Variation of the speed of light and a minimum speed in the scenario of an inflationary universe with accelerated expansion
aa r X i v : . [ phy s i c s . g e n - ph ] S e p Variation of the speed of light and a minimum speed in the scenario of aninflationary universe with accelerated expansion
Cl´audio Nassif Cruz and Fernando Antˆonio da Silva *CPFT: Centro de Pesquisas em F´ısica Te´orica, Rua Rio de Janeiro 1186/s.1304,Lourdes, 30.160-041, Belo Horizonte-MG, [email protected], [email protected]
In this paper we aim to investigate a deformed relativistic dynamics well-known as SymmetricalSpecial Relativity (SSR) related to a cosmic background field that plays the role of a variablevacuum energy density associated to the temperature of the expanding universe with a cosmicinflation in its early time and an accelerated expansion for its very far future time. In this scenario,we show that the speed of light and an invariant minimum speed present an explicit dependence onthe background temperature of the expanding universe. Although finding the speed of light in theearly universe with very high temperature and also in the very old one with very low temperature,being respectively much larger and much smaller than its current value, our approach does notviolate the postulate of Special Relativity (SR), which claims the speed of light is invariant in akinematics point of view. Moreover, it is shown that the high value of the speed of light in theearly universe was drastically decreased and increased respectively before the beginning of theinflationary period. So we are led to conclude that the theory of Varying Speed of Light (VSL)should be questioned as a possible solution of the horizon problem for the hot universe.
Keywords : cosmic inflation, speed of light, Planck temperature, Planck mass, vacuum en-ergy density, cosmological constant.
PACS numbers: 11.30.Qc, 04.20.Cv, 04.20.Gz, 04.20.Dw, 04.90.+e
I. INTRODUCTION
The advent of Varying Speed of Light (VSL) theories[1][2][3][4][5][6] seems to shake the foundations of SpecialRelativity (SR) theory, since the speed of light c in vacuum is no longer constant. However, we must takecare to investigate the veracity of such proposals. To do that, first of all we should consider a new DeformedSpecial Relativity (DSR) so-called Symmetrical Special Relativity (SSR)[7][8][9][10][11][12][13]that presents anew causal structure of spacetime where there emerges an invariant minimum speed V connected to a universalbackground field given by the vacuum energy, so that we have V and the speed of light c as being the invariantspeeds for lower and higher energies respectively. So in view of SSR theory, this paper aims to go beyond byinvestigating the speed of light c and the universal minimum speed V in the modern cosmological scenario with acosmic inflation in the early universe and a very rapid accelerated expansion for a so far future time. Due to thecosmic inflation at much higher temperatures T ≤ T P ( M P c /K B ∼ K ) (Planck temperature) and the rapidaccelerated expansion for a far future time at much lower temperatures T ≥ T mim ( M P V /K B ∼ − K )[13],i.e., the minimum temperature related to the cosmological horizon, we should take into account an extendedSSR with the presence of an isotropic background field with temperature T , where all the particles are movingwith respect to a preferred reference frame that plays the role of a universal thermal bath with a temperature T decreasing in the time of the expanding universe. Thus we are considering that the energy scale at which acertain particle is subjected has a nonlocal origin by representing the background thermal energy of the wholeuniverse, i.e., the particle should be coupled to the background field with temperature T . Such a backgroundthermal effect that leads to a correction on its total energy ( E = m c p − V /v / p − v /c [11]) is muchmore pronounced during the inflationary period and a too far future time governed only by vacuum.The background thermal effect will allow us to obtain the speed of light with an explicit dependence on thetemperature of the universe. So we will be able to preserve the postulate of constancy of the speed of lightand extend it just for the implementation of the temperature of the expanding universe. In this sense, we havea function c ( T ) and so we will find an enormous value for the speed of light in the early universe when itstemperature was extremely high close to the Planck temperature T P . In addition, we will also conclude that FIG. 1: The external and internal conical surfaces represent respectively the speed of light c and the unattainableminimum speed V , which is a definitely prohibited boundary for any particle. For a point P in the world line of aparticle, in the interior of the two conical surfaces, we obtain a corresponding internal conical surface, such that wemust have V < v p ≤ c . The 4-interval S is a time-like interval. The 4-interval S is a light-like interval (surface of thelight cone). The 4-interval S is a space-like interval (elsewhere). The novelty in spacetime of SSR are the 4-intervals S (surface of the dark cone) representing an infinitly dilated time-like interval, including the 4-intervals S , S and S inside the dark cone for representing a new space-like region (see ref.[11]). the very high speed of light was rapidly damped to a value much closer to its current value even before thebeginning of the cosmic inflation. This result will lead us to question VSL theory as an alternative explanationfor the horizon problem. II. A BRIEF REVIEW OF SYMMETRICAL SPECIAL RELATIVITY
