To fit Newtonian gravitational constant with microscopic physical constants Satya Seshavatharam U V, Lakshminarayana S, U Seshavatharam, S Lakshminarayana To cite this version: Satya Seshavatharam U V, Lakshminarayana S, U Seshavatharam, S Lakshminarayana. To fit New- tonian gravitational constant with microscopic physical constants . 2017. hal-01657304v2 HAL Id: hal-01657304 https://hal.archives-ouvertes.fr/hal-01657304v2 Preprint submitted on 15 Dec 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. To fit Newtonian gravitational constant with microscopic physical constants U. V. S. Seshavatharam1 and S. Lakshminarayana2 1Hon. Faculty, I-SERVE, S. No-42, Hitex Rd., Hitech city, Hyderabad-84, Telangana, India 2Department of Nuclear Physics, Andhra University, Visakhapatnam-03, AP, India Corresponding Emails: [email protected]; [email protected]; Abstract: By considering two virtual gravitational constants assumed to be associated with electromagnetic and strong interactions, in a theoretical and verifiable approach, we make an attempt to estimate the Newtonian gravitational constant from microscopic elementary physical constants. With respect to weak coupling constant and root mean square radius of proton, estimated value of the Newtonian gravitational constant is 6.67454X10^(-11) m3/kg/sec2 and our estimated value is 6.679856X10^(-11) m3/kg/sec2. Keywords: Final unification; Newtonian gravitational constant; Virtual electromagnetic gravitational constant; Virtual nuclear gravitational constant; 1. Introduction gravitational constant associated with nuclear or strong interaction [1,8,9] The most desirable cases of any unified description are: 11 3 -1 -2 a) To implement gravity in microscopic physics and to 8) GN 6.619384 to 6.679856 10 m kg sec = estimate the magnitude of Newtonian gravitational Estimated (virtual) gravitational constant associated constant. with gravitational interaction b) To develop a model of microscopic quantum gravity. 9) G 1.44021 1062 J.m 3 = Estimated Fermi’s weak c) To simplify the complicated issues of known physics. F coupling constant d) To predict new effects, arising from a combination of the 2 15 fields inherent in the unified description. 10) Gsp m c 0.6196455 10 m = Estimated In this context, in our earlier publication [1] and references characteristic nuclear radius therein, we suggested the role of two new gravitational constants associated with strong and electromagnetic 3. Two basic assumptions of final unification interactions. In this paper, we make a bold attempt to inter- relate the Fermi’s weak coupling constant [2,3] and With reference to our earlier publication [1], we propose the Newtonian gravitational constant [4,5,6] via the two following two modified assumptions. proposed electromagnetic and nuclear gravitational constants. We would like to appeal that, with respect to Assumption-1: In Hydrogen atom, ground state potential String theory models, Quantum gravity models [7] and energy of electron can be given by, proposed assumptions, it is possible to show that, weak interaction is a natural manifestation of microscopic 22 ee quantum gravity [3]. Epot ground 4 Gm2 2 0 ee 40 Gsp m c (1) 2. Nomenclature and magnitudes 2 4 Gm Gmsp Bohr radius, a 0 ee 0 22 1) e 1.602176565 1019 C = Elementary charge ec 27 2) mp 1.672621777 10 kg = Rest mass of proton where, we choose or define, 31 3) me 9.10938291 10 kg = Rest mass of electron 2 mp 40Gmee -34 2 (2) 4) h 2 1.054571726 10 J.sec = Reduced m e2 e Planck’s constant 8 -1 5) c 2.99792458 10 m.sec = Speed of light. 2 c Gs m p m e G e m e 37 3 -1 -2 6) Ge 2.374335472 10 m kg sec = Defined virtual m 2 (3) gravitational constant associated with electromagnetic p e 2 hc G m interaction [1] sp me 40 28 3 -1 -2 7) Gs 3.32956081 10 m kg sec = Estimated virtual 1 1 E) Neutron star mass and radius Note: Considering as a probability of finding n2 ‘electron shell’ in any orbit labeled with n 1,2,3,.. further 1) If MmNS, n represent the masses of neutron star [10] research can be carried out. and neutron, then, Assumption-2: With reference to Planck scale, GN M NS m n G s MMNS 3.175 (9) cGN 6 Gs m p m p G N (4) cc23m Note: By considering , mass of neutron star can be e 2 GN estimated to be 1.5875M .This is just greater than the where = Planck length. 