Three Fundamental Masses Derived by Dimensional Analysis

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1 Amerian Journal of Spae Siene (): 4-49, 0 ISSN: Siene Publiations doi:0.844/ajssp Published Online () 0 ( Three Fundamental Masses Derived by Dimensional Analysis Dimitar Valev Department of Stara Zagora, Spae Researh and Tehnology Institute, Bulgarian Aademy of Sienes, P.O. Box 7, 6000 Stara Zagora, Bulgaria Reeived 0-0-9; Revised 0-0-0; Aepted ABSTRACT Three new mass dimension quantities have been derived by dimensional analysis, in addition to the famous Plank mass m P ~0 8 kg. These masses have been derived by means of fundamental onstants-the speed of light (), the gravitational onstant (), the Plank onstant (ħ) and the Hubble onstant (H). The enormous mass m ~0 kg pratially oinides with the Hoyle-Carvalho formula for the mass of the observable universe. The extremely small mass m ~0 ev has been identified with the minimum quantum of energy, whih seems lose to the graviton mass. It is noteworthy that the Plank mass appears geometri mean of the masses m and m. The mass m ~0 7 ev ould not be unambiguously identified at present time. Besides, the order of magnitude of the total density of the universe has been estimated by this approah. Keywords: Plank Mass, Dimensional Analysis, Mass of The Universe, Minimum Quantum of Energy Siene Publiations. INTRODUCTION h The Plank mass m P ~ has been introdued from Plank (99) by means of three fundamental onstantsthe speed of light in vauum (), the gravitational onstant () and the redued Plank onstant (ħ). Sine the onstants, and ħ represent three very basi aspets of the universe (i.e., the relativisti, gravitational and quantum phenomena), the Plank mass appears to a ertain degree a unifiation of these phenomena. The Plank mass have many important aspets in modern physis. One of them is that the energy equivalent of h Plank mass EP = mp ~ ~0 9 ev appears unifiation energy of the fundamental interations (eorgi et al., 974). Also, the Plank mass an be approximately derived by setting it as a mass, whose Compton wavelength and Shwartzhild radius are equal (Bergmann, 99). The Plank mass formula has been derived by dimensional analysis using fundamental onstants, and ħ. The dimensional analysis is a oneptual tool often applied in physis to understand physial situations 4 involving ertain physial quantities (Bridgman, 9; Kurth, 97; Bhaskar and Nigam, 990; Petty, 00). It is routinely used to hek the plausibility of the derived equations and omputations. When it is known, the ertain quantity with whih other determinative quantities would be onneted, but the form of this onnetion is unknown, a dimensional equation is omposed for its finding. In the left side of the equation, the unit of this quantity q 0 with its dimensional exponent has been plaed. In the right side of the equation, the produt of units of the determinative quantities q i rise to n [ ] n i the unknown exponents n i has been plaed [q ]~ q 0 i i= where n is positive integer and the exponents n i are rational numbers. Most often, the dimensional analysis is applied in the mehanis and other fields of the modern physis, where there are many problems with a few determinative quantities. Many interesting and important problems related to the fundamental onstants have been onsidered from (Levy-Leblond, 977; Duff, 00; Duff et al., 00; Barrow, 00; Fritzsh and Stodolsky, 009). The disovery of the linear relationship between reessional veloity of distant galaxies and distane v =

2 Dimitar Valev / Amerian Journal of Spae Siene (): 4-49, 0 Hr from Hubble (99) introdues new fundamental onstant in physis and osmology-the famous Hubble onstant (H). Even seven years before, Friedman (9) derived his equations from the Einstein (96) field equations, showing that the universe might expand at a rate alulable by the equations. Hubble onstant determines the age of the universe H, the Hubble distane H, the ritial density of the universe H (Peebles, 97) and other large-sale ρ = π 8 properties of the universe. Beause of the importane of the Hubble onstant, in the present paper we inlude H in the dimensional analysis together with, and h aiming to find the new mass dimension quantities Siene Publiations m ~ i nj qj, where every triad j= q,q,q onsists of three onstants,, h and H. Thus, the Hubble onstant will represent the osmologial phenomena in new derived fundamental masses. Aording to the reent osmology, the Hubble onstant slowly dereases with the age of the universe, but there are indiations that other onstants, espeially gravitational and fine struture onstants also vary with omparable rate during the expansion (Dira, 97; Wu and Wang, 986; Webb et al., 00). That is why, the Hubble onstant ould deserve being treated on an equal level with the other three onstants used by Plank.. THREE FUNDAMENTAL MASSES DERIVED BY DIMENSIONAL ANALYSIS Below, we obtain a mass dimension quantity m onstruted from the fundamental onstants-the speed of light (), the ravitational onstant () and the Hubble onstant (H) using dimensional analysis. A quantity m having mass dimension