are nuclear and gavitational forces of same nature

8/8/2019 Are Nuclear And Gavitational Forces of Same Nature

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Are Nuclear and Gravitational fields of the same nature?Jose N Pecina-Cruz 1

University of Texas Pan-American1201 W. University Drive, Edinburg Texas 78539-2999

[email protected]

This article suggests that nuclear and gravitational forces may be of the same nature. The ratiobetween the free fall distance of an object and half of the Schwarzschild radius of a black hole isidentified with the nuclear mass number A= r/m . This substitution, in the free fall equations,allows one to reproduce the curve of the Binding Energy per Nucleon (BEN) of a nucleus. Ablack holes formation is the appropriate model to execute this task. By calculating the energyper particle falling in a black hole the BEN curve is recreated. Therefore, the nuclei are identifiedwith black holes in the process of forming. Primordial black hole formations could be thebeginning of the nuclear particles zoo.

Introduction

The equivalence principle claims that the acceleration inertial and the gravitational accelerationthat experiment a free falling mass are indistinguishable [1]. Newtons Second Law of Motionclaims force equals mass times acceleration. It is remarkable that the application of this law doesnot depend on the nature of the force. It could be a gravitational, magnetic, etc. type of force. Asa physicist, one attempts to model each force that one observes with a mathematical formula.However, instead of thinking in many different forces, with a law for each one, let us apply theMach principle of economy of thoughts by claiming that all forces are of inertial nature.Therefore, one is tempted to generalize the example of Einsteins elevator. To account for themagnetic field, whose presence is explained by the relative motion of the electrical charges [3].And so think in a similar mechanism might occur in the generation of nuclear force. If thisconjecture were true for any of the gravitational forces, then we have to explain why gravitationis so unique.Following the above assumption, that is, there is a unification of forces (fields) through theconcept of inertial forces, one proceed to perform the calculation of the curve of binding energyper nucleon by using Einstein gravitational equations with the Schwarzschild metric as a solution[1].A curve of average binding energy per nucleon was empirically derived by Weizscker [4]. Laterwas improved by the Liquid Drop Model of the nucleus [3]. Also Bethe and Bacher [2]contributed to a clearer understanding of this formula [2]. In Section 1 we will discuss thederivation of the equation of the Binding Energy per Nucleon from Einsteins gravitationalequations and Schwarzschild metric. Section 2 is dedicated to the comparison of the leastsquared fitting of the nuclear masses obtained from experimental data [8][9] with the energy pernucleon obtained from GR formula. In the application of the least square technique one termresponsible of the electromagnetic interaction was added. A very rudimentary Coulomb

1 On leave: 1501 Camellia Ave., McAllen, Texas 78501


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interaction was used. However, the LSQR fitting of data was quite good. Quantum correctionswere not considered at this time (in He for instance). However, one was able to calculate theexact formulae for the free fall object in a massive body. This formula (20) was derived overpure GR bases and is exact.

1. From General Relativity to Nuclear Field

In the calculations for free falling objects in the neighboring area of a very dense mass one usesEinsteins equations for a matter-free space, which are given by [7]

0. =

(1)

For a static spherically symmetric and asymptotically flat, empty space-time, a solution wasproposed by Schwarzschild by using the metric

22222212222 sin) / 21() / 21( d r d r dr r mdt cr md c = (2)

Where

2 . MG

mc

=

(3)

The equations of free fall are

2

2 0d x dx dxdk dk dk

+ = . (4)

Here k is a parameter that describes the trajectory of the particle, on its journey to the center of the black hole.

Using the definition of the Christoffel symbols and the components of the Schwarzschild metric,equation (4) can be written

2.. . .2 / 0,

1 2 / m r

t t r m r

=

(5)

.2 2 2 22.. . . . .

2 2 2 / / (1 2 / ) (1 2 / ) (1 2 / ) sin 0,1 2 /

m r r c m r m r t r r m r r m r

m r + =

(6)


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.2.. . . .2

sin cos 0,r r

+ = (7)

.

.. . . . .2 2cot 0.r r

+ + = (8)

The solutions of the differential equations (5), (6), (7) and (8) provide us with the free fallvelocity of an object moving under the attraction of a black hole. For instance, integrating eq. (6)we get

1,

1 2 / dt cdk m r

=

(9)

where, c1 stands for a constant of integration. If the free fall is radial 0. = =

Substituting eq.(9) into eq. (6) and applying these conditions we obtain

2 2 22

2

/ 1 .