The breakdown of Lorentz symmetry for very low energies[10][11] generated by the presence of a backgroundfield is due to an invariant mimimum speed V and also a universal dimensionless constant ξ [11], working like agravito-electromagnetic constant, namely: ξ = Vc = r Gm p m e πǫ q e ~ c , (1) V being the minimum speed and m p and m e are respectively the mass of the proton and electron. Such aminimum speed is V = 4 . × − m/s. We have found ξ = 1 . × − [11], where Dirac’s large numberhypothesis (LNH) were taken into account in obtaining ξ , i.e., F e /F g = q e / πǫ Gm p m e ∼ [11].It was shown[11] that the minimum speed is connected to the cosmological constant in the following way: V ≈ s e m p Λ (2). Therefore the light cone contains a new region of causality called dark cone [11], so that the speed of a particlemust belong to the following range: V (dark cone) < v < c (light cone) (Fig.1). FIG. 2: In this special case of (1 + 1) D , the referential S ′ moves in x -direction with a speed v ( > V ) with respect to thebackground field connected to the ultra-referential S V . If V → S V is eliminated (empty space), and thus the galileanframe S takes place, recovering Lorentz transformations. The breaking of Lorentz symmetry group destroys the properties of the transformations of Special Relativity(SR) and so generates an intriguing kinematics and dynamics for speeds very close to the minimum speed V ,i.e., for v → V , we find new relativistic effects such as the contraction of the improper time and the dilationof space[11]. In this new scenario, the proper time also suffers relativistic effects such as its own dilation withrespect to the improper one when v → V , namely:∆ τ r − V v = ∆ t r − v c , (3)which was shown in the reference[11], where it was also made experimental prospects for detecting such newrelativistic effect close to the invariant minimum speed V , i.e., too close to the absolute zero temperature.Since the minimum speed V is an invariant quantity as the speed of light c , V does not alter the value of thespeed v of any particle. Therefore we have called ultra-referential S V [10][11] as being the preferred (background)reference frame in relation to which we have the speeds v of any particle. In view of this, the reference frametransformations change substantially in the presence of S V , as follows:a) The special case of (1 + 1) D transformations in SSR[7][8][9][10][11] with ~v = v x = v (Fig.2) are x ′ = Ψ( X − vt + V t ) = θγ ( X − vt + V t ) (4)and t ′ = Ψ (cid:18) t − vXc + V Xc (cid:19) = θγ (cid:18) t − vXc + V Xc (cid:19) , (5)where θ = p − V /v and Ψ = θγ = p − V /v / p − v /c .b) The (3 + 1) D transformations in SSR (Fig.3)[11] are ~r ′ = θ (cid:20) ~r T + γ (cid:18) ~r // − ~v (cid:18) − Vv (cid:19) t (cid:19)(cid:21) = θ h ~r T + γ (cid:16) ~r // − ~vt + ~V t (cid:17)i (6)and t ′ = θγ " t − ~r · ~vc + ~r · ~Vc . (7) FIG. 3: S ′ moves with a 3 D -velocity ~v = ( v x , v y , v z ) in relation to S V . For the special case of 1 D -velocity ~v = ( v x ), werecover Fig.2; however, in this general case of 3 D -velocity ~v , there must be a background vector ~V (minimum velocity)with the same direction of ~v as shown in this figure. Such a background vector ~V = ( V /v ) ~v is related to the backgroundreference frame (ultra-referential) S V , thus leading to Lorentz violation. The modulus of ~V is invariant at any direction. Of course, if we make V →
0, we recover the well-known Lorentz transformations.Although we associate the minimum speed V with the ultra-referential S V , this frame is inaccessible forany particle. Thus, the effect of such new causal structure of spacetime generates an effect on mass-energybeing symmetrical to what happens close to the speed of light c , i.e., it was shown that E = m c Ψ( v ) = m c p − V /v / p − v /c , so that E → v → V [10][11]. We notice that E = E = m c for v = v = √ cV [11]. It was also shown that the minimum speed V is associated with the cosmological constant,which is equivalent to a fluid (vacuum energy) with negative pressure[10][11].The metric of such symmetrical spacetime of SSR is a deformed Minkowski metric with a global multiplicativefunction (a scale factor with v -dependence) Θ( v ) working like a conformal factor[12]. Thus we write dS = Θ η µν dx µ dx ν , (8)where Θ = Θ( v ) = θ − = 1 / (1 − V /v ) ≡ / (1 − Λ r / c ) [12], working like a conformal factor and η µν is theMinkowski metric.We can say that SSR geometrizes the quantum phenomena as investigated before (the origin of the UncertaintyPrinciple)[9] in order to allow us to associate quantities belonging to the microscopic world with a new geometricstructure that originates from Lorentz symmetry breaking. Such a geometry should be investigated in the future. A. SSR-metric as a conformal metric in a DS-scenario and dS-metric
Let us consider a spherical universe with Hubble radius r u filled by a uniform vacuum energy density. On thesurface of such a sphere (frontier of the observable universe), the bodies (galaxies) experience an acceleratedexpansion (anti-gravity) due to the whole “dark mass (energy)” of vacuum inside the sphere. So we could thinkthat each galaxy is a proof body interacting with the big sphere of dark energy (dark universe) like in the simplecase of two bodies interaction. However, we need to show that there is an anti-gravitational interaction betweenthe ordinary proof mass m and the big sphere with a “dark mass” of vacuum ( M ). To do that, let us first startfrom the well-known simple model of a massive proof particle m that escapes from a newtonian gravitationalpotential φ on the surface of a big sphere of matter, namely E = m c (1 − v /c ) − / ≡ m c (1 + φ/c ), where E is its relativistic energy. Here the interval of escape velocity 0 ≤ v < c is associated with the interval ofpotential 0 ≤ φ < ∞ , where we stipulate φ > S V connected tothe dark energy that fills the sphere has origin in a non-classical (non-local) aspect of gravity that leads to arepulsive gravitational potential ( φ < E = m c q − V v q − v c = m c (cid:18) φc (cid:19) , (9)from where we obtain φ = c q − V v q − v c − , (10)where m is the mass of the proof particle, v being its input speed or also its escape velocity from the sphere. Ifthe sphere is governed by vacuum as occurs in the universe as a whole, then v should be understood as an inputspeed in order to overcome anti-gravity, and thus the factor p − V /v prevails for determining the potential.However, for a sphere of matter, v is the well-known escape velocity, so that the Lorentz factor takes place.From the above equation, we observe two regimes of gravitational potential, namely: φ = φ Q : − c < φ ≤ V (= ξc ) < v ≤ v ,φ att : 0 ≤ φ < ∞ for v (= √ ξc = √ cV ) ≤ v < c, (11)where v = √ cV ∼ − m/s[11]. φ att and φ Q are respectively the attractive (classical) and repulsive (non-classical or quantum) potentials.We observe that the strongest repulsive potential is φ = − c , which is associated with vacuum energy for theultra-referential S V (consider v = V in Eq.