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A New Large Number Numerical Coincidences April,2013 PROGRESSINPHYSICS Volume2 A New Large Number Numerical Coincidences Alexander Kritov E-mail: [email protected] In this article, the author gives a set of new hypothesis wherein he presents new, ex- act and simple relations between physical constants and numbers. The author briefly analyses the discovered coincidences in terms of their accuracy and confidence, while leaving himself aside any physical explanation of the presented formulas. Important: all the found relations have a common nature of the “power of two”. The exact nature of this remains unknown for yet, so it requires further research. The presented material may also be viewed as a logical continuation and development of Dirac’s and Edding- ton’s Large Numbers Hypothesis (LNH). However, in contrast to Dirac’s LNH, two of the presented ratios are not approximate but manifest exact equality. This allows a the- oretical prediction of the Universe’s radius as well as a calculation of the exact value of Newtonian gravitational constant G, which all fall within the range of the current mea- surement data and precision. The author formulates these Large Number Numerical (LNN) coincidences by realizing that further discovery of their meaning may lead to a significant change in our understanding of Nature. In this work, SI units are used. Introduction 1 Cosmological coincidence 40 Many attempts of bringing together physics and numerology The relation is analogous to famous Dirac’s ratio RU /re 10 had been done before but a very important step was done in which relates the Universe radius with classical∼ 1938 by Arthur Eddington. According to Eddington’s pro- electron radius. However, Dirac’s ratio is actually valid only posal the number of protons in the entire Universe should be approximately (with precision of “the same order of mag- exactly equal to: N = 136 2256 1079 [1,2,17]. So, it nitude”), in opposite, the suggested replacement is an exact Edd × ≈ was hypothesised that square root of NEdd should be close to equation given as follows: Dirac’s Big number N √136 2256 = √136 2128. Later ≈ × × R on, Eddington changed 136 to 137 and insisted that the fine U = 2128, (1) structure constant has to be precisely 1/137, and then his the- λe ory seemed to fail at this cornerstone. However, Eddington’s where R is value for the radius of the observable Universe statement also had the number (2128)2 which has been left U and λ = ~/m c 3.86 10 13 (m) is electron’s reduced without proper attention. Actually, few years earlier, in 1929, e e − Compton wavelength≈ (De× Broglie wave). The relation (1) it was German physicist R. F¨urth who proposed to use 1632 provides us with precise size and age of the observable Uni- (which is also 2128) in order to connect gravitation to atomic verse. So it leads to exact value for the Universe radius of constants [10]. However, all these coincidences have been R = 1.314031 1026 meters corresponding to the Universe left unexplained until present time. As G.Gamov said [16]: U age of 13.8896 billion× years. “Since the works of Sir Arthur Eddington, it has become cus- Recently F.M.Sanchez, V.Kotov, C.Bizouard discovered tomary to discuss from time to time the numerical relations that the use of the reduced electron Compton wavelength is between various fundamental constants of nature”. For exam- decisive for the compatibility of the Hubble-Lemaitre length ple, another interesting attempt to use “a log-base-2 relation” with 2128 [13–15]. They use this length unit because of pro- between electromagnetic and gravitational coupling constant posed holographic relation involving it. Here, the author in- was made by Saul-Paul Sirag, the researcher from San Fran- dependentlydevelopesthis idea suggesting that (1) is an exact cisco in 1979 [12]. Particularly, as noted, power of 2 should relation. have significant role in numerical relations for physics con- The age of the Universe, according to the Wilkinson Mi- stants according to the author’s idea. crowave Anisotropy Probe (WMAP) 7-year results, is 13.75 Suggested four Large Number Numerical (LNN) relations 0.13 billion years [9]. Latest NASA observation by Hubble± or coincidences are presented below. These coincidences are gives the age of the Universe as 13.7 billion years [3]. It is not dependent and related to each other, so prove or disprove very close to the obtained value and lies in the existing er- of one of them does not mean the same for the others. They ror range. So, the coincidence (1) seems to define the exact all have common number of 2128. First two relations seem Universe elapsed life time as: to be exact