Variation of the fundamental constants over the cosmological time: veracity of Dirac's intriguing hypothesis
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Variation of the fundamental constants over the cosmological time: veracity of Dirac’sintriguing hypothesis
Cl´audio Nassif** and A. C. Amaro de Faria Jr*[email protected]** (retired professor**), [email protected] ∗ (Dated: February 11, 2021)We investigate how the universal constants, including the fine structure constant, have variedsince the early universe close to the Planck energy scale ( E P ∼ GeV) and, thus, how they haveevoluted over the cosmological time related to the temperature of the expanding universe. Accordingto a previous paper[1], we have shown that the speed of light was much higher close to the Planckscale. In the present work, we will go further, first by showing that both the Planck constantand the electron charge were also too large in the early universe. However, we conclude that thefine structure constant ( α ∼ = 1 / G )at the Planck scale. Thus, we will be able to verify the veracity of Dirac’s belief about the existenceof “coincidences” between dimensionless ratios of sub-atomic and cosmological quantities, leadingto a variation of G with time, i.e., the ratio of the electrostatic to gravitational force between anelectron and a proton ( ∼ ) is roughly equal to the age of the universe divided by an elementarytime constant, so that the strength of gravity, as determined by G , must vary inversely with timejust in the approximation of lower temperature or for times very far from the early period, in orderto compensate for the time-variation of the Hubble parameter ( H ∼ t − ). In short, we will show thevalidity of Dirac’s hypothesis only for times very far from the early period or T ≪ T P ( ∼ K).
PACS numbers: 03.30.+p, 11.30.Qc, 06.20.Jr, 98.80.Es
I. INTRODUCTION
There are many theoretical proposals for variation ofthe fundamental constants of nature, including the vari-ation of the fine structure constant α [2][3][4][5][6]. Fur-thermore, many evidences behind recent claims of spatialvariation in the fine structure constant, due to ground-based telescopes for the observations of ion absorptionlines in the light of distant quasars, have led to muchdiscussion because of the controversial results about howdifferent telescopes should observe distinct spatial varia-tions on α [7]. Variation over cosmological time has alsobeen conjectured[8][9][10]. In view of all this, we shouldbe careful to investigate the veracity of such controversialresults. For this, we will start from a significant result ofthe extended relativistic dynamics due to the presence ofan isotropic background field with temperature T , whichhas been addressed in a previous Brief Report[1], wherewe have found the dependence of the speed of light withtemperature of the expanding universe. Starting fromthis result[1], the present work goes further in order toobtain the variation of the Planck constant with temper- ∗ *CBPF : Centro Brasileiro de Pesquisas F´ısicas, Rua Dr.XavierSigaud 150, Urca, 22.290-180, Rio de Janeiro-RJ, Brazil.Residence address: Rua Rio de Janeiro n.1186, ap.1304, 30.160-041(whose name of fantasy is CPFT: Centro de Pesquisas em F´ısicaTe´orica, i.e., a non-profit name of fantasy), Belo Horizonte-MG,Brazil. *IEAv : Instituto de Estudos Avancados, Rodovia dos TamoiosKm 099, 12220-000, S˜ao Jos´e dos Campos-SP, Brazil. ature, the behavior of the electron charge with tempera-ture, the electron (or proton) mass which has varied withthe background temperature and, finally, the behavior ofthe gravitational constant at very high temperatures inthe early universe and also at lower temperatures for longtimes, confirming Dirac’s hypothesis ( G ∼ t − ).Hence, we will show that the veracity of Dirac’s in-triguing hypothesis about the existence of “coincidences”between