(10)) (Fig.4).By considering the simple model of spherical universe with a radius r u and a uniform vacuum energy density ρ , we find the total vacuum energy inside the sphere, i.e., E Λ = ρV u = − pV u = M c , V u being its volume and M the total dark mass associated with the dark energy. Therefore, we are able to get a repulsive (negative)gravitational potential φ on the surface of such a sphere (universe) filled by dark “mass” (dark energy), namely: φ = − GMr u = − GρV u r u c = 4 πGpr u c , (12)where p = − ρ , ρ being the vacuum energy density and V u = 4 πr u / ρ = Λ c / πG , we write the repulsive potential as follows: φ = φ (Λ , r u ) = − Λ r u , (13)from where, for any radius r of the expanding universe, we generally write φ = − Λ r / v such that V < v ≤ v (= √ ξc ), namely: φc = − Λ r c = r − V v − , (14) V v c v -c in ɸ v esc v FIG. 4: This graph shows the potentials of SSR representing the function in Eq.(10) that presents two regimes, namely:a) The attractive (classical) regime is well-known as Lorentz sector for describing gravity of a source of matter like asphere of mass having a proof particle with mass m on its surface. This particle escapes from this gravity with an escapevelocity v ≤ v esc < c according to the attractive (positive) potential 0 ≤ φ att < ∞ . b) The repulsive (quantum) regimeis the sector that provides the signature of SSR for describing anti-gravity of a source of dark energy (vaccum energy)like an exotic sphere of dark mass having a proof particle of matter with mass m on its surface. In this quantum sector,the escape velocity from anti-gravity should be understood as the input velocity V < v in ≤ v according to the negative(quantum) potential − c < φ Q ≤
0, such that the proof particle with mass m is able to penetrate the dark spherewhose anti-gravity pushes it far away. Here we should observe that there is an intermediary velocity v = √ cV , whichcorresponds to the point of phase transition between these two regimes in such a way that the general potential φ = 0.This means that v can represent both escape and input velocities, which depends on the sector we are considering. Aswe are just interested in the quantum sector (anti-gravity) of SSR, we have v in = v for φ = φ Q = 0 and v in = V for φ = φ Q = − c , since we just take into account the sector of negative potential for treating the extended dS-relativity. such that, if v = V , we find φ ( V ) /c = −
1, so that φ ( V ) = − c . We have − c < φ < (cid:0) − V v (cid:1) = 1 (cid:16) φc (cid:17) = 1 (cid:0) − Λ r c (cid:1) , (15)where we see that there are three equivalent representations for Θ.Substituting Eq.(15) in Eq.(8), we write the spherical metric of SSR in the following way: d S = − c dt (cid:0) − Λ r c (cid:1) + dr (cid:0) − Λ r c (cid:1) + r ( dθ ) + r sin θ ( d Φ) (cid:0) − Λ r c (cid:1) . (16)We should note that φ/c = − Λ r / c , where we have − c < φ ≤
0. So it is interesting to realize that, forthe approximation φ >> − c or | φ Q | << c , we are in the regime v >> V , which means a weakly repulsiveregime that corresponds to the present time of the universe whose temperature T ( ≈ . v is still so far from T = T min ( ∼ − K )[13] connected to the minimum speed V .As Eq.(16) encompasses all types of vacuum, specially the ideal vacuum given for a too long time ( r → r horizon )when the universe (vacuum energy density ρ ) will be in a very strong repulsive regime with a very negativecurvature R = − πGρ/c (1 − V /v ) ≡ − πGρ/c (1 − Λ horizon r / c ) → −∞ [12], now we can realizethat only the approximation for a weakly repulsive regime is able to generate a special metric well-similar toDS-metric. So, in order to see this special metric (like DS-metric) given only for weak anti-gravity, we just makethe expansion of the denominator of Θ-factor in Eq.(16), so that we take into account only the first order term,since we are considering Λ r / << c such that we have (1 − Λ r / c ) ≈ (1 − r / c ) = (1 − Λ r / c )[12].Finally, in doing this approximation in SSR-metric [Eq.(16)], we find DS-metric, namely: d S ≈ dS DS = − c dt (cid:0) − Λ r c (cid:1) + dr (cid:0) − Λ r c (cid:1) + r dΩ (cid:0) − Λ r c (cid:1) , (17)where dΩ is dΩ = ( dθ ) + sin θ ( d Φ) (18)We realize that the validity of DS-metric remains only in a weak anti-gravity regime as occurs in the case ofthe tiny positive cosmological constant given in the present time of the expanding universe. III. DEFORMED ENERGY EQUATION OF A PARTICLE IN SPECIAL RELATIVITY DUE TOTHE PRESENCE OF A THERMAL BACKGROUND FIELD