1 2 / d r c m r r

cdk m r c

=

(10)

This equation may be integrated to get

2 1/2[ 1 2(1 2 / )] ,dr c c c m r dk

= (11)

c2 is a new constant of integration. From eq. (10)

(1/ 1)(1 2 / )dk c m r dt = , (12)

Therefore

.) / 21(1) / 21(2 / 1

21

2 = r mccr mc

dt dr (13)

When the object is released 20 2 / 1 1.dr

r c cdt

> => = => = the integration constants c2/c1 2=1

are evaluated. Therefore the radial velocity of the free falling object is


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( )2 3/22 2 / .v mc r m r = (14)

v can be written in terms of A= r/m

2 / 3

)2(2 A

Acv

= (15)

The energy for a falling particle is given by the eigenvalue equations of the Poincare Lie group

,422 cm p = (16)

where pr

is a four-vector, which components are generators of the Poincare Lie algebra.

In order to find a meaning for the mass number A lets follow old Bohrs quantization of theHydrogen atom. Postulating that the angular momentum of the falling nucleon is quantized, andgiven by (see appendix I))

hn L = (17)

pn

r h

=n

(18)

1n nr r =(19)

Therefore,

number mass Anr r

r m

_1

n ==== (20)

Equations (17) to (19) show how in a natural way quantum mechanics appears in the scenery.Actually the mass number, A, is a quantum number. A is the orbit number in the old formalismof Bohrs quantization of the atom[11]. This argument explains why the nucleons do not collidewith each other, within such small space inside a nuclear shell. Waves can penetrate each other.This explains the superconductivity and superfluity in the nuclei. The same is true for the magicnumbers. It may explain hot superconductivity/where the formation of Copper pairs would beindependent of temperature.The Standard Model is entirely based on SU(3) symmetry. In the Standard Model asymptoticfreedom occurs. The same phenomenon is observed in this model since a very short distance thenuclear particles are free. As Bohr, philosophically express the systems in nature repeats. Theplanetary system as well as the atomic system is a clear of this. Therefore, the same argumentcould be used in favor of a nuclear system[12].


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2. Nuclear Binding Energy Approximation from Black Hole Formalism

Substituting v given by eq. (15) into eq. (16), one obtains

2 / 11

32

2

2 / 32 / 10 1882

121

2)(

+

+

=

A A A A A E A f , (20)

where, )( A f is the energy for the free falling particle. It seems in some way natural to identifythe mass number A with a multiple integer (quantum) of the Schwarzschild radius ( A=r/m ); sincethe height of a free fall particle is directly related to its energy. A=r/ m could be identified withenergy packets or discrete mass particles. With this analogy ) ( A f is the binding energy of anucleon bound to a nucleus of mass number A. Other physical interpretation is obtained byconsidering this partition r/m, of the energy as energy quanta which become particles or nucleifragments after a very energetic collision. In analogy to a free fall object attracted by a massivebody.It must also be included in the equation for the binding energy per nucleon the electromagneticinteraction and the quantum effects. The deviation of the curve due to a quantum interaction forinstance, for He is discussed with detail by J. Sakurai [7]. In the next equation the last term andthe first are part of the electromagnetic force. The potential energy was calculated for Z particlesin the nucleus interacting with the free falling nucleon with Z=1.Finally, one does not need any numerical expansion to determine the free fall energy of an objectin the neighborhood of a black hole. Because the semi-empirical mass formula of Bethe andBacher, in p. 183 [2] is quadratic in Z, there is an atomic number Z which minimizes M which iscalled the nuclear charge of the most stable isobar.The Weizsckers semi-empirical formula for the total energy of a nucleus is slightly simplifiedby the authors of Ref 2. It is given by

3 / 120

23 / 22 ) / (5 / 3 / )( +++++= A Z r e A A Z N A ZM NM M pn . (21)

This is a quadratic equation in Z then for each fixed value of A there is a value of Z thatminimizes M.

0)(21

53)(2

2 / 10

2

=+

=

n p M M

Ar Z e

A Z N

Z M

. (22)

By substituting A=200 in eq. (20) it is found that the Z value that minimizes M, which is thenuclear charge of the most stable isobar, denoted Z 0=80.The most stable nucleus of atomic weight 200 has the nuclear charge 80, correspond to Hg 200 .


