . R H = cT ≈ . × m ,which leads to Λ ≈ × − s − . This result is very closeto the observational results[31][32][33][34][35]. The tiny vacuum energy density[36][37] shown in (111) for R H is ρ (Λ ) ≈ × − g/cm ,which is also in agreement with observations. For scale of the Planck length,where R u = l P = ( G ~ /c ) / ,from (110) we find Λ = Λ P = 6 c /G ~ ∼ s − , and from (111) ρ (Λ) = ρ (Λ P ) = T vac.P =Λ P c / πG = 3 c / πG ~ ∼ J/m (= 3 c / πl P G ∼ kgf /S P ∼ atm ∼ g/cm ). So just at thatpast time,Λ P or ρ (Λ P ) played the role of an inflationary vacuum field with 122 orders of magnitude[38] beyondof those ones (Λ and ρ (Λ ) ) for the present time.It must be emphasized that our assumption for obtaining the tiny value of Λ starts from new fundamentalprinciples in the space-time. So it does not depend on detailed adjustments with cosmological models.The study of competition between gravity and anti-gravity (Λ) during the expansion of the universe will betreated elsewhere. IX. CONCLUSIONS AND PROSPECTS
We have introduced a space-time with symmetry,so that
V < v ≤ c , where V is an inferior and unattainablelimit of speed associated with a privileged inertial reference frame of universal background field. So we haveessentially concluded that the space-time structure where gravity is coupled to electromagnetism at quantumlevel naturally contains the fundamental ingredients for comprehension of the quantum uncertainties throughthat mentioned symmetry ( V < v ≤ c ),where gravity plays a crucial role due to the minimum velocity V ( ∝ G / )related to the minimum length (Planck scale) of DSR[20][21][22][23] [24][25] by Magueijo,Smolin,Camelia,et al.We have studied the cosmological implications of S V ,by estimating the tiny values of the vacuum energydensity ( ρ ( Λ) = 10 − g/cm ) and the current cosmological constant (Λ ∼ − s − ),which are still not wellunderstood by quantum field theories for quantum vacuum[38],because such theories foresee a very high valuefor Λ,whereas,on the other hand,exact supersymmetric theories foresee an exact null value for it,which also doesnot agree with Reality.The present theory has various implications which shall be investigated in coming articles. A new trans-formation group for such a space-time will be explored in details. We will propose the development of a newrelativistic dynamics, where the energy of vacuum (ultra-referential S V ) plays a crucial role for understandingthe origin of the inertia,including the problem of mass anisotropy.Another relevant investigation is with respect to the problem of the absolute zero temperature in thermo-dynamics of a gas. We intend to make a connection between the 3rd. law of Thermodynamics and the newdynamics,through a relationship between the absolute zero temperature ( T = 0 K ) and the minimum aver-age speed ( h v i N = V ) for N particles. Since T = 0 K is thermodynamically unattainable,this is due to theimpossibility of reaching h v i N = V from the new dynamics standpoint. This leads still to other importantimplications,such as for example,Einstein-Bose condensate and the problem of the high refraction index of ul-tracold gases,where we intend to estimate that the speed of light would approach to V inside the condensatemedium for T → K . So the maximum refraction index would be n max = c/v min = c/V = ξ − = σ ∼ to beshown elsewhere. Thus we will be in a condition to propose an experimental manner of making an extrapolationin order to obtain v lightMin. = c ′ min → V for T → K ,through a mathematical function obtained by the theoryapplied to ultracold systems.In sum,we begin to open up a new fundamental research field for various areas of Physics,by includingcondensed matter,quantum field theories,cosmology (dark energy and cosmological constant) and specially anew exploration for quantum gravity at very low energies (very large wavelengths). Acknowledgedments
This research has been supported by Prof.J.A.Helayel-Neto,from
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