According to the relativistic dynamics of Special Relativity (SR), the relativistic mass of a particle is m = γm ,where γ = 1 / p − v /c and m is its rest mass. On the other hand, according to Newton second law appliedto its relativistic momentum, we find F = dP/dt = d ( γm v ) /dt = ( m γ ) dv/dt = m (1 − v /c ) − / dv/dt ,where m γ represents an inertial mass ( m i ) that is larger than the relativistic mass m (= γm ); i.e., we have m i > m .The mysterious discrepancy between the relativistic mass m ( m r ) and the inertial mass m i from Newtonsecond law is a controversial issue[14][15][16][17][18][19][20]. Actually the Newtonian notion about inertia asthe resistance to acceleration ( m i ) is not compatible with the relativistic dynamics ( m r ) in the sense thatwe generally cannot consider ~F = m r ~a . An interesting explanation for such a discrepancy is to take intoconsideration the influence of an isotropic background field that couples to the particle, by dressing its relativisticmass ( m r ) in order to generate an effective (dressed) mass m ∗ (= m effective ) working like the inertial mass m i ( > m r ) in accordance with the Newtonian concept of inertia, where we find m ∗ = m i = γ m r = γ m . Inthis sense, it is natural to conclude that m ∗ has a nonlocal origin; i.e., it comes from a kind of interaction witha background field connected to a universal frame[11], which is within the context of the ideas of Sciama[21],Schr¨odinger[22] and Mach[23].If we define the new factor γ = Γ, we write m ∗ = Γ m, (19)where Γ provides a nonlocal dynamic effect due to the influence of a universal background field over the particlemoving with speed v with respect to such a universal frame. According to this reasoning, the particle is notcompletely free, since its relativistic energy is now modified by the presence of the whole universe, namely: E ∗ = m ∗ c = Γ mc (20)As the modified energy E ∗ can be thought as being the energy E of the free particle plus an increment δE ofnonlocal origin, i.e., E ∗ = Γ E = E + δE , let us now consider that δE comes from the thermal background fieldof the whole expanding universe instead of simply a dynamic effect of a particle moving with speed v in thebackground field, in spite of the fact that there should be an equivalence between the dynamical and thermalapproaches for obtaining the modified energy. To show this, we make the following consideration inside thefactor Γ, namely:Γ( v ) = (cid:18) − v c (cid:19) − ≡ Γ( T ) = − m P v K B m P c K B ! − , (21)from where we find Γ( T ) = (1 − T /T P ) − , T being the background temperature. T P (= m P c /K B ∼ K) isthe Planck temperature in the early universe with Planck radius R P ∼ − m. E P (= m P c ∼ GeV) is thePlanck energy and m P ( ∼ − g) is the Planck mass. From the thermal approach, if T → T P , Γ( T ) diverges.Now we rewrite Eq.(20) as follows: E ( T ) = Γ( T ) mc = γm c − TT P . (22)As the factor Γ( T ) has a nonlocal origin and is related to the background temperature of the universe, let usadmit that this factor acts globally on the speed of light c , while the well-known factor γ acts locally on therelativistic mass of the particle. In view of this, we should redefine Eq.(22) in the following way: E = [ γ ′ m ][Γ( T ) c ] = γ ′ m c ′ = mc ( T ) = mc ′ , (23)where now we have m = γ ′ m , so that γ ′ = 1 q − v c ′ (24)And from Eq.(23) we extract c ′ = c ( T ) = c q − TT P , (25)where c ( T ) = p Γ( T ) c = γ T c , with γ T = 1 / p − T /T P .From Eq.(25) we find that the speed of light was infinite in the initial universe when T = T P . As the universewas expanding and getting colder, the speed of light had been decreased to achieve c ( T ) ≈ c for T << T P .Currently we have c ( T ) = c , with T ≈ . δc = c ′ − c , namely: δc = ( γ T − c = q − TT P − c, (26)where, for T << T P , we have δc ≈ c ′ for a given temperature remains invariant only withrespect to the motion of massive particles, but not with respect to the age and temperature of the universe. Inother words, we say that, although the speed of light has decreased rapidly during the initial expansion of theuniverse and thereafter with a smooth variation as shown in Fig.5, its value for a given temperature ( c ( T )) isstill a maximum limit of speed that is invariant only with respect to the motion of all subluminal particles.The modified spatial momentum of a particle moving in the presence of a cosmic background field withtemperature T is the following: R(t) TtT TT T T T Tt t t
P 1 2 3 4 p 5 65 6
GUTEPOCH HADRONEPOCHCOSMICINFLATIONEPOCH c c c c ,c R c(T) c(T)~cc(T)~cc(T)>c A vacuumenergyeffect ? R FIG. 5: This figure shows two graphics, namely R ( t ), which is the size (radius) of the universe as a function of time,and c ( T ), representing the speed of light with dependence on the temperature of the universe according to Eq.(25).At the beginning of the universe when it was a singularity with a minimum radius of the order of the Planck radius,i.e., R P ∼ − m, having the Planck energy scale E p ∼ GeV which corresponds to the Planck temperature T P ∼ K and the Planck time t P ∼ − s, the speed of light c ′ was infinite since there was no spacetime [seeEq.(25) for c ( T )]. But immediately after, when T = 10 K, the speed of light had already assumed a value closeto the current value as shown by Eq.(25) for c ( T ), and therefore a cone of light (a spacetime) had been formed; i.e.,with c = 2 . × m/s for the present time, then, according to the function c ( T ), we find c = c ( T ) =3 . × m/s (see the figure). Subsequently, for T = 10 K, we find c = c ( T ) = 3 . × m/s . For T = 10 K ⇒ c = c ( T ) = 2 . × m/s. And for T = 10 K ⇒ c = c ( T ) = 2 . × m/s. Finally,for T = 10 K ⇒ c = c ( T ) = 2 . × m/s. From this temperature T = 10 K, when t = t ∼ − s,corresponding to the energy scale of the Grand Unified Theory (GUT) with 10 GeV, the universe inflated very quickly,starting with a radius R ∼ − m and reaching R ∼ m at the time t ∼ − s; i.e., the size of the universeincreased rapidly 50 orders of magnitude. Since the speed of light c ≈ c , VSL should be put in doubt. Hence, perhapsthe vacuum energy had played a fundamental role in that epoch. In the next section, only when we will treat SSR witha thermal background field, this question about the vacuum energy in the inflationary period and also in the period ofa far future time with accelerated expansion will be clarified. P ( T ) = γ ′ γ T m v = γ ′ m v q − TT P (27)Before obtaining the modified energy-momentum relation, we first introduce the modified 4-velocity, namely: U ′ µ = γ ′ c q − TT P , γ ′ v α q − TT P , (28)where µ = 0 , , , α = 1 , ,
