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 1062 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 40 Gsp m c    (1) 2. Nomenclature and magnitudes 2  4 Gm Gmsp Bohr radius, a 0 ee  0 22 1) e 1.602176565 1019 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 40Gmee -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 40  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) cG 6 N Gs m p m p G   N (4)  cc23m Note: By considering , mass of neutron star can be e 2 G where N = Planck length. estimated to be 1.5875M .This is just greater than the 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  1015 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 1015 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 the above points, quantitatively, we noticed that, 1.66054 1027 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 Gc3 12 G  F (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 1011 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 mGesc 11 3 -1 -2 G  6.679856  10 m kg sec N mG2 pemp 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 8kTB 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 1062 J.m 3 , F Neutron life time [15] can be fitted with: 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]. 2Gm 2GGc 9. Understanding nuclear stability and binding energy R s npl ss 2  0.722 fm (19) npl 2 2 3 c cGs c a) Proton-Neutron stability: Based on relations (15, 16 and 17), Let,

3 Gs m p m e c 3 GFF c1 G s   1.604637101  10 (25) me  (20) 2 2 c Gmee 4 Gs RGnpl s 

3

Using s , proton-neutron stability relation can be fitted 22 Nss Z kZA directly in the following way [17].  (29) 33Z  2 2 kZAs As 2 Z  s 2 Z  2 Z   4 s Z (30) (26) BAA s  9.5 MeV s 3 2ZZ 0.00641855 2 Based on this strange and simple relation and first four

terms of the semi empirical mass formula (SEMF), close to where As is the estimated stable mass number of Z. It is the beta stability line, for Z  2 to 100 , it is possible to very interesting to note that, close to beta stability line,   show that, considering 4sk  0.00641855, nuclear binding energy

[18,19] can be estimated very easily. One very interesting kA N Z observation is that, 13 ss (31) BAAB ss   1  0  10.09 MeV As 3.40  mmnp 11 For relation (31), see figure 1 (dashed red curve) for the exp   4 (27) m4 s k estimated binding energy per nucleon close to the beta e stability line of Z= 2 to 100 compared with first four terms of the SEMF (Green curve) where: a 15.77 MeV, b) Nuclear binding energy: v as 18.34 MeV, aa  23.21 MeV and ac  0.71 MeV.  being the strong coupling constant [2], characteristic s Figure 1: Binding energy per nucleon close to beta stability line nuclear binding energy potential can be expressed with the from Z= 2 to 100 following relation [19,20]. 10,0 8,0 11 e2   e 2 c 2 B    10.09 MeV 6,0 0     s 48 0R 0   s   0 G s m p 4,0 (28) 2,0 Note: With reference to , it is possible to consider, 0,0

2 nucleon(MeV) 2 12 22 32 42 52 62 72 82 92 2 2 Gmsp 11 Gm energyBinding per  . Considering sp 2.9464, Proton number c s 0.1152 c   it is possible to define the existence of a strongly interacting 10. Discussion and conclusion 2 Gm nuclear elementary charge of magnitude, ee sp . s c We appeal that,  2 a) We presented a number of applications connecting es Based on this idea,  2 and micro-macro physical systems and finally developed 4 G m m 0 s p e arithmetic relations for understanding the role of the ec22 Newtonian gravitational constant [22] in microscopic s  10.09 MeV. Proton and neutron magnetic  physics. Following this kind of computational approach, 80Gmsp  it is certainly possible to reproduce another set of es arithmetic relations by using which, in near future; in a diploe moments can be addressed as,  p  and verifiable approach, it may be possible to find a set of 2mp absolute relations and GN can be estimated. For ee  s  example, based on the recommended values of n  respectively. For details, readers are 2mn 62 3 Rp  0.8751 fm and GF 1.435850984 10 J.m , encouraged to see our preprint [21].  With reference to relations (2, 3 and 5 or 14), 11 3 -1 -2 Based on the new integrated model proposed by GN 6.679856 10 m kg sec . N. Ghahramany et al [18] and with reference to relation  With reference to relations (2, 7 and 14), 11 3 -1 -2 (26), for Z  40 to 83 , close to the beta stability line, if GN 6.6706246 10 m kg sec .

