Cosmological Origin of Gravitational Constant

Transcription

1 Apeiron, Vol. 5, No. 4, October Cosmological Origin of Gravitational Constant Maciej Rybicki Sas-Zbrzyckiego 8/7 3-6 Krakow, oland rybicki@skr.pl The base nits contribting to gravitational constant are spposed to be related to mass and size of niverse and to the speed of light. We test this idea in respect to its conformity with estimations concerning the present parameters of niverse. Considering Hbble s law, the size-dependence of gravitatonal constant determines its variability. In trn, this involves the variability of lanck nits containing G. rovided that mass of niverse and lanck mass coincident at lanck epoch, we arrive at new parameters of niverse in qantm state, described by the newly derived lanck nits of mass, size, time and temperatre. Keywords: niverse; gravitational constant; lanck nits; lanck epoch; lanck constant. Introdction The fndamental qestions concerning physical constants are still nsolved. What is the relationship between initial conditions of niverse, laws of physics and physical constants? Do constants follow from (hypothetic) fndamental laws, or determine these laws? Are all 8 C. Roy Keys Inc.

2 Apeiron, Vol. 5, No. 4, October of constants really constant? As long as these qestions are not properly recognized, the restrictions pt on the constant s vales by the present state of niverse cannot be treated as ltimate. For the last few decades, the main attention in that connection has been focsed on the Sommerfeld fine-strctre constant α. If it increases in time as some of recent observations seem to sggest [] then at least one of its components mst be variable, so that it affects the whole ratio. The main sspected became the speed of light, spposed to be time dependent: c t () Another case of that kind concerns the gravitational constant. The conjectre of its variability has been advanced by.a.m. Dirac in 937, in the framework of his large nmbers hypothesis []. Comparing two ratios: ) radis of visible niverse / radis of qantm particle, and ) gravitational force / electric force that, with 4 the (spposed) accracy to the order of magnitde, amont and 4 respectively, Dirac dedced that the vale of gravitational constant decreases in proportion to the age of niverse: G t () The conjectre sbmitted in this paper (hereafter called in short the conjectre ) also consists in relating the gravitational constant to some of fndamental parameters describing the niverse. However, the exact content of the conjectre essentially differs from Dirac s hypothesis. Namely, we assme that gravitational constant is a physically meaningfl qantity of variable vale, related in a definite way to the size and mass of niverse and to the speed of light. 8 C. Roy Keys Inc.

3 Apeiron, Vol. 5, No. 4, October hysical premises for introdcing the conjectre All of the nits describing dimensional constants refer to the comprehensible physical qantities sch as mass, length, velocity, charge or action. Meanwhile, according to the crrent state of the art, the only sensible physical explanation of G nit amonts to the need of completing the Newton s law of gravitation, so as to obtain the right dimension of force. Let s take a closer look at G from that point of view. Its sally applied nit is Nm kg (3) It can be also expressed by 3 m kgs (4) Both forms are mathematically eqivalent, yet preferable is the latter since it reflects the qantm-mechanical connotation of gravitational constant, thoght to have fndamental significance. Namely, G relates to lanck length, lanck mass and lanck time as 3 G = m t This relationship seems to contradict the claim that physical meaning of gravitational constant s dimension is nclear or nrecognized. However, since G contribtes to all of the above lanck nits: G = 3 ; m c c = G ; t G = 5 c (5) (6) 8 C. Roy Keys Inc.

4 Apeiron, Vol. 5, No. 4, October so, in fact, obtaining G from lanck nits consists in drawing ot that what has been previosly inserted to them; namely, rewritting (5) gives 3 G c G = cg= 3 5 c G c G (7) Nevertheless, Eq. 5 brings in two essential informations. The first one is that, mathematically, this eqation does not determine the vale of G. Let the vale of gravitational constant be different from the sal one. What are the conseqences of sch assmption to lanck nits? Let s start from lanck mass. By definition, it is the mass for which the Schwarzschild radis r S eqals the Compton wavelength λ divided by π. Sbstitting lanck mass to Compton wavelength gives 3 c G 3 λ = = = = (8) π mc G c which conforms to the vale obtained by sbstitting lanck mass to Schwarzschild radis: 3 Gm c G rs = = G c = = (9) c G c If, therefore, we assme gravitational constant eqal to ξ G, with ξ the factor of proportionality of a free vale, then we get the lanck mass: c ξ m = ξg () 8 C. Roy Keys Inc.