3. If T → T P , the 4-velocity diverges.The modified 4-momentum is P ′ µ = m U ′ µ . So, from Eq.(28) we find P ′ µ = γ ′ m c q − TT P , γ ′ m v α q − TT P , (29)0where E ( T ) = c ′ P ′ = m γ ′ c ′ = mc / (1 − T /T P ). The spatial components ( α = 1 , ,
3) of Eq.(29) representsthe spatial momentum P ′ = P ( T ) = m γ ′ v/ p − T /T P .From Eq.(29), by performing the quantity P ′ µ P ′ µ , we obtain the modified energy-momentum relation asfollows: P ′ µ P ′ µ = [ E ( T )] c ′ − [ P ( T )] = m c ′ , (30)from which we get [ E ( T )] = m c (cid:16) − TT P (cid:17) = c ′ [ P ( T )] + m c ′ , (31)where c ′ = c/ p − T /T P and m = γ ′ m .It is curious to notice that the Magueijo-Smolin doubly special relativity equation ( mc / − E/E P )[24]reproduces Eq.(22) when we just replace E by K B T and E P by K B T P in the denominator of their equation. IV. ENERGY EQUATION OF A PARTICLE IN SYMMETRICAL SPECIAL RELATIVITY WITHTHE PRESENCE OF A THERMAL BACKGROUND FIELD
Let us first consider a force applied to a particle, in the same direction of its motion. More general caseswhere the force is not necessarily parallel to velocity will be treated elsewhere. In our specific case ( ~F || ~v ), therelativistic power P ow (= vdp/dt ) of SSR is given as follows: P ow = v ddt " m v (cid:18) − V v (cid:19) (cid:18) − v c (cid:19) − , (32)where we have used the momentum in SSR, i.e., p = m v Ψ( v ).After performing the calculations in Eq.(32), we find P ow = (cid:16) − V v (cid:17) (cid:0) − v c (cid:1) + V v (cid:0) − v c (cid:1) (cid:0) − V v (cid:1) dE k dt , (33)where E k = m v .If we make V → c → ∞ in Eq.(33), we simply recover the power obtained in newtonian mechanics,namely P ow = dE k /dt . Now, if we just consider V → P ow = (1 − v /c ) − / dE k /dt . We notice that such a relativistic power tends to infinite( P ow → ∞ ) in the limit v → c . We explain this result as an effect of the drastic increase of an effective inertialmass close to c , namely m eff = m (1 − v /c ) k ′′ , where k ′′ = − /
2. We must stress that such an effectiveinertial mass is the response to an applied force parallel to the motion according to Newton second law, and itincreases faster than the relativistic mass m = m r = m (1 − v /c ) − / .The effective inertial mass m eff that we have obtained is a longitudinal mass m L , i.e., it is a response to theforce applied in the direction of motion. In SR, for the case where the force is perpendicular to velocity, we canshow that the transversal mass increases like the relativistic mass, i.e., m = m T = m (1 − v /c ) − / , whichdiffers from the longitudinal mass m L = m (1 − v /c ) − / . So, in this sense, there is anisotropy of the effectiveinertial mass to be also investigated in more details by SSR in a further work.In the previous section, it was already notice that the mysterious discrepancy between the relativisticmass m ( m r ) and the longitudinal inertial mass m L from Newton second law [Eq.(33)] is a controversial1issue[14][15][16][17][18][19][20]. Actually it is already known that the newtonian notion about inertia as theresistance to an acceleration ( m L ) is not compatible with the relativistic dynamics ( m r ) in the sense that wegenerally cannot consider ~F = m r ~a . The dynamics of SSR aims to give us a new interpretation for the inertiaof the newtonian point of view in order to make it compatible with the relativistic mass. This compatibilityis possible only due to the influence of the background field that couples to the particle and “dresses” its rel-ativistic mass in order to generate an effective (dressed) mass in accordance with the newtonian notion aboutinertia from Eq.(32) and Eq.(33). This issue will be clarified in this section.From Eq.(33), it is important to observe that, when we are closer to V , there emerges a completely new result(correction) for power, namely: P ow ≈ (cid:18) − V v (cid:19) − ddt (cid:18) m v (cid:19) , (34)given in the approximation v ≈ V . So, we notice that P ow → ∞ when v ≈ V . We can also make thelimit v → V for the general case [Eq.(33)] and so we obtain an infinite power ( P ow → ∞ ). Such a newrelativistic effect deserves the following very important comment: Although we are in the limit of very lowenergies close to V , where the energy of the particle ( mc ) tends to zero according to the approximation E = mc ≈ m c (1 − V /v ) k with k = 1 /
2, on the other hand the power given in Eq.(34) shows us that thereis an effective inertial mass that increases to infinite in the limit v → V , that is to say, from Eq.(34) we getthe effective mass m eff ≈ m (1 − V /v ) k ′ , where k ′ = − /
2. Therefore, from a dynamical point of view, thenegative exponent k ′ (= − /
2) for the power at very low speeds [Eq.(34)] is responsible for the inferior barrierof the minimum speed V , as well as the exponent k ′′ = − / c according to Newton second law. Actually, due to the drastic increaseof m eff of a particle moving closer to S V , leading to its strong coupling to the vacuum field in the backgroundframe S V , thus, in view of this, the dynamics of SSR states that it is impossible to decelerate a subatomicparticle until reaching the rest. This is the reason why there is a unattainable minimum speed V .In order to see clearly both exponents k ′ = − / V ) and k ′′ = − / c ), let us write the general formula of power [Eq.(33)] in the following alternative way after some algebraicmanipulations on it, namely: P ow = (cid:18) − V v (cid:19) k ′ (cid:18) − v c (cid:19) k ′′ (cid:18) − V c (cid:19) dE k dt , (35)where k ′ = − / k ′′ = − /
2. Now it is easy to see that, if v ≈ V or even v << c , Eq.(35) recovers theapproximation in Eq.(34). As V << c , the ratio V /c ( <<