dimensionless ratios of sub-atomic and cosmo-logical quantities, linking the micro and macro-universe,leads to a variation of G with time, i.e., we will verify thatthe ratio of the electrostatic force ( F e ) to gravitationalforce ( F g ) between an electron and a proton ( ∼ ) isroughly equal to the age of the universe divided by anelementary time constant, so that the strength of grav-ity, as determined by G , must vary inversely with timejust in the approximation of lower temperature or fortimes very far from the early period, in order to com-pensate for the time-variation of the Hubble parameter( H ∼ t − ). Furthermore, at the end of the last section,we will also verify that such a ratio of 10 , as a dimen-sionless number, does not vary with temperature, i.e., F e /F g = F ′ e /F ′ g = F e ( T ) /F g ( T ) ∼ .We conclude that the fine structure constant α , as adimensionless number as well as the ratio F e /F g , hasalso remained invariant with the cosmic time scale (tem-perature). Thus, we will show that α ′ = α ( T ) = α = q e / πǫ ~ c , where α − ≈ . α has increased over cosmologicaltime, where the combined analysis over more than 100quasar systems has produced a value of a relative changeof ∆ α/α = − . ± . × − , which is at the 5 σ sig-nificance level[14], in contrast, we have laboratory teststhat cover only a short time span and they have foundno indications for the time-variation of α [15]. Their ad-vantage, however, is their great accuracy, reproducibilityand unequivocal interpretation. II. ENERGY EQUATION OF A PARTICLE INA THERMAL BACKGROUND FIELD
According to the relativistic dynamics, the relativisticmass of a particle is m = γm , where γ = 1 / p − v /c and m is its rest mass. On the other hand, according toNewton second law applied to 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 in-ertial mass ( m i ) that is larger than the relativistic mass m (= γm ); i.e., we have m i > m .The mysterious discrepancy between the relativisticmass m ( m r ) and the inertial mass m i from Newton sec-ond law is a controversial issue[16][17][18][19][20][21][22].Actually, the Newtonian notion about inertia as the re-sistance to acceleration ( m i ) is not compatible with therelativistic dynamics ( m r ) in the sense that we gener-ally cannot consider ~F = m r ~a . An interesting expla-nation for such a discrepancy is to take into considera-tion the influence of an isotropic background field[1][23]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 conceptof inertia, where we find m ∗ = m i = γ m r = γ m . Inthis sense, it is natural to conclude that m ∗ has a non-local origin; i.e., it comes from a kind of interaction witha background field connected to a universal frame[23],which is within the context of the ideas of Sciama[24],Schr¨odinger[25] and Mach[26].If we define the new factor γ = Γ, then we write m ∗ = Γ m, (1)where Γ[1] provides a non-local dynamic effect due to the influence of a universal background field on the particlemoving with speed v with respect to such a universalframe[25]. According to this reasoning, the particle isnot completely free, since its relativistic energy is nowmodified by the presence of the whole universe, namely: E ∗ = m ∗ c = Γ mc (2)As the modified energy E ∗ can be thought of as beingthe energy E of the free particle plus an increment δE of non-local origin, i.e., E ∗ = Γ E = E + δE , then let usnow consider that δE comes from a thermal backgroundbath due to the whole expanding universe instead of adynamic effect ( m ∗ ) of a particle moving with speed v inthe background field, in spite of the fact that there shouldbe an equivalence between the dynamical and thermaleffects for obtaining the modified energy. To show this[1],we make the following assumption inside the factor Γ,namely:Γ( v ) = (cid:18) − v c (cid:19) − ≡ Γ( T ) = − m P v K B m P c K B ! − , (3)from where we find Γ( T ) = (1 − T /T P ) − [1], T being thebackground temperature. T P (= m P c /K B ∼ K) isthe Planck temperature in the early universe with Planckradius 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 simply rewrite Eq.