5 Apeiron, Vol. 5, No. 4, October In conseqence, the eqality between Compton wavelength and Schwarzschild radis wold take place for lanck length of the vale ξ, and lanck time eqal to ξ c. The second information obtained from Eq. 5 is the following. Since t = c then the relationship between gravitational constant and lanck nits can be also written in the form: c G = () m We conjectre that R = () m M where M is the (constant) mass of niverse, and R is its (present) size. Ths, the gravitational constant can be expressed as R c G = (3) M Considering that R is, in the general case, variable accordingly to the Hbble s law, we may define gravitational constant as def ξ R c ξ G = (4) M where ξ is the dimensionless factor, sch that < ξ, provided that ξ = refers to the present epoch. 8 C. Roy Keys Inc.

6 Apeiron, Vol. 5, No. 4, October The gravitational constant and the magnitde of niverse An ideal test of the conjectre shold consist in sbstitting the right vales describing the present size and mass of niverse to the new formla of gravitational constant (Eqs. 3, 4), so as to obtain the proper vale of G : G = 6,6748 ±,67 m kgs (5) ( ) 3 It is clear, however, that we cannot expect the exact otcome on this way since all avaliable data concerning the mass and size of the whole (or visible) niverse are merely approximations. Nevertheless, the recently applied observational methods are sfficiently advanced to obtain the credible reslts with the accracy to the order of magnitde. The lower bond for the the size of niverse, obtained from the 6 7 WMA data, is 4 Gpc. This gives in SI nits 7, m [3]. There are convincing argments in favor of spposition that the real size does not considerably exceed this limit. 5 The estimations of the mass of niverse range from 3 kg to 6, 6 kg (ignoring the assmption of infinite mass) [4]. Sch a big discrepancy is, to a large extent, determined by diversity of the applied cosmological assmptions. However, the most reliable valations referred to the whole niverse range from 53 kg to 55 kg. They conform to the valations of mass of visible niverse 5 that amont approximately 3 kg, with the age of niverse estimated by varios methods (inclding Hbble s law, WMA and astrophysical data) for abot 3,7 billion years. Considering the above mentioned vale of size and the estimation of density 8 C. Roy Keys Inc.

7 Apeiron, Vol. 5, No. 4, October ( ρ kg m ), it seems reasonable to regard in calclations the 54 mass vale M kg. 7 Considering c m s, we may calclate for ξ = 7 7 Rc 3 G = 54 ( m kgs ) (6) M Taking into accont that the right approximation to the order of magnitde of gravitational constant is G, we may conclde that the conjectre and the recently obtained estimates concerning mass and size of niverse, validate each other with the fair accracy. 4. Coincidence of lanck mass and the mass of niverse at lanck epoch If gravitational constant relates to the (increasing ) size of niverse then all lanck nits containing G also become variable. Let s write the for (originally proposed by M. lanck [5]) base natral nits, in the form inclding factor ξ. We se the sperscript to differentiate the general (new) form of lanck nit from the sal form that, according to the conjectre, applies to the present instant only: m c = ξg ξg = 3 (7) (8) c 8 C. Roy Keys Inc.