1) in Eq.(35) is a very small dimensionless constant ξ = V /c ∼ − [11]. So ξ can be neglected in Eq.(35).So, from Eq.(35) we get the effective inertial mass m ∗ = m eff of SSR, namely: m ∗ = m eff = m (cid:18) − V v (cid:19) − (cid:18) − v c (cid:19) − . (36)By taking into account the same reasoning as used before (Section 3) to interpret m ∗ within the context of athermal background field, we also realize that the effective (inertial) mass m ∗ has a nonlocal origin, which nowpresents a natural connection with SSR due to the existence of a preferred background frame related to theuniversal minimum speed[11]. Thus SSR with a thermal background field will be able to predict the variationof the speed of light with temperature for a too far future cosmic time.In order to obtain the variation of the speed of light for a very cold cosmological horizon close to a minimumtemperature T mim (= m P V /K B ∼ = 3 . × − K )[13], we first must get the factor Ω that transforms the inertialmass m to the generalized inertial mass m ∗ of SSR given in Eq.(36). So, let us now write the equivalent formof Eq.(36) in the following way:2 E ∗ = m ∗ c = Ω m c = "(cid:18) − V v (cid:19) − (cid:18) − v c (cid:19) − m c ≡ (Γ SSR Ψ) m c , (37)where we already know that Ψ m c = mc = E . So we must stress that E ∗ = E = mc , since Ω = Ψ. As, indeformed SR, we have written E ∗ = Γ mc = (Γ γ ) m c with Γ = Γ SR = γ , in an analogous way, from Eq.(37)we are able to obtain the factor Γ SSR , so that we realize thatΩ = Γ
SSR
Ψ = (cid:18) − V v (cid:19) − (cid:18) − v c (cid:19) − . (38)As we have Ψ = p − V /v / p − v /c , from Eq.(38) we findΓ SSR = Γ SR (cid:18) − V v (cid:19) − = (cid:18) − v c (cid:19) − (cid:18) − V v (cid:19) − . (39)In the same way that the deformation factor Γ SR is due to the presence of a thermal background field thatincreases significantly the energy E of a particle only for higher temperatures close to Planck temperature( T P ) in the early universe, i.e., E ∗ = Γ SR E with Γ SR = Γ SR ( T ) = (1 − T /T P ) − and E = mc , the generaldeformation factor Γ SSR is also able to predict the influence of a very cold thermal background field on theenergy of a particle.According to a previous paper[13], we have demonstrated the existence of a universal minimum temperature T min (= m P V /K B ∼ − K ), which is related to a ultra-cold cosmological horizon in a too far future timeof a very old universe. We have shown the following thermal equivalence relation for lower energies which areassociated to a ultra-cold background thermal bath, namely: (cid:18) − V v (cid:19) − = − (cid:16) m P V K B (cid:17)(cid:16) m P v K B (cid:17) − ≡ (cid:18) − T min T (cid:19) − . (40)So by substituting Eq.(21) and Eq.(40) in Eq.(39), we find the general deformation factor Γ SSR in its equiva-lent thermal form for representing the general effect of any thermal background field on the energy of a particle,as follows: Γ
SSR ( T ) = (cid:18) − T min T (cid:19) − (cid:18) − TT P (cid:19) − . (41)By substituting Eq.(41) in Eq.(37), we finally obtain the deformed energy E ∗ of a particle in SSR due to thetemperature of the universal background field, namely: E ( T ) = Γ SSR ( T ) mc = Ψ m c (cid:0) − T min T (cid:1) (cid:16) − TT P (cid:17) . (42)If we make V → T min →
0, and thus Eq.(22) is recovered as a special case just forhigher background temperatures of an early universe governed by the cosmic inflation.Before continuing to investigate the important implications of Eq.(42), it is interesting to note that SSRwithout temperature ( E = Ψ m c ) remains valid for a long period of cosmic time when T min << T << T P .In a similar reasoning that was used to interpret the factor Γ SR ( T ) in Eq.(22) as being of non-local origin byacting globally on the speed of light c , then the general factor Γ SSR ( T ) in Eq.(42) should be also interpreted as3 FIG. 6: This graph for representing Eq.(45) shows that the speed of light diverges for both limits of temperature, namelyPlanck temperature T P ( ∼ K ) for the scale of Planck L P ( ∼ − m ) in the early (too hot) universe, and a minimumtemperature T min ( ∼ − K ) in a ultra-cold universe connected to a horizon radius r h >> r u ( ∼ m ). being of non-local origin by acting on the speed of light c , since its non-local aspect is due to the backgroundtemperature of the whole universe. Thus let us admit that this thermal factor acts globally on the speed oflight c , while the kinematics factor Ψ acts locally on the relativistic mass of the particle. In view of this, weshould redefine Eq.(42) as follows: E ( T ) = [Ψ ′ m ][Γ SSR ( T ) c ] = Ψ ′ m c ′ = mc ( T ) = mc ′ , (43)where we have m = Ψ ′ m , so that we write Ψ ′ = q − V ′ v q − v c ′ , (44)where c ′ = c ( T ) will be obtained soon and V ′ = V ( T ) will be obtained in the next section.By inserting Eq.(41) into Eq.(43), we finally extract c ( T ), namely: c ′ = c ( T ) = c q − T min T q − TT P . (45)In Eq.(45), if we make T min →
0, we recover Eq.(25) that represents the particular case of c ( T ) in the deformedSR with the presence of a background thermal field as it was well investigated in the previous section wherewe have shown that the drastic decreasing of the speed of light for T < T P is not able to explain the horizonproblem (background isotropy) in the hot universe (Fig.5). This result calls into question the VSL theories thatcounteract the inflationary model that aims to explain the background isotropy (horizon problem), although theinflationary model does not still provide a clear explanation for the origin of the cosmic inflation field usuallyso-called “inflaton” .It is very curious to notice that Eq.