ould be onstruted by means of the fundamental onstants, and H: m k H n n n = () where, n, n and n are unknown exponents to be determined by mathing the dimensions of both sides of the equation and k is dimensionless parameter of an order of magnitude of a unit. As a result we find the system of linear Equation : n + n = 0 n n n = 0 n = () 46 The unique solution of the system is n =, n =-, n = -. Replaing obtained values of the exponents in Equation we find formula () for the mass m Equation : m ~ () H First of all, the formula () has been derived by dimensional analysis from Valev (009). This formula pratially oinides with the Hoyle formula for the mass of the observable universe M= (Kragh, 999) and H perfetly oinides with Carvalho (99) formula for the mass of the observable universe, obtained by totally different approah. Evidently, the Hoyle formula oinides with the mass of the Hubble sphere M H, i.e., mass of the sphere having radius equal to the Hubble distane H and density equal to the total density of the universe ρ ρ Equation 4: M 4 H = π = H 8 π H H (4) The reent experimental values of, and H are used from Mohr and Taylor (999): = m s, = m kg s and H 70 km s Mps from Mould et al. (000). Replaing this values in () we obtain m ~.76 0 kg. Therefore, the enormous mass m would be identified with the mass of the observable universe. Analogously, by means of the fundamental onstants, h and H, a quantity m having dimension of a mass ould be onstruted: m k n n n = h H () We determine the exponents n =-, n =, n = by the dimensional analysis again. Replaing the obtained values of the exponents in Equation we find formula (6) for the mass m Equation 6: m ~ Replaing the reent values of the onstants, h and H in (6) we obtain m ~ kg =. 0 ev. This exeptionally small mass oinides with the minimal measurable gravitational self energy of a partile (Sivaram, 98) whih is aepted as minimum quantum of energy h H~0 ev from Alfonso-Faus (6)

3 Dimitar Valev / Amerian Journal of Spae Siene (): 4-49, 0 (0; Alfonso-Faus et al., 0). This quantity takes substantial plae in the estimations of total information and entropy of the universe (kigkitzis et al., 0; Haranas and kigkitzis, 0). Thus, the mass m seems lose to the graviton mass obtained by different methods (Woodward et al., 97; ershtein et al., 997; Valev, 00; Alves et al., 009). The mass m is in several orders of magnitude smaller than the upper limit of the graviton mass, obtained by astrophysial onstraints from (oldhaber and Nietto, 974). From Equation 6 we find that the redued Compton wavelength D of this mass is equal to the Hubble distane H Equation 7 and 8: h 6 D = = H ~. 0 m (7) m From formulae () and (6) we find an interesting relation (8): H m m h = = H h mp = Siene Publiations kg (8) Therefore, the Plank mass appears geometri mean of the Hubble mass and the mass of the observable universe. As the physial quantity mass is among the most important properties of the matter, the formula (8) hints at a deep relation of the miro partiles and the entire universe. Besides, the ratios (9) take plae Equation 9: m mp H H = = = = m m r t P P P ~ h where r P = is the Plank length, tp is the Plank time, H is the Hubble distane and H is the Hubble time. The third quantity m, having mass dimension ould be onstruted by means of the fundamental onstants, ħ and H: m k n n n = h H (0) (9) We determine the exponents n =,n =,n = by dimensional analysis again. Replaing the obtained 47 values of the exponents in Equation 0 we find formula () for the mass m Equation : Hh m ~ () Replaing the reent values of the onstants, ħ and H, the mass m takes value m ~ kg ev. This mass is a dozen of orders of magnitude lighter than the Plank mass and several orders of magnitude heavier than the heaviest known partiles like the top quark m t 74. ev (Mangano and Trippe, 000). On the other hand, the energy m ~ ev appears medial for the important UT sale E UT ~ 0 6 ev and eletroweak sale E EW ~0 ev. Therefore, the mass/energy m ould not be unambiguously identified at the present time and it ould be onsidered as heuristi predition of the suggested approah. Below, we demonstrate the heuristi power of the suggested approah approximately estimating the total density of the universe by dimensional analysis. Atually, a quantity ρ having dimension of density ould be onstruted by means of the fundamental onstants, and H Equation : n n n ρ= k H () where, k is a dimensionless parameter of the order of magnitude of unit. By the dimensional analysis, we have found the exponents n = 0, n =, n =. Therefore Equation : H ρ ~ kg m () The reent Cosmi Mirowave Bakground (CMB) observations show that the total density of the universe ρ is (Balbi et al., 000; De Bernardis et al., 000; Spergel et al., 00): H a ρ=ωρ ρ = ~ 0-6 kg m - (4) 8 π Evidently, the density ρ derived by means of the fundamental onstants, and H oinides with formula (4) for the total density of the universe with an auray of a dimensionless parameter of an order of magnitude of a unit. Besides, the formula () ould be derived by means of other triad of fundamental onstants, namely, h and H.

