could nucleons fall as newton's apples do

8/9/2019 Could Nucleons Fall as Newton's Apples Do

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Could nucleons fall as Newtons apples do?

J.N. Pecina-Cruz1

University of Texas Pan-American

University Drive, 1201 W. Edinburg Texas [email protected]

This article suggests that nuclear and gravitational forces may be of the same nature. InThis manuscript the ratio between the free fall distance of an object and half of the

Schwarzschild radius of a black hole is identified 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. A black holes formation is the

appropriate model to execute this task. By calculating the energy per particle falling in ablack hole the BEN curve is recreated. Therefore, the nuclei are identified with blackholes in the process of forming. Primordial black hole formations could be the beginning

of the nuclear particles zoo.

Introduction

The equivalence principle claims the acceleration inertial and the acceleration

gravitational that experiment a free falling mass are indistinguishable [1]. NewtonsSecond Law of Motion claims force equals mass times acceleration. It is remarkable that

the application of this law does not depend on the nature of the force. It could be agravitational, magnetic, etc. type of force. As a physicist, one attempts to model each

force that one observes with a mathematical formula. However, instead of thinking inmany different forces, with a law for each one, let us apply the Mach principle of

economy of thoughts by claiming that all forces are of inertial nature. Therefore, one istempted to generalize the example of Einsteins elevator. To account the magnetic field,

whose presence is explained by the relative motion of the electrical charges [3]. And sothink a similar mechanism might occur in the generation of nuclear force. If thisconjecture were true for the gravitational force, then we have to explain why gravitation

enjoys such a privilege. Nature always refuses to give special treatment to any physicalentity.

Following this assumption, that there is a unification of forces (fields) through theconcept of inertial forces, one proceed to perform the calculation of the curve of binding

energy per nucleon by using Einstein gravitational equations with the Schwarzschildmetric as a solution [1].

A curve of average binding energy per nucleon was empirically derived by Weizscker[4]. Later was improved by the Liquid Drop Model of the nucleus [3]. Also Bethe and

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


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Bacher [2] contributed to a clearer understanding of this formula [2]. In Section 1 we willdiscuss the derivation of the equation of the Binding Energy per Nucleon from Einsteins

gravitational equations and Schwarzschild metric. Section 2 is dedicated to thecomparison of the least squared fitting of the nuclear masses obtained from experimental

data [8][9] with the energy per nucleon obtained from our GR formula merged by least

square fitting to the electromagnetic energy due to the proton charge. In the application ofthe least square technique one term responsible of the electromagnetic interaction wasadded. A very rudimentary Coulomb interaction was used. However, the LSQR fitting of

data was quite good. Quantum corrections were not considered at this time (in He forinstance). However, one was able to calculate the exact formulae for the free fall object in

a massive body. This formula (20) was derived over pure 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 uses Einsteins 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

was proposed by Schwarzschild by using the metric

22222212222 sin)/21()/21( drdrdrrmdtcrmdc = (2)

Where

2.

MGm

c=

(3)

The equations of free fall are

2

20

d x dx dx

dk dk dk

+ = . (4)

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

Using the definition of the Christoffel symbols and the components of the Schwarzschild

metric, equation (4) can be written


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2.. . .2 /0,

1 2 /

m rt t r

m r =

(5)

.2 2 2 22.. . . . .

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

1 2 /

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

m r

+ =

(6)

.2.. . . .2

sin cos 0,rr

+ = (7)

... . . . .2

2cot 0.rr

+ + = (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 c

dk 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 c

dk 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


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4

.)/21(1)/21(

2/1

2

1

2

= rm

c

crmc

dt

dr(13)

When the object is released 20 2 / 1 1.drr c cdt

> => = => = the integration constants

c2/c12=1 are evaluated. Therefore the radial velocity of the free falling object is

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

v can be written in terms ofA= 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 cmp = (16)

where p&

is a four-vector, and its components generators of the Poincare Lie algebra.

In order to find a meaning for the mass number A lets follow old Bohrs quantization of

the Hydrogen atom. Postulating that the angular momentum of the falling nucleon isquantized, and given by

!nL =

(17)

p

nr

!=n

(18)

1n nrr =(19)

Therefore,

numbermassAnr

r

r

m_

1

n ==== \ (20)

This article shows 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

Bohrs quantization of the atom[11]. This argument explains why the nucleons do not


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collide each other, within such small space inside a nuclear shell. Waves can penetrateeach other. This explains the superconductivity and superfluity in the nuclei. The same is

true for the magic numbers. It may explain hot superconductivity/where the formationof Copper pairs would be independent of temperature.

The Standard Model is entirely based on SU(3),SU(2) symmetries and Gauge Theory. It

is a nuclear theory for high energy physics asymptotic freedom, is observed in this modelsince a very short distance the nuclear particles are free. As Bohr, philosophically expressthe systems in nature repeats. The planetary system as well as the atomic system is a