(2) as follows: E = Γ( T ) mc = mc − TT P (4)It is curious to notice that the equation of Magueijo-Smolin in their doubly special relativity ( mc / − E/E P )[27] reproduces Eq.(4) [1] when we just replace E by K B T and E P by K B T P in the denominator of theirequation.As the factor Γ( T ) has a non-local origin and is relatedto the background temperature of the universe, let us ad-mit that this factor acts globally on the speed of light c ,while the well-known factor γ acts locally on the rela-tivistic mass of the particle. In view of this, we shouldredefine Eq.(4) in the following way: E = [ γ ′ m ][Γ( T ) c ] = γ ′ m c ′ = mc ( T ) = mc ′ , (5)where now we have m = γ ′ m , so that γ ′ = 1 q − v c ′ (6)And, from Eq.(5) we extract c ′ = c ( T ) = c q − TT P , (7)where c ( T ) = p Γ( T ) c = γ T c , with γ T = 1 / p − T /T P .So the change in the speed of light is δc = c ′ − c , i.e., δc = ( γ T − c = (1 / p − T /T P − c . For T << T P ,we get δc ≈
0. When T = T P , c ′ has diverged.From Eq.(7), we find that the speed of light was infinitein the early universe when T = T P . As the universe isexpanding and getting colder, the speed of light had beendecreased to achieve c ( T ) ≈ c for T << T P . Currentlywe have c ( T ) = c , with T ≈ . c ′ given at a certain backgroundtemperature still remains invariant only with respect tothe motion of massive particles, but not with respect tothe temperature and so the cosmological time. III. INVARIANCE OF THE FINE STRUCTURECONSTANT WITH COSMOLOGICAL TIME
From a modified relativistic dynamic of a particle mov-ing in a space-time with an isotropic background field ata given temperature, it has been shown the dependenceof the speed of light with temperature of the expandinguniverse, shown in Eq.(7). So we have found that theenergy of a particle moving under the influence of such athermal background field is E = Γ( T ) mc = mc ′ [1].Now, let us consider the energy of a photon modifiedby the presence of a given background temperature. Sowe write E = mc ′ = p ′ c ′ , (8)where p ′ = mc ′ = mc ( T ) = mγ T c , which represents themodified momentum of the photon.On the other hand, we already know that the energyof a photon is E = hν = ~ w , where ~ = h/ π and w = 2 πν = 2 πc/λ , λ being the wavelength of the photonand ν (= c/λ ) being its frequency. Now, if we considerthis energy modified by the background temperature, wefind E = ~ ′ w ′ = h ′ ν ′ = h ′ c ′ λ , (9)where ν ′ = c ′ /λ . Comparing Eq.(8) to Eq.(9), we write E = mc ′ = λ − h ′ c ′ , (10)or simply p ′ = mc ′ = λ − h ′ = λ − h ( T ) , (11)which represents the de-Broglie equation for the photonin the presence of the background temperature.According to Eq.(7), Eq.(11) is written as m ( γ T c ) = λ − h ′ . Since both the wavelength λ and the relativisticmass m of the photon are not corrected with temper-ature, only the universal constants c and h , as globalquantities, are influenced by the background tempera-ture, so that we have c ′ (= γ T c ) and h ′ [= h ( T )] in Eq.(11).Thus, from Eq.(11), we conclude that the Planck con-stant should be corrected in the same way of the speedof light in Eq.(7). So we find h ′ = h ( T ) = γ T h = h q − TT P , (12)or else ~ ′ = ~ ( T ) = γ T ~ , with ~ = h/ π . So, fromEq.(11), we find p ′ = m ( γ T c ) = λ − ( γ T h ) ⇒ λ = h/mc ,in such a way to preserve the de-Broglie equation and,thus, the wavelength (frequency) of the photon. In theearly universe, when T = T P , h ′ has diverged.It is known that c = 1 /µ ǫ , where µ is the magneticpermeability of vacuum and ǫ is the electric permittivityof vacuum. Thus, based on Eq.(7), by correcting thisMaxwell relation with temperature of the universe, wewrite c ′ = 1 µ ′ ǫ ′ = c − TT P = 1 µ ǫ (cid:16) − TT P (cid:17) , (13)from where we extract µ ′ = µ ( T ) = µ r − TT P , (14)and ǫ ′ = ǫ ( T ) = ǫ r − TT P , (15)since the electric ( ǫ ) and magnetic ( µ ) aspects of radia-tion are in equal-footing.Based on the electric interaction energy ∆ E e betweentwo point-like electrons separated by a certain distance r , we write ∆ E e = ∆ m e c = e r = q e πǫ r , (16) xc(T)0 1 2 3 4 5 6 7 8 9 10x 10 −34 xh(T)0 1 2 3 4 5 6 7 8 9 10x 10 −19 xq e (T) FIG. 1:
