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1 Chapte 3 Is Gavitation A Results Of Asymmetic Coulomb Chage Inteactions? Jounal of Undegaduate Reseach èjurè Univesity of Utah è1992è, Vol. 3, No. 1, pp. 56í61. Jeæey F. Gold Depatment of Physics, Depatment of Mathematics Univesity of Utah Abstact Many attempts have been made to equate gavitational foces with manifestations of othe phenomena. In these emaks we exploe the consequences of fomulating gavitational foces as asymmetic Coulomb chage inteactions. This is contay to some established theoies, fo the model pedicts diæeential acceleations dependent on the elemental composition of the test mass. The pedicted diæeentials of acceleation of vaious elemental masses ae compaed to those diæeentials that have been obtained expeimentally. Although the model tuns out to fail, the constuction of this model is a useful intellectual and pedagogical execise. 1
2 CHAPTER 3. ASYMMETRIC COULOMB CHARGE INTERACTIONS 2 Intoduction The similaities in the expessions fo Newtonian gavitation and Coulomb electostatic inteactions, ç ~Fg =,G m1m2 ^ ~F 2 c = k q1q2 ^ 2 ae appaent, paticulaly the invesesquae natue of both foces and the binay poduct of the masses and chages. Based on this similaity, and the physicists' goal to unify all fundamental foces of natue, it is easonable to conjectue that gavitation is some manifestation of electostatic phenomena. With these consideations in mind, we constuct a simple gavity model; howeve, the model fails. The model pedicts diæeential acceleations dependent on the elemental composition of the test mass. This is in keeping with vaious FifthFoce èhypechageè hypotheses and eevaluations of the Eíotvíos expeiments which postulate the existence of nonnewtonian and bayondependent foces ove shot anges. Such hypotheses ae pesently contovesial, at best. T.M. Niebaue et al. have shown that the diæeential acceleation æg, of two diæeent mateials, is less than 5 pats in of the gavitational acceleation g = cmès 2. Expeiments veifying the Equivalence Pinciple have been pefomed by Dicke et al., demonstating that gavitational mass and inetial mass ae equivalent to within 1 pat in These expeiments ule out Fifth Foce hypotheses as well as ou simple gavity model. Ou model will show that the calculated diæeential acceleation æg calc, fo the same mateials, is on the ode of 8.37 cmès 2, in clea contadiction to the limits established by the expeiments of Niebaue et al. The oiginal motivation fo this fomulation of gavitation was to calculate the appoximate change æk, of the electostatic constant k = 1è4çæ 0 = 8:9876 æ10 9 Nm 2 èc 2, in an asymmetic Coulomb chage inteaction, necessay to account fo gavitational foces. By this we mean that foces between like chages ae expessed as ~F 1;2 = k q 1q 2 ^ ; è3.1è 2 while foces between unlike chages ae expessed as ~F 1;2 =èk + ækè q 1q 2 2 ^ ; è3.2è with æk é 0. We will show that æk=k is on the ode of 10,37. The idea of expessing gavity as a Coulomb inteaction is not new; it had been exploed by Wilhelm Webe è1804í1891è of Gíottingen and Fiedich Zíollne è1834í1882è of Leipzig. Because Webe and Zíollne pefomed thei ëgedanken" expeiment befoe the discovey of atomic paticles, they wee unable to foesee the consequences of combining chages in a densely packed nucleus. In these emaks we takeinto account the mass defects due to the binding enegies of nuclei.