8 Apeiron, Vol. 5, No. 4, October 8 47 ξg = 5 t (9) c T Considering < ξ it follows 5 c = ξgk c = lim m ( ξ ) = ξ ξg () () ξg = ( ξ ) = ξ 3 lim () c ξg = t ( ξ ) = ξ 5 lim (3) c 5 c = limt ( ξ ) = ξ ξgk (4) Certainly it is not correct to mix qantm terms with infinitesimal vales. Let s notice, however, that, if lanck nits refer to the natre, the lanck mass sholdn t exceed the mass of niverse. Considering that, we may assme that at lanck epoch the lanck mass coincidences with the mass of niverse: m M (5) Hence 8 C. Roy Keys Inc.

9 It follows Apeiron, Vol. 5, No. 4, October M c = ξg The lanck length at lanck epoch is therefore (6) c ξ = (7) GM G c G M 3 3 = ξ = = (8) c G c M c The analogos derivations for lanck time and lanck temperatre at lanck epoch ( t, T ) are: t G c G M 5 5 G = ξ = = c c M c (9) 5 c M c T ξ = = (3) Gk k 54 By sbstitting nmerical vales: M, 34 8, c, 7 3 c, k to Eqs. 8, 9, 3, we obtain respectively (see the conting rle expressed in [6]): ( m) (3) 54 8 t ( s) (3) 8 C. Roy Keys Inc.

10 Apeiron, Vol. 5, No. 4, October T The density at lanck epoch wold be then 54 M ρ = ( K ) (33) ( kg m ) C. Roy Keys Inc. (34) Let s call the newly derived parameters the basic lanck nits. Hence, the lanck epoch that we postlate here is defined by the basic lanck nits of length, time, temperatre and density. The nmerical vale of the factor ξ at lanck epoch ( ξ ) can be obtained either directly from ξ = cgm (Eq. 7) or, considering m ( ) c G =, from It follows ( m M ) ξ = (35) ξ (36) Since we assme that ξ is linearly related to R so we shold expect that R ξ = (37) Sbstitting the nmbers obtained for and for ξ, gives R ( m ) (38)

11 Apeiron, Vol. 5, No. 4, October which confirms the conjectre. We may write then c ξ R c G = = m M (39) where G stands for gravitational constant at lanck epoch. Nmerically, this gives G 54 ( m kgs ) (4) Ths G 4 G (4) From the variability of gravitational constant it follows the variable (in time) vale of the gravitational force. However, if the distances between atracting masses are eqal as measred in lanck nits relevant to given epochs, then gravitational force does not change, i.e. Gmm Gmm = = F const. (4) Rewriting the lanck length (Eq. 8) gives 3 G G c = = (43) Let s consider the lanck constant from the point of view of the obtained reslts. It relates to lanck nits as = mc (44) Rewriting lanck mass and lanck length, we obtain 8 C. Roy Keys Inc.

12 Apeiron, Vol. 5, No. 4, October ξg c 3 c = (45) ξg c which means that Eq. 44 redces to the identity, and therefore holds for any vale of ξ. This can be written as = mc (46) For m and (the lanck epoch) it takes the form = M c (47) Nmerically, this gives J s (48) ( ) According to the assmption m M, the lanck energy at lanck epoch is E E = M c (49) From E t = ω, where ω =, we get E = t (5) which is the energy eqation for the niverse at lanck epoch. Note that the niverse at lanck epoch, as well as the lanck particle, satisfies the Schwarzschild eqation for the black hole: r = G m c. Namely S r ( ) = S ( ) m (5) 7 In all probability, the present niverse also flfils the Schwarzschild condition: 8 C. Roy Keys Inc.

13 Apeiron, Vol. 5, No. 4, October r ( ) = R S 7 ( m) (5) We may, therefore, calclate the entropy of the niverse at lanck epoch, and in present epoch, by applying the Bekenstein-Hawking formla [7], [8] for the entropy of black hole: 3 Akc S = (53) 4G where A is the srface area of the event horizon and k is the Boltzmann constant. For the spherically symmetrical black hole we A = m 8π G c, and therefore (53) takes the form: get ( ) S = m π kcg (54) Ths, for the lanck epoch we get kcg S( ) = M π (55) which gives Jm S( ) 34 (56) Ks This vale is eqal to the vale of entropy for the lanck particleblack hole of the mass m. Namely kcg 7 Jm π 34 S( ) = m (57) Ks Instead, for the present niverse we have 8 C. Roy Keys Inc.