(45) provides a similarity between the too hot universe ( T ≈ T P ) closeto the Planck scale L P and the ultra-cold universe ( T ≈ T min ) close to the horizon radius ( r h = √ c/ √ Λ h [12])in the sense that the speed of light diverges for both limits. This leads us to think that there emerges anotherinflation very close to the ultra-cold horizon ( r h ), i.e., there emerges a very rapid stretching of a ultra-cold spacewhose temperature is too close to T mim , which is responsible for the drastic increasing of the speed of lightbeing dragged by very cold inflation itself. This novelty is provided only by SSR with the presence of a thermalbackground field. In this sense, the theory shows that both inflationary periods, i.e., the initial inflation andthe final rapid accelerated expansion, have the same origin related to a too hot and cold vacuum respectively.The current vacuum energy is related to the well-known cosmological constant Λ ∼ − s − , but in a veryfar future time, the theory predicts that the temperature will decrease until it will approach T mim when the4cosmological constant will also decrease until approaching to a horizon cosmological constant Λ h for r → r h .So a new inflation will begin due to the appearance of an infinitely negative curvature, i.e., there will emergea Big Rip of the spacetime tissue for a ultra-cold vacuum, thus leading to a very rapid increasing of the speedof light that will instantly illuminate the whole exploding universe. To show all these effects in such a limit ofultra-cold universe, we should realize that there is an equivalence between Eq.(15) and Eq.(40), so that we find c ′ = c q − T min T ≡ c (cid:16) − Λ h r c (cid:17) , (46)where we can see that, for T → T min or for r → r h , the speed of light diverges so rapidly that a ultra-coldinflation of the whole universe begins. At a first sight, such an inflation seems to lead to the so-called BigRip of space-time tissue, however, as the minimum temperature T min and the horizon radius r h are bothunattainable, there could be strong fluctuations of vacuum during the rapid expansion so that the temperaturecould drastically fluctuates in many small parts of space, thus leading to an enourmous number of very hotinflationary bubbles that would emerge of such parts and thus many expanding “baby universes” working likebubbles would be created from the final inflation and so on. Then, in a distant past, probably our universeenerged from drastic fluctuations of a small part in the order of Planck scale in the scenario of a previousexpanding universe (“mother universe”). This a reasonable conjecture according to the present theory, and itis in a certain accordance with some other theories about “mother universes” and “baby universes”. However,according to such conjectures, some puzzles arise, namely: Was there an uncreated primordial universe fromwhich all the mother and baby universes have been arisen? And if such a primordial “universe” is the firstcause of all others, then how did this first creation take place? These intriguing questions are still on hold. Inthe next section, we intend to go deeper into to these questions with the present theory.According to a previous paper[12], we have obtained the curvature R of an extended DS-space governed byall kind of vacuum (cosmological constants), where the fundamental vacuum is given by the minimum speedrelated to a horizon cosmological constant. So, now taking into account Eq.(40) and Eq.(15), we can write R [12] in the following way: R = − πGρc (cid:0) − T min T (cid:1) ≡ − πGρc (cid:16) − Λ h r c (cid:17) . (47)If T → T min or r → r h , this implies that the scalar curvature of the universe governed only by ultra-coldvacuum becomes infinitely negative,i.e., R → −∞ . Such an infinitely negative curvature is responsible by aninfinite anti-gravity that stretches drastically the space by dragging the light so that its speed increases to theinfinite. In this sense, we realize that there is a direct connection between the scalar curvature and the variationof the speed of light with temperature of the expanding universe close to the cosmological horizon. To showthis, we just compare Eq.(46) with Eq.(47), and so we find R ( T ) = − πGρc [ c ( T )] , (48)where c ( T ) = c p − T min /T .According to Eq.(45), the change in the speed of light is δc = c ′ − c , namely: δc = (Γ SSR − c = q − T min T q − TT P − c, (49)where, for T min << T << T P , we have δc ≈ U ′ µ = Ψ ′ c q − T min T q − TT P , Ψ ′ v α q − T min T q − TT P , (50)where µ = 0 , , , α = 1 , ,
3. If T → T P or even T → T min , the 4-velocity diverges.The modified 4-momentum is P ′ µ = m U ′ µ . So, from Eq.(50) we obtain P ′ µ = Ψ ′ m c q − T min T q − TT P , Ψ ′ m v α q − T min T q − TT P , (51)where E ( T ) = c ′ P ′ = m Ψ ′ c ′ = mc / (1 − T min /T )(1 − T /T P ).From Eq.(51),by performing the quantity P ′ µ P ′ µ , we find the following modified energy-momentum relationas follows: P ′ µ P ′ µ = [ E ( T )] c ′ − [ P ( T )] = m c ′ (cid:18) − V ′ v (cid:19) , (52)from which we get [ E ( T )] = m c (cid:0) − T min T (cid:1) (cid:16) − TT P (cid:17) = c ′ [ P ( T )] + m c ′ (cid:18) − V ′ v (cid:19) , (53)where we have the spatial momentum of deformed SSR, namely P ( T ) = Ψ ′ m v/ p (1 − T min /T ) p (1 − T /T P )and c ′ = c/ p − T min /T p − T /T P [Eq.(45)].Eq.(53) represents the dispersion relation of deformed SSR with the presence of a thermal background fieldin the cosmological scenario of the expanding universe, where both inflationary primordial and final epochs aretaken into account. V. VARIATION OF THE UNIVERSAL MINIMUM SPEED IN BOTH SCENARIOS OF ANINFLATIONARY UNIVERSE AND FINAL ACCELERATED EXPANSIONA. The concept of reciprocal velocity in SSR: the uncertainty on position