3 c famous Chandrasekhar mass limit of 1.4M . 4. Important and interesting relations 2) If R represents the neutron star radius [11], then, NS A) Rest mass of proton: RG NS s R 8.06 km (10) 1 NS 3 GN 7 Gcs mp Gs m 23 13 e GGN e (5) 5. Fitting Fermi’s weak coupling constant and 1 electron rest mass 6 GN c mm peGG Fitting the gravitational constant with elementary physical eN constants is a very challenging issue. According to G. Rosi c et al [3]: “There is no definitive relationship where = Planck mass. G between G and the other fundamental constants, and N N there is no theoretical prediction for its value, against which B) Nuclear charge radius: to test experimental results. Improving the precision with which we know GN has not only a pure metrological 2Gm interest, but is also important because of the key role R sp 1.239291 1015 m (6) 0 2 c that GN has in theories of gravitation, cosmology, particle physics and astrophysics and in geophysical models”. C) Root mean square radius of proton: In this context, we would like to stress that, by considering the Fermi’s weak coupling constant, in a verifiable approach, it is certainly possible to explore the 2Gmsp 15 Rp 0.8763111 10 m (7) back ground physics of the role of the Newtonian c2 gravitational constant in microscopic physics. It may be This can be compared with the recommended value [2] of noted that, according to Roberto Onofrio [3], 0.8751 0.0061 1015 m. 1) Weak interactions are peculiar manifestations of quantum gravity at the Fermi scale. D) Characteristic atomic radius of Hydrogen atom: 2) Fermi’s weak coupling constant is related with the Newtonian constant of gravitation. 2 G G m 3) At atto-meter scale, Newtonian gravitational constant s e atom 22 Rhydrogen 33 picometers (8) seems to reach a magnitude of 8.205 10 c2 3 -1 -2 m kg sec . where matom is the unified atomic mass, With reference to the proposed assumptions and based on 27 the above points, quantitatively, we noticed that, 1.66054 10 kg. This can be compared with radius of 1 2 hydrogen atom associated with covalent bond. 2 3 2Gmsp G G m22 G m (11) F e p N p 2 (https://en.wikipedia.org/wiki/Covalent_radius) c 2 Based on relation (11), 7. To understand proton’s melting point 3 12 GcF G (12) N 2 6 12 With reference to Hawking black hole temperature formula 64Ge G s m p [12], melting point of proton [13,14] can be understood 3 with: GF GN (13) 3 GG26 c 12 es Tproton 0.15 10 K (21) 62 3 8 kB G s m p If, recommended GF 1.435850781 10 J.m , obtained G 6.619384 1011 m 3 kg -1 sec -2 . Based on this relation and with reference to up quark, other N quark melting points can be expressed with the following kind of relation. With reference to proposed assumptions, GN can be 3 expressed with, mq c Tquark (22) 9 m8 k G m up B s up mGesc 11 3 -1 -2 G 6.679856 10 m kg sec N mG2 pemp where mmq up represents the ratio of mass of any quark (14) to mass of up quark. Based on this relation, for up quark of Based on relations (11) and (14), rest energy 2 MeV, its corresponding Tup 69 Tera K and 28 3 -1 -2 8kTB up 236 MeV. This can be compared with currently If, Gs 3.32956081 10 m kg sec 22 believed QCD energy scale (170 to 270) MeV [14]. 4 Gmse 62 3 GF 1.44021 10 J.m (15) c3 8. To fit neutron’s life time If, G 1.435850781 1062 J.m 3 , Neutron life time [15] can be fitted with: F 3 GcF 28 3 -1 -2 Gs 3.324518 10 m kg sec (16) 2 2 Ge G s m n 4 me t n G 3 N mnp m c Electron rest mass can be fitted with, (23) G G G m2 s e s n 896.8 sec 3 33 G m m c G c1 G c N np m FF (17) e 2 where, m = Rest mass of neutron, 4 Gs 2Gs n Ge 23 5.9645176 10 Avogadro number, N A 16 6. Nuclear Planck mass and its Schwarzschild radius GN With reference to Planck mass, nuclear Planck mass can be By considering the unified atomic mass unit, 27 expressed with: matom 1.66054 10 kg , c 2 mcnpl 546.62 MeV/ (18) G G G G m2 s s e s atom tn 881.5 sec (24) G m m c3 With reference to Schwarzschild radius of a black hole, N np Schwarzschild radius of nuclear Planck mass can be This value can be compared with the recommended value expressed with: [2] and results of bottle experiments [16].