3 CHAPTER 3. ASYMMETRIC COULOMB CHARGE INTERACTIONS 3 Asymmetic Coulomb Chage Inteactions Suppose two spheical masses, m 1 and m 2, ae sepaated by a distance connecting thei centes of mass. Each massm i is composed of p i positive chages and e i negative chages. Let F ++ denote the foce between the positive chages of mass m 1 and the positive chages of mass m 2 ; then the thee emaining foces F +,, F,+, and F,, ae labelled appopiately to epesent thetype of inteaction between the chages of m 1 and m 2. We expess the positive fundamental chage as q and negative fundamental chage as,q in ode to maintain the pope oientation of these cental foces ë1ë. The foces on one of the masses ae 8 ~F ++ =,k q2 p 1p 2 ^ é ~F,, 2 =,k q2 e 1e 2 ^ 2 ~F +, = èk + ækè é: q2 p 1e 2 è3.3è ^ 2 ~F,+ = èk + ækè q2 e 1p 2 ^ : 2 If we combine these foces ë2ë, ~F tot =,k q2 èp 1 p 2 + e 1 e 2 è 2 o, upon eaanging the tems, we ænd that ~F tot =,k q2 èp 1, e 1 èèp 2, e 2 è 2 ^ +èk + ækè q2 èp 1 e 2 + e 1 p 2 è 2 ^ ; è3.4è ^ + æk q2 èp 1 e 2 + e 1 p 2 è 2 ^ : è3.5è An examination of the æst tem of equation è3.5è eveals that it is a efomulation of the Coulomb foce equation. Note that this tem vanishes if eithe mass is electically neutal. The last tem, æk q2 èp 1 e 2 + e 1 p 2 è 2 ^ ; è3.6è does not change sign, egadless of the p i and e i ; like gavity, itis always attactive. Next we wish to ænd what the p i and e i ae fo each mass. This is simpliæed if we assume that m 1 and m 2 ae electically neutal masses. This implies that e 1 = p 1 and e 2 = p 2, and that ~F =2æk q2 p 1 p 2 2 ^ : è3.7è When chages ae combined, a vey small faction of the constituent paticle masses is lost to the binding enegy; this is efeed to as a mass defect. We
4 CHAPTER 3. ASYMMETRIC COULOMB CHARGE INTERACTIONS 4 denote the actual mass of an atom by m i, and its mass defect by B i. elationships of these quantities to the constituent paticle masses ae m 1 + B 1 = Z 1 èm p + m e è+n 1 m n m 2 + B 2 = Z 2 èm p + m e è+n 2 m n : The è3.8è Hee Z i epesents the numbe of potons, and n i epesents the numbe of neutons contained in the electically neutal mass m i ;thequantities m p, m e, and m n epesent the masses of potons, electons, and neutons, espectively. It is cucial to ou model that each neuton is composed of one positive and one negative chage as indicated in the disintegation: n! p + e, + ç e,. This implies that p 1 = Z 1 + n 1 è3.9è p 2 = Z 2 + n 2 : The numbe of neutons can be expessed as a function of the numbe of potons ë3ë, that is, n i = ç i Z i, whee ç i =èa i, Z i è=z i. Substituting ç 1 Z 1 fo n 1 and ç 2 Z 2 fo n 2 in equations è3.9è, we ænd p 1 = Z 1 è1 + ç 1 è p 2 = Z 2 è1 + ç 2 è : è3.10è By eplacing n i with ç i Z i in equations è3.8è, we have m 1 + B 1 = Z 1 èm p + m e + ç 1 m n è m 2 + B 2 = Z 2 èm p + m e + ç 2 m n è : è3.11è Solving fo Z i in equations è3.11è, and substituting these, in tun, into equations è3.10è, we have p 1 = èm1+b1èè1+ç1è èm p+m e+ç 1m nè è3.12è p 2 = èm2+b2èè1+ç2è èm : p+m e+ç 2m nè Finally, equation è3.7è becomes ~F = èm p + m e è 2 ç è1 + ç1 èè1 + ç 2 èè1 + ç 1 èè1 + ç 2 è è1 + ç 1 æèè1 + ç 2 æè whee ç i = Bi m i, ç i = Ni,Zi Z i,andæ= mn m p+m e =1: ç m1 m 2 2 ^ ; è3.13è If we assume the binding enegies ae negligible, that is, ç i =0,and that æ ç 1, then equation è3.13è becomes ~F = èm p + m e è 2 m 1 m 2 2 ^ : è3.14è
5 CHAPTER 3. ASYMMETRIC COULOMB CHARGE INTERACTIONS 5 This allows us to equate the constant, G =6:672 æ 10,11 Nm 2 èkg 2,toænd èm p+m eè 2, with the gavitational constant, æk = G æ èm p + m e è 2,27 Nm2 =3:639 æ 10 : è3.15è 2q 2 C 2 Dividing this value by k, we conclude that ækèk ç 10,37. Diæeential Acceleations Unlike Newtonian o Einsteinian gavitation, ou model pedicts small diæeences in the acceleations of diæeent test masses. These elative diæeences in g ae dependent on the mass defects B i, and the neutonèpoton atios, ç i. The foce exeted by the eath, M eath, on a test mass m 1,is ~F = èm p + m e è 2 ç è1 + çeath èè1 + ç 1 èè1 + ç eath èè1 + ç 1 è ç Meath m 1 2 ^ : è1 + ç eath æè è1 + ç 1 æè è3.16è If anothe test mass, m 2, of diæeing composition, is acceleating unde the same conditions, the atio of the acceleations of masses m 1 and m 2 is j~a 1 j j~a 2 j = ç çç 1+ç1 1+ç1 1+ç 2 1+ç 2 çç ç 1+ç2 æ 1+ç 1 æ : è3.17è In the feefall expeiment ofniebaueet al., the test masses epesented extemes in elemental composition. Onetestmasswas 40.0 gams of coppe è 63 29Cuè and the othe was gams of depleted Uanium è Uè. The following table fo ç i and ç i is deived fom these data ë4ë: Table I. Coppe è 63 29Cuè Uanium è 238 test mass èmè 40.0 gm gm atomic mass èm a è amu amu ç ç Uè Substituting the values of Table èiè into equation è3.17è, we ænd j~a 1 j j~a 2 j =1:00854 ; è3.18è so that æg calc =g = Next we constuct the following table fo the elements aluminum èalè and gold èauè used in the expeiments of Dicke et al.