14 Apeiron, Vol. 5, No. 4, October kcg S ( ) = M π (58) present which gives Jm S( ) present 34 (59) Ks Hence, the increase of the entropy of niverse conforms the factor ξ and amonts S( present) G 4 = (6) S G Conclsion ( ) The revealed dependence of gravitational constant from the size of niverse involves the redefinition of lanck nits that eventally appears inconstant. The base parameters of niverse (inclding mass) coincident with basic lanck nits at lanck epoch, which means that initial state of niverse can be considered in terms of a single qantm. This indces to interprete lanck time rather as a prely technical parameter related to the Heisenberg ncertainty principle than as a hypothetic period ranging from the zero instant to the end point of lanck epoch. The conjectre seems to entail profond conseqences to physics and cosmology. It brings soltion to the fndamental cosmological qestions concerning the very early niverse: the problem low entropy, the horizon problem and the flatness problem. The thermodynamic arrow of time is well defined by the initial vale of 7 entropy ( 7 ) and its present vale ( ). The possibility of casal contact within the whole volme of niverse at lanck epoch follows 8 C. Roy Keys Inc.

15 Apeiron, Vol. 5, No. 4, October from ct =. In trn, the ratio between the density at lanck epoch (dramatically greater in comparison with the hitherto estimations) and the relevant spatial crvatre (that is constant as related to lanck particle /Eq. 43/) explains fairly well the apparent flatness of the present niverse. Besides, the conjectre confirms the time invariability of lanck constant and the speed of light. This, in trn, speaks against the hypothesis of the varying fine strctre constant as related to inconstant c. Last bt not least, the conjectre may also help in solving the problem of qantm gravity. The variable vale of gravitational constant involves specific predictions referring to the creation of different cosmic objects sch as stars, galaxes and black holes in the deep past. This makes the idea of inconstant G experimentally testable. The same refers to the varying ratio between gravity and the electric forces. Since, according to GRT, gravitation is recognized as a phenomenon basically connected with spacetime, then it sholdn t be very srprising in general that evolving spacetime determines the (variable) properties of gravitation. Nevertheless, the postlated dependence of gravitational constant from the size of niverse demands physical jstification. Only then, in or opinion, the conjectre wold be able to evolve into a matred theory. References: [] Webb, J. K.; et al. Search for Time Variation of the Fine-Strctre Constant, hysical Review Letters 8 (5): (999) [] Dirac,.A.M.; A New Basis for Cosmology, roceedings of the Royal Society of London, Series A, Mathematical and hysical Sciences, Vol. 65, Isse 9, 99-8 (937) [3] Cornish, N. J.; et al. Constraining the Topology of the Universe, hysical Review Letters, Vol. 9, 3 (4) 8 C. Roy Keys Inc.

16 Apeiron, Vol. 5, No. 4, October 8 48 [4] The hysics Factbook, Edited by Glenn Elert [5] lanck, M.; Über irreversible Strahlngsvorgänge, Sitzngsberichte der reßischen Akademie der Wissenschaften, Vol. 5, p. 479 (899) [6] We apply the following approximation rle in calclations. While m n mltiplying a b that, by the reason of ncertainty as to the ab vale, m n is approximated to, we calclate: m+ n+ if both indices are positive, and m+ n if both indices are negative. It follows from the probability that ab exceeds 5, and ths the whole otcome (while approximated) increases by the order of magnitde. This does not concern the 7 case of mltiplying by ( c ) since this vale is already ronded to the higher order of magnitde. [7] Bekenstein, J.; Black Holes and Entropy, hysical Review D7: (97) [8] Hawking, S.W.; article Creation by Black Holes, Commnications in Mathematical hysics 43: 99- (975) 8 C. Roy Keys Inc.