The three graphs above provide variations in c , h and q e from top to bottom respectively, where there are divergencesat Planck scale ( ∼ K). where e = q e / πǫ . ∆ E e (= ∆ m e c ) is the relativisticrepresentation for such an interaction energy of electricorigin, which decreases to zero when r → ∞ , i.e., we have∆ m e → r → ∞ .Now, by correcting Eq.(16) due to the presence of athermal background field according to Eq.(7), we write∆ m e c ′ = ∆ m e c ( T ) = ∆ m e c − TT P = e ′ r = q ′ e πǫ ′ r , (17)from where we get e ′ = e ( T ) = e − TT P , (18)or else q ′ e πǫ ′ = q e πǫ (cid:16) − TT P (cid:17) (19)Inserting Eq.(15) into Eq.(19), we find q ′ e πǫ q − TT P = q e πǫ (cid:16) − TT P (cid:17) , (20)which implies q ′ e = γ T q e = q e q − TT P , (21)or q ′ e = q e ( T ) = q e q − TT P (22)The fine structure constant without temperature is α = e / ~ c = q e / πǫ ~ c = q e µ c/ h . Now, by taking −1012 T α −30 Tm e −10 TG FIG. 2:
The first graph above shows the invariance of the finestructure constant with temperature of the universe. The othertwo graphs below show the divergences of m e and G close tothe Planck scale ( ∼ K). into account a given temperature of the expanding uni-verse, we have α ( T ) = α ′ = e ′ ~ ′ c ′ = q ′ e πǫ ′ ~ ′ c ′ = q ′ e µ ′ c ′ h ′ (23)Finally, by inserting Eq.(7)( c ′ ), Eq.(12)( ~ ′ ) andEq.(18)( e ′ ) into Eq.(23), or by inserting c ′ , ~ ′ , ǫ ′ [Eq.(15)] and q ′ e [Eq.(21)] into Eq.(23), or even by in-serting c ′ , h ′ , µ ′ [Eq.(14)] and q ′ e [Eq.(21)] into Eq.(23),we find e ′ ~ ′ c ′ = e ~ c , (24)or q ′ e πǫ ′ ~ ′ c ′ = q e πǫ ~ c (25)or q ′ e µ ′ c ′ h ′ = q e µ c h , (26)that is, α ′ = α ≈ . , (27)which reveals to us the invariance of the fine structureconstant with temperature of the expanding universe andthus its invariance over cosmological time. IV. VARIATION OF THE GRAVITATIONALCONSTANT IN THE EARLY UNIVERSE
Now, by considering the very low energy of gravita-tional interaction ∆ E g between two point-like electronsseparated by a certain distance r , we write∆ E g = ∆ m g c = Gm e r , (28)where ∆ E g (= ∆ m g c ) is the relativistic representationfor such an interaction energy of gravitational origin,which is much lower than the interaction energy of elec-tric origin, i.e., we have ∆ m g ≪ ∆ m e (∆ E g ≪ ∆ E e ),where ∆ m g also decreases to zero when r → ∞ , i.e., wehave ∆ m g → r → ∞ .Now, by correcting Eq.(28) due to the presence of thethermal background field according to Eq.(7), we write∆ m g c ′ = ∆ m g c ( T ) = ∆ m g c − TT P = G ′ m ′ e r , (29)from where we get G ′ m ′ e = G ( T ) m e ( T ) = Gm e − TT P , (30)and from where we extract separately: m ′ e = m e ( T ) = m e q − TT P (31)and G ′ = G ( T ) = G q − TT P , (32)where we have admitted that m e (Eq.31) and q e (Eq.22)are on equal footing with respect to their variation withtemperature, i.e., the ratio m e /q e has been remained in-variant with the cosmological time, since the dimension-less number α g = Gm e / ~ c should be also invariant like α = e / ~ c . Thus we can realize that m e and q e divergedlikewise close to the Planck temperature.According to Eq.(32), G diverges for T = T P . A. Veracity of Dirac’s intriguing hypothesis