6 CHAPTER 3. ASYMMETRIC COULOMB CHARGE INTERACTIONS 6 Table II. Aluminum è 27 13Alè Gold è Auè atomic mass èm a è amu amu ç ç The diæeential acceleation of aluminum and gold pedicted by ou model is æg calc èg = Both values of æg calc, namely cmès 2 and cmès 2, fa exceed the limit æg é4:9033 æ 10,7 cmès 2 expeimentally found by Niebaue et al. Conclusion The model pedicts diæeential acceleations that cannot be econciled with those obtained by expeiment. This does not peclude some elation o uniæcation of gavitation and electomagnetism, but it does show that the simple model of Coulomb asymmety is not viable. Acknowledgements I thank Pofesso Richad H. Pice fo the immense assistance povided in the pepaation of this manuscipt. Thanks ae also extended to Vince Fedeick fo his assistance in ænding vaious bibliogaphic souces. Special thanks ae extended to Pofesso Don H. Tucke fo the suppot I have eceived fo this and many othe pojects. Notes 1. The magnitudes of the fundamental positive and negative chages ae equivalent to within 1 pat in Note that we have not taken into account any Coulombic shielding that would occu. 3. Typical values of ç =èa, Zè=Z ç 1í1:5, whee A is the bayon numbe and Z is the atomic èo potonè numbe. 4. 1amu èatomic mass unitè = 1:661 æ 10,24 gams.
7 CHAPTER 3. ASYMMETRIC COULOMB CHARGE INTERACTIONS 7 Bibliogaphy Emsley, J. è1989è The Elements, èpp. 12í13, pp. 54í55, pp. 78í79, pp. 202í203è. Claendon Pess: Oxfod. Eíotvíos, R. V., Pekça, D., and Fekete, E. è1922è Ann. Phys. èleipzigè 68, 11. Fischbach, E., Sudasky, D., Szafe, A., Talmadge, C., and Aonson, S. è1986è Phys. Rev. Lett. 56, 3. King, J. G. è1960è Phys. Rev. Lett. 5, 562. Kittel, C., Knight, W., Rudeman, M., Helmholz, A., and Moye, B. è1973è Mechanics, Bekeley Physics Couse, Volume 1, Second Edition, èpp. 398í408è. McGawHill Book Company, Inc.:New Yok. Livesey, D. L. è1966è Atomic and Nuclea Physics, èpp è. Blaisdell Publishing Company: Massachusetts. Niebaue, T., McHugh, M., and Falle, J. è1987è Phys. Rev. Lett. 59, 6. Petley, B.W.è1988èThe Fundamental Physical Constants and the Fontie of Measuement, èpp. 282í287è. Adam Hilga: Bistol. Pucell, E. M. è1985è Electicity and Magnetism, Bekeley Physics Couse, Volume 2, Second Edition, èpp. 5í6è. McGawHill Book Company, Inc.: New Yok. Roll, P. G., Kotkov, R., and Dicke, R. H. è1964è Ann. Phys. èn.y.è 26, 442. Stubbs, C., Adelbege, E., Raab, F., Gundlach, J., Heckel, B., McMuy, K., Swanson, H., and Watanabe, R. è1987è Phys. Rev. Lett. 58, 11. Thiebege, P. è1987è Phys. Rev. Lett. 58, 11. Wheele, J. A. è1990è A Jouney into Gavity and Spacetime, èpp. 24í27è. Scientiæc Ameican Libay: New Yok. Whittake, E. è1960è A Histoy of the Theoies of Aethe & Electicity, Vol. II, èpp. 144í196è. Hape & Bothes: New Yok.
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