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All the elements from Z=1 to Z=80 at least have one stable nuclide. The first 82 from hydrogento lead except for technetium (Z=43) and promethium (Z=61). Elements with Z>82 only haveradioactive isotopes.Therefore if we take as a probability energy distribution function f(A,Z)=f(A)/A or M/A

A A Z r e A A Z N A ZM NM A M Z A f pn / )) / (5 / 3 / )(( / ),(3 / 12

023 / 22

+++++== (23)

According to Ref. 2, p. 87, the interval of integration can be replaced by the limits between O 16 and Hg 200 where, const A M / . Therefore

2001

1),(200

0

200

0

==>== k kdAdA Z A f (24)

Hence equation (17) must be divided by 200A, since f(A)/A is the energy per nucleon.

2 / 1

2

1

32

2

2 / 52 / 30 8821

212

200),(

+

+

=

A A A A A A

E Z A f (25)


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Figure 1. This figure depicts the experimental data for stable nuclides (with Z 82).


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Figure 2. The Figure shows the Binding Energy per Nucleon obtained from eq. (22) noCoulomb force is included. It is obtained from pure GR.


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Figure 3. The figure displays the Least Square Fitting of equation (22) including the Coulombinteraction. And compare it with the experimental data of the Binding Energy per Nucleon


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Figure 4. This figure shows all the three curves for the Binding Energy per Nucleon displayedabove.


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Figure 5. This figure displays the maxima in the plots of the average Binding Energy perNucleon data.

The curve of binding energy derived by using GR is displaced from its negative values(attractive potential) to positive values in order to compare with the experimental data.

Conclusion

This manuscript presents a curious coincidence (A= r/m) that allows the achievement of theunification of the two forces, gravity and nuclear forces. Quantization imposes a constraint in anucleus, equation (19) in this paper.What we are suggesting that the primordial black holes in formation are nucleons. These black holes in formation never reach the final fate of a black hole since quantum mechanics preventsthe occurrence of this event [10]. Hot superconductivity may be explained by superconductivitystates in nuclei.

Acknowledgments

My most sincere acknowledgment goes for Roger M. Pecina for his sharp suggestions during thepreparation of this manuscript.


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Appendix I

By Bohrs quantization method argument, given in Reference [13], the energy of the fallingobject is according to Plank

nhf E =

. (26)

Where E is the total energy of the free following object

nh f E

Plank =

| (27)

pf r dr r rf m

r d F E r

vmF

r r

)2()2( 22

====>=

rr(28)

Therefore

LSystem f E )2(| = . (29)

And from equations (27) and (29)

One obtains

hn L =

References

[1] A. Einstein, The Meaning of Relativity , Princeton University Press, Centennial Edition(1979); S. Weinberg, Gravitation and Cosmology , John Wiley & Sons, p. 61-63 (1972).[2] H. Bethe and RF Bacher, Rev of Mod Phys 8 82 (1936).

[3] L. Landau A. Lifshitz, Teoria Clasica de Campos Vol. 2 , Ed. Reverte S.A. (1973); Purcell, Electricity and Magnetism , McGraw-Hill Book Company, NY (1971).

[4] N. Bohr, J.A. Wheeler, Phys. Rev. 56, 426 (1939).[5] Von C.F. Weizscker, Z. Physik 96, 431 (1935).[6] J. Foster and J.D. Nightingale, A short course in General Relativity , Longman, Inc. NYp.107-111.

[7] J.J. Sakurai, Modern Quantum Mechanics , Addison Wesley Pub. Co., Inc. (1994).[8] J.H.E. Mattauch, E.Thiele, A.H.Wapstra, Nucl. Phys. 67, 1 (1965); E.U. Condon, H.Odishaw, Handbook of Physics 9-65 to 9-86; McGraw-Hill-Book Co. (1958).[9] A. Sonzogni, "Interactive Chart of Nuclides". National Nuclear Data Center: Brook havenNational Laboratory. http://www.nndc.bnl.gov/chart/.[10] Pecina-Cruz J.N., Quantum Mechanics and Black Holes, arXiv:physics/0510163 [11] N. Bohr, Essays 1958-1962 on atomic physics and human knowledge , John Willey & Sons,1963.


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[12] A. Bohr, B.R. Motelson, Nuclear Structure Vol I, Amsterdam: North-Holand 1970.[13] J.H.O. Sales, A.T. SuzUki, B.S. Donafe, arXiv:physics/0608102v1 [physics.atom-ph] 9Aug 2006.

are nuclear and gavitational forces of same nature

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