 AZNss , we noticed that,

4

 With reference to relations (2, 14 and 16), [3] Roberto Onofrio. On Weak Interactions as Short- Distance G 6.6697396 1011 m 3 kg -1 sec -2 . Manifestations of Gravity. Modern Physics Letters A, Vol. N 28, No. 7, 1350022 (2013)  With reference to relations (2, 5, and 7), [4] G. Rosi, F. Sorrentino, L. Cacciapuoti, M. Prevedelli and G. 11 3 -1 -2 M. Tino1. Precision measurement of the Newtonian GN 6.6660136 10 m kg sec . gravitational constant using cold atoms. Nature 510, 518-521.  With reference to relations (2, 5 and 16), (2014) 11 3 -1 -2 GN 6.664687 10 m kg sec . [5] S. Schlamminger and R.D. Newman. Recent measurements of  With reference to relations (2, 3 and 12), the gravitational constant as a function of time. Phys. Rev. D 11 3 -1 -2 91, 121101 (2015) GN 6.619384 10 m kg sec . [6] Ideas Lab: Measuring “Big G” Challenge.  With reference to relations (2, 7 and 12), https://www.nsf.gov/pubs/2015/nsf15591/nsf15591.htm. 11 3 -1 -2 [7] K. Becker, M. Becker and J. H. Schwarz. String Theory and GN 6.674538 10 m kg sec . th th M-theory: A Modern Introduction. Cambridge University b) The ultimate aim of ‘Ideas Lab’ (16 May to 26 Press, (2006) October 2016) [6] organized by the Physics Division of [8] Salam A, Sivaram C. Strong Gravity Approach to QCD and the Mathematical and Physical Sciences Directorate at Confinement. Mod. Phys. Lett., v. A8(4), 321–326. (1993) the National Science Foundation (NSF), was to [9] O. F. Akinto, Farida Tahir. Strong Gravity Approach to QCD facilitate the development of new experiments designed and General Relativity. arXiv:1606.06963v3 [10] N. Chamel et al. On the Maximum Mass of Neutron Stars. Int. to measure Newton’s gravitational constant GN with J. Mod. Phys. E22 (2013) 1330018 relative uncertainties approaching or surpassing one [11] Sebastien Guillot et al. Measurement of the Radius of Neutron part in 100,000. In this context, we humbly and Stars with High S/N Quiescent Low-mass X-ray Binaries in sincerely request NSF to consider and encourage our Globular Clusters. Astrophys.J. 772 (2013). proposed method of estimating the Newtonian [12] S. Hawking, Particle Creation by Black Hole, Commun. Math. Phys. 43, 199-220(1975) gravitational constant with possible support. [13] Helmut Satz. The Quark-Gluon Plasma.Nucl.Phys.A862- c) As it is inevitable to unite gravity and other three 863:4-12,2011. atomic interactions, if one is willing to explore the [14] Horst Stoecker et al. Glueballs amass at RHIC and LHC possibility of incorporating the proposed assumptions Colliders! - The early quarkless 1st order phase transition either in String theory models or in Quantum gravity at T=270 MeV - from pure Yang-Mills glue plasma to models, certainly, background physics assumed to be GlueBall-Hagedorn states. J. Phys. G 43 (2016) 1, 015105. connected with proposed semi empirical relations can [15] C. L. Morris et al. A new method for measuring the neutron lifetime using an in situ neutron detector. Report number: be understood and in near future, a ‘workable’ or LAUR-16-27352. https://arxiv:1610.04560. ‘practical’ model of “everything” can be developed. [16] B. Andreas et al. An accurate determination of the Avogadro Based on relations (11) to (14), Fermi’s weak coupling constant by counting the atoms in a 28Si crystal. Phys. Rev. constant and the three gravitational constants can be Lett. 106, 030801. fitted in a unified approach and finally, in a verifiable [17] Chowdhury, P.R. et al. Modified Bethe-Weizsacker mass approach, Newtonian gravitational constant can be formula with isotonic shift and new driplines. Mod. Phys. Lett. A20 p.1605-1618. (2005). estimated accurately with microscopic physical [18] N.Ghahramany et al. New approach to nuclear binding energy constants. in integrated nuclear model. Journal of Theoretical and Applied Physics 6:3 (2012). Acknowledgements [19] U. V. S Seshavatharam, S. Lakshminarayana. On the role of strong coupling constant and nucleons in understanding Author Seshavatharam U.V.S is indebted to professors brahmasri nuclear stability and binding energy. Journal of Nuclear Dr. M. Nagaphani Sarma, Chairman, shri K.V. Krishna Murthy, Sciences. (2017); 4(1): 7-18. former chairman, Institute of Scientific Research in Vedas [20] Seshavatharam, U.V.S.; Lakshminarayana, S.To unite nuclear (I-SERVE), Hyderabad, India and Shri K.V.R.S. Murthy, former and sub-nuclear strong interactions. International Journal of scientist IICT (CSIR), Govt. of India, Director, Research and Physical Research, 5 (2) (2017) 104-108. Development, I-SERVE, for their valuable guidance and great [21] Seshavatharam, U.V.S.; Lakshminarayana, S. To Develop a support in developing this subject. Virtual Model of Microscopic Quantum Gravity. Preprints 2017,2017110119, 23 pages. References (doi: 10.20944/preprints201711.0119.v1). [22] Brandenburg J. E. The GEM Unification Theory of the Vacuum: Did Dimensional Collapse Trigger the Big Bang. [1] U. V. S. Seshavatharam and S. Lakshminarayana. Towards a International Journal of Astrophysics and Space Science. workable model of final unification. International Journal of Special Issue: Quantum Vacuum, Fundamental Arena of the Mathematics and Physics Vol. 7, No. 1, 117 (2016). Al-Farabi Universe: Models, Applications and Perspectives. Vol. 2, No. Kazakh National University. 6-1, pp. 24-38. (2014). [2] C. Patrignani et al. (Particle Data Group), Chin. Phys. C, 40, 100001 (2016) and 2017 update

5