4 Dimitar Valev / Amerian Journal of Spae Siene (): 4-49, 0 Siene Publiations. CONCLUSION Three new mass dimension quantities m i have been derived by dimensional analysis, in addition to the h Plank mass m P ~ ~ kg. Four fundamental onstants-the speed of light in vauum (), the gravitational onstant (), the redued Plank onstant (h ) and the Hubble onstant (H) have been involved in the dimensional analysis. The first derived mass dimension quantity m ~ ~0 kg pratially H oinides with the Hoyle-Carvalho formula for the mass of the universe obtained by totally different approah. The exeptionally small mass dimension quantity m ~ ~0 ev has been identified with the minimum quantum of energy, whih seems lose to the graviton mass. It is amazing that the Plank mass appears geometri mean of the masses m and m, i.e., mp = mm. The third derived mass Hh m ~ ~ 0 7 ev ould not be identified unambiguously at present time. The identifiation of the two derived masses reinfores the trust in the suggested approah. Aording to the big bang osmology, the Hubble onstant dereases with the age of the universe. Therefore, the mass of the universe m ~ H inreases, whereas the Hubble mass m ~ and mass Hh m ~ derease with time. Nevertheless, the Plank mass remains a geometri mean of the Hubble mass and mass of the observable universe. 4. REFERENCES Alfonso-Faus, A., 0. Universality of the self gravitational potential energy of any fundamental partile. Astrophys. Spae Si., 7: 6-6. DOI: 0.007/s x Alfonso-Faus, A., M.J. Fullana and I. Alfonso, 0. Cosmi Bakground Bose Condensation (CBBC). Astrophys. Spae Si., 47: DOI: 0.007/s Alves, M.E.S., O.D. Miranda and J.C.N. De Araujo, 009. Can massive gravitons be an alternative to dark energy. Phys. Lett. B, 700: DOI: 0.06/j.physletb Balbi, A., P. Ade, J. Bok, J. Borrill and A. Bosaler et al., 000. onstraints on osmologial parameters from MAXIMA-. Astrophys. J., 4: L-L4. DOI: 0.086/7 Barrow, J.D., 00. The Constants of Nature: From Alpha to Omega. st Edn., Jonathan Cape, London, ISBN-0: 0748, pp:. Bergmann, P.., 99. The Riddle of ravitation. st Edn., Dover Publiations, New York, ISBN-0: , pp: 4. De Bernardis, P., P.A.R. Ade, J.J. Bok, J.R. Bond and J. Borrill et al., 000. A flat Universe from highresolution maps of the osmi mirowave bakground radiation. Nature, 404: DOI: 0.08/000 Bhaskar, R. and A. Nigam, 990. Qualitative physis using dimensional analysis. Artifiial Intelligene, 4: 7-. DOI: 0.06/ (90)9008- Bridgman, P.W., 9. Dimensional Analysis. st Edn.,Yale Univ. Press, Yale, ISBN-0: , pp: 8. Carvalho, J.C., 99. Derivation of the mass of the observable universe. Int. J. Theor. Phys., 4: DOI: 0.007/BF Dira, P.A.M., 97. The osmologial onstants. Nature, 9: -. DOI: 0.08/9a0 Duff, M.J., 00. Comment on time-variation of fundamental onstants. Duff, M.J., L.B. Okun and. Veneziano, 00. Trialogue on the number of fundamental onstants. J. High Energy Phys. DOI: 0.088/6-6708/00/0/0 Einstein, A., 96. Die grundlage der allgemeinen Relativitätstheorie. Annalen Der Physik, 4: DOI: 0.00/andp Friedman, A., 9. Uber die krummung des raumes. Z. Physik, 0: DOI: 0.007/BF080 Fritzsh, H. and. Stodolsky, 009. The Fundamental Constants, a Mystery of Physis. st Edn., World Sientifi Publishing Company, Singapore, ISBN- 0: , pp: 4.