As already discussed in a previous paper[9], SSR generates a kinematics of non-locality as also proposed inthe emergent gravity theories[25].In order to see more clearly the aspect of non-locality of SSR due to the stretching of space when v ≈ V , weshould take into account the idea of the so-called reciprocal velocity v rec , which has been already well exploredin a previous paper[9]. Thus, here we will just reintroduce such idea in a more summarized way. To do that, letus use Eq.(3) and first multiply it by c at both sides, and after by taking the squared result in order to obtain (cid:18) c − c V v (cid:19) (∆ τ ) = ( c − v )(∆ t ) , (54)where the right side of Eq.(54) related to the improper time ∆ t provides the velocity v of particle (∆ t → ∞ for v → c ) and, on the other hand, the left side gives us the new information that shows that the propertime in SSR is not an invariant quantity as is in SR, so that the proper time goes to infinite (∆ τ → ∞ )6 FIG. 7: The figure shows that the dark cone and light cone are opposite aspects of a same transcent state working likea Newtonian space where c ′ → ∞ and V ′ → T P and T min . when v → V , which leads to a too large stretching of the proper space interval c ∆ τ in this limit of muchlower speed, giving us the impression that the particle is well delocalized due its “high internal speed” that isso-called reciprocal velocity that appears at the left side of the equation as being v rec = ( cV ) /v = v /v . Sowe have ( c − v rec )(∆ τ ) = c − v )(∆ t ) . Now we can perceive that the reciprocal velocity v rec representsa kind of inverse of v such that, when v → V , we get v rec → c , i.e., the “internal motion” is close to c , thusleading to the new effect of proper time dilation associated to a delocalization that was shown as being anuncertainty on position[9] within the scenario of spacetime in SSR. In this scenario, it was shown that thedecreasing of momentum close to zero ( v ≈ V ) leads to a delocalization of the particle, which is justified bythe increasing of v rec → c and the dilation of the proper time ∆ τ → ∞ . As the uncertainty on position is∆ x = v rec ∆ τ = ( v /v )∆ τ [9], for v → V , we find ∆ x = c ∆ τ → ∞ . And, on the other hand, the large increasingof momentum for v → c leads to the well-known dilation of the improper time ∆ t (right side of Eq.(54)), so thatwe find ∆ τ << ∆ t and the minimum reciprocal velocity v rec → v /c = V , which provides a small uncertaintyon position, since ∆ x = V ∆ τ .Therefore Eq.(54) or Eq.(3) can be rewritten in the following way:∆ τ r − v rec c = ∆ t r − v c , (55)where we find v rec /c = V /v . We have V < v < c and
V < v rec < c , where V is the reciprocal of c andvice-versa.As the minimum speed is the reciprocal of c , i.e., we have V = v /c , then we simply obtain the V ( T ), namely: V ( T ) = v c ( T ) , (56)where we must call attention to the fact that v = p c ( T ) V ( T ) = √ cV is a universal fixed point that doesnot have dependence on temperature because it represents the unique point where occurs the phase transitionbetween gravity (ADS-space with positive curvature for v > v or Φ > < v < v or Φ < > v is the perfectNewtonian regime where the curvature is exactly null, i.e., it is a perfectly flat space where temperature doesnot make sense, however such a point do not have stability in any spacetime, since any spacetime is necessarilythe result of the existence of two barriers given by a dark cone for a certain minimum limit V ′ = V ( T ) and alight cone for certain maximum limit c ′ = c ( T ) (see Fig.7). Otherwise, it would not be a physical space. Inthis sense based on SSR, we conclude that a Newtonian space would merely be a non-physical idealization of aperfectly inertial “space” (see letter a in Fig 8 representing such flat space).By introducing Eq.(45) for c ( T ) into Eq.(56), we obtain its reciprocal speed V ( T ) (Fig.7), namely: V ( T ) = V r − T min T r − TT P , (57)7 FIG. 8: A Newtonian or flat space works like an uncreated primordial universe. For an unknown reason, this “serenelake” (null curvature) is in the eminence of being disturbed. An infinite negative curvature arises generating a vacuumwith a very strong anti-gravity which creates a high peak at the Planck scale. The temperature that increases drasticallyleads to the emergence of an inflationary bubble that will generate our universe. where it is easy to verify that v is in fact a fixed (invariant) Newtonian point, since we get c ( T ) V ( T ) = cV = v when multiplying Eq.(45) by Eq.(57).Both Eq.(45) and Eq.(57) show respectively that c ( T P ) and c ( T min ) diverge, while V ( T P ) and V ( T min ) vanish(no dark cone). Of course, the absence of the dark cone would be the result of the absence of a light cone as thespeed of light becomes infinite, so that there would be no light to cast darkness, and so this dialectical (dual)idea of thesis X anti-thesis based on a dynamical symmetry would be overcame by an absolute permanentstate for representing the non-physical (Newtonian) flat space without temperature or even at a zero absolutetemperature ( T = 0 K ).Figure 7 shows clearly an abysmal gap from a non-physical state (a flat space or a space without fluctuations:see letter a of Fig.8) to a dynamical spacetime where the emergence of temperature (vibration) leads to the finitespeeds of light and the non-zero minimum speeds until reaching their well-known current values that generateour expanding spacetime.To summarize, SSR theory makes us rethink deeply that such an ethernal dialectical process of creation,expansion, and destruction of universes is sustained by an even more fundamental permanent (Newtonian) statefrom which there was an abysmal leap that provided a perturbation in the flat space (the letters b and c in Fig.8)and thus a bubble on the Newtonian absolute empty space has emerged for representing our universe (letter d in Fig.8), although a multiverse may also arise mysteriously from this non-physical Newtonian state, whichseems to be a First Cause since there are no fluctuations (letter a in Fig.8). The misterious passage betweensuch a Newtonian absolute state (uncreated primordial “universe”) and the spacetime seems to bring back atranscendent aspect that surpasses the dialectical materialism of the modern cosmology of the cyclic universes,where such an ideal state of Newtonian (flat) space associated to a fixed point v = v = p c ( T ) V ( T ) = √ cV isstill completely neglected by the idea of dialectical materialism. VI. CONCLUSIONS AND PROSPECTS
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