equations, and second two are valid with defined uncertainty. Because of that their nature is more hypothetical, λ T = e 2128. (1.1) so second two relations are also called “weak”. U c Alexander Kritov. A New Large Number Numerical Coincidences 25 Volume2 PROGRESSINPHYSICS April,2013 2 1 Importantto note, that having (1.1), initial Dirac’s relation following: Ee = mc . The factor 2 appears if one pos- may be expressed in the following form: tulates that electromagnetic energy (Eem) of the elec- tron or proton is just a half of particle’s rest mass en- RU 1 128 = 1 2 N1 = = α− 2 , (1.2) ergy as: Eem 2 mc . There are two possible alterna- re tive explanations for this: 1 where α− = 137.036 is inverted fine structure constant and 1. The Virial Theorem that tells us that the potential 2 2 re = ke /(mec ) — classical electron radius with eliminated energy inside a given volume is balanced by the numerical factor (i.e. equal to unity) and N1 is exact value for kinetic energy of matter and equals to half of it. the large number introduced by Dirac (4.66 1040). As we So if one considers electromagnetic energy as ki- × know for sure that the Universe is expanding and RU (t) is de- netic and rest mass as potential energy we would 1 2 pendenton time, so the equation (1) suggestthat oneor few of have: Eem = 2 mc ; the fundamental constants (h, c, me) should also vary in time. 2. Simply assuming that half of total energy may be However, current uncertainty in RU measurement still leaves magnetic energy or of another nature. One may a room for other alternative ideas and possible coincidences. also propose that there could be no 1 in classical 1 2 For example, noting that m /m 40/3 α , relation (1) 129 p e ∼ × − proton radius definition, but there is 2 instead can have another form: of 2129 in formula (2). From the author’s point 2 of view this does not correspond to reality, and mp 1 3 ke 128 RU = 2 (1.3) particularly the number 2128 should have strong m 4 10 m c2 ! e e presence in all numerical expressions of Nature. which would correspond to 13.95809 Gyr. As this value is It can be easily seen that rp = (me/mp)re, so another way currently out of the present WMAP data frame, therefore it is to rewrite (2) is: not supported by the author here. r mp e = 2128. (2.1) rg m 2 Electron-proton radius coincidence e e And this leads to another possible representation of the initial Another interesting idea connects the classical proton radius formula as: and gravitational radius of the electron by an exact equation re = 2128, (2.2) as follows: rgp rp 128 = 2 , (2) where r is classical electron radius, r is gravitational rge e gp (Schwarzschild) radius of the proton. The expression (2.2) 1 3 2 2 where rp = 2 5 ke /(mpc ) — classical proton radius and is very similar to (2). So, we may actually combine them into 2 rgp = 2Gme/c — gravitational electron radius (i.e. the another interesting equation: Schwarzschild radius for the electron mass). Of course some 1 3 comments are required regarding coefficients 2 and 5 . Usu- rp rgp = re rge. (2.2a) ally numerical factors are ignored or assumed to be close to unity when defining classical (electron) radius. However, The precision of the Electron-protoncoincidencegiven by 3 (2)is smaller than 0.02%. From the author’s pointof view this suggested new definition has exact numerical factor 10 = 3 1 deviation originates from current uncertainty in gravitational 5 2 , so it is obvious to have the following explanations for that× one by one: constant (G) measurement. If we consider that the relative G uncertainty nowadays is around and not less than 0.02% Ratio 3 is classical proton radius definition. The only 5 then we must accept this amazing and unexplained coinci- • important difference with modern representation of the dence that allows us to predict the exact value for the grav- classical radius is the coefficient 3 . It is well known 5 itational constant (G). So, this finding suggests that the fol- from electrostatics that the energy required to assem- lowing possible consequences are valid. Firstly, because of ble a sphere of constant charge density of radius r and 3 ratio proton or electron still may be considered as classical charge r is E = 3 ke2/r. Usually these factors like 3 or 5 5 5 particle with uniform charge density inside its radius. And 1 are ignored while defining the classical electron ra- 2 secondly, directly from (2) one can express the value of the dius. Surprisingly, the coincidence advices the use of 3 Newtonian constant of gravitation (G) exactly as follows: 5 which means that charge is equally spread within the sphere of the electron (or proton) radius.
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