5 Dimitar Valev / Amerian Journal of Spae Siene (): 4-49, 0 eorgi, H., H.R. Quinn and S. Weinberg, 974. Hierarhy of interations in unified gauge theories. Phys. Rev. Lett., : DOI: 0.0/PhysRevLett..4 ershtein, S.S., A.A. Logunov and M.A. Mestvirishvili, 997. The upper limit on the graviton mass. kigkitzis, I., I. Haranas and S. Kirk, 0. Number of information and its relation to the osmologial onstant resulting from Landauer s priniple. Astrophys. Spae Si. DOI: 0.007/s oldhaber, A.S. and M.M. Nieto, 974. Mass of the graviton. Phys. Rev. D, 9: 9-. DOI: 0.0/PhysRevD.9.9 Haranas, I. and I. kigkitzis, 0. Bekenstein bound of information number N and its relation to osmologial parameters in a universe with and without osmologial onstant. Mod. Phys. Lett. A, 8: DOI: 0.4/S Hubble, E., 99. A relation between distane and radial veloity among extra-galati nebulae. Pro. Nat. Aad. Si., : DOI: 0.07/pnas...68 Kragh, H., 999. Cosmology and Controversy: The Historial Development of Two Theories of the Universe. st Edn., Prineton University Press, Prineton, ISBN-0: X, pp:. Kurth, R., 97. Dimensional Analysis and roup Theory in Astrophysis. st Edn., Pergamon Press, Oxford, ISBN-0: , pp:. Levy-Leblond, J.M., 977. On the oneptual nature of the physial onstants. Riv. Nuovo Cim., 7: DOI: 0.007/BF Mangano, M. and T. Trippe, 000. The top quark. Europ. Phys. J. C, : 8-9. DOI: 0.007/BF0684 Mohr, P. and B. Taylor, 999. CODATA reommended values of the fundamental physial onstants 998. J. Phys. Chem. Ref. Data, 8: 7-8. DOI: 0.06/.6049 Mould, J.R., J.P. HuhraWendy, L. FreedmanLaura Ferrarese and J.A. raham et al., 000. The hubble spae telesope key projet on the extragalati distane sale. XXVIII. Combining the onstraints on the hubble onstant. Astrophys. J., 9: DOI: 0.086/0804 Peebles, P.J., 97. Physial Cosmology. Prineton Univ. st Edn., Press, Prineton, ISBN-0: , pp: 96. Petty,.W., 00. Automated omputation and onsisteny heking of physial dimensions and units in sientifi programs. Software Pratie Experiene, : DOI: 0.00/spe.40 Plank, M., 99. The Theory of Heat Radiation. st Edn., Dover Publiations, New York, ISBN-0: 484, pp: 4. Sivaram, C., 98. Cosmologial and quantum onstraint on partile masses. Am. J. Phys., 0: DOI: 0.9/.870 Spergel, D.N., L. VerdeH. V. PeirisE. Komatsu and M.R. Nolta et al., 00. First-year Wilkinson Mirowave Anisotropy Probe (WMAP) observations: Determination of osmologial parameters. Astrophys. J. Suppl. Series, 48: DOI: 0.086/776 Valev, D., 00. Neutrino and graviton mass estimations by a phenomenologial approah. Valev, D., 009. Determination of total mehanial energy of the universe within the framework of Newtonian mehanis. Webb, J.K., M.T. Murphy, V.V. Flambaum, V.A. Dzuba and J.D. Barrow et al., 00. Further evidene for osmologial evolution of the fine struture onstant. Phys. Rev. Lett., 87: DOI: 0.0/PhysRevLett Woodward, J.F., R.J. Crowley and W. Yourgrau, 97. Mah's priniple and the rest mass of the graviton. Phys. Rev. D, : DOI: 0.0/PhysRevD..7 Wu, Y. and Z. Wang, 986. Time variation of Newton's gravitational onstant in superstring theories. Phys. Rev. Lett., 7: DOI: 0.0/PhysRevLett Siene Publiations 49