In analogous way to the reasoning above for the gravi-tational interaction between two electrons, now if we con-sider the gravitational interaction between two protons inthe presence of such a thermal background field, we find: m ′ p = m p ( T ) = m p q − TT P , (33) where m p is the proton mass.In his intriguing hypothesis, Dirac[28] noted that, forsome unexplained reason, the ratio of the electrostatic togravitational force between an electron and a proton isroughly equal to the age of the universe divided by anelementary time constant, implying that F e F g = αhcGm p m e ∼ m p c Hh ∼ , (34)where H is the Hubble parameter. And, it is easy toverify that the ratio F e /F g remains invariant with tem-perature, i.e., F e /F g = F ′ e /F ′ g = α ′ h ′ c ′ /G ′ m ′ p m ′ e ∼ .Eq.(34) led Dirac to argue that the strength of gravity,as determined by G , must vary inversely with time, tocompensate for the time-variation of the Hubble param-eter H . Indeed, we will show that this Dirac’s hypothesiscan be confirmed within a certain approximation only for T ≪ T P , when we obtain H ′ [= H ( T )] in such a way asto preserve the coincidence and invariance of the ratio of10 for any temperature in Eq.(34)[29][30]. To do that,we simply write m ′ p c ′ H ′ h ′ ∼ (35)In order to preserve the invariance of the ratio of 10 with temperature in Eq.(35), we first insert Eq.(7) ( c ′ ),Eq.(12) ( h ′ ) and Eq.(33) ( m ′ p ) into Eq.(35) and, after, weshould perform the calculations, such that we must find H ′ = H ( T ) = H (cid:18) − TT P (cid:19) − ≈ H (cid:18) TT P (cid:19) , (36)which is only valid when T ≪ T P , becoming a linearfunction of the form H ′ = H + aT , a being a constant.In order to make the approximation given in Eq.(36)to be consistent with the time-variation of the Hubbleparameter, i.e., H ∼ t − , which is also an approximationvalid just for long times, thus, we must extract the fol-lowing approximation, namely T ∼ t − for T ≪ T P . So,finally, by doing this, we can write Eq.(32) as follows: G ′ = G ( T ) = G (cid:18) − TT P (cid:19) − ≈ G (cid:18) TT P (cid:19) , (37)when T ≪ T P , such that we get a linear function G ′ = G + bT , b being a constant. Hence, since T ∼ t − , we cansimply write G ∼ t − , confirming Dirac’s hypothesis. [1] C. Nassif and A. C. Amaro de Faria Jr., Phys. Rev. D , 027703 (2012): arXiv:gr-qc/1205.2298. [2] J. D. Bekenstein, Phys. Rev. D , 1527 (1982). [3] Ph. Brax, C. van de Bruck, A.C. Davis and C.S. Rhodes,Astrophysics and Space Science , 627 (2012).[4] E. Calabrese, E. Menegoni, C.J.A.P. Martins, A. Mel-chiorri and G. Rocha, Phys. Rev. D , 023518 (2011).[5] A. V. Ivanchik, A.Y. Potekhin and D.A. Varshalovich,Astronomy and Astrophysics , 439 (1999).[6] W. Marciano, Phys. Rev. Lett. , 489 (1984).[7] J. K. Webb, J.A. King, M.T. Murphy, V.V. Flambaum,R.F. Carswell and M.B. Bainbridge, Phys. Rev. Lett. , 191101 (2011).[8] L. D. Thong, N.M. Giao, N.T. Hung and T.V. Hung,EPL. , 69002 (2009).[9] J. K. Webb, V.V. Flambaum, C.W. Churchill, M.J.Drinkwater and J.D. Barrow, Phys. Rev. Lett. , 884(1999).[10] J. K. Webb, M.T. Murphy, V.V. Flambaum, V.A. Dzuba,J.D. Barrow, C.W. Churchill, J.X. Prochaska and A.M.Wolfe, Phys. Rev. Lett. , 091301 (2001).[11] R. Bouchendira, P. Clad ´ e, S. Guellati-Kh ´ elifa, F. Nezand F. Biraben, Phys. Rev. Lett. , 080801 (2011).[12] E. Cameron and T. Pettitt, arXiv: astro-ph.CO/1207.6223.[13] J. Bahcall, W. Sargent and M. Schmidt, The Astrophys-ical Journal Vol. , July (1967).[14] M. T. Murphy, J. K. Webb and V. V. Flambaum, Mon.Not. Roy. Astron. Soc., , 609 (2003). [15] H. L. Bethlem and W. Ubachs, Faraday Discuss. ,25-36 (2009).[16] C. J. Adler, Am. J. Phys. , 739 (1987).[17] R. P. Feynman , R. B. Leighton and M. Sands, The Feyn-man Lectures on Physics (Addison-Wesley,Reading, MA,1963), Vol.1.[18] V. L. Okun , Physics Today No.6, 31 (1989).[19] T. R. Sandin, Am. J. Phys. , 1032 (1991).[20] W. Rindler, Introduction to Special Relativity (ClarendonPress, Oxford 1982),pp. 79-80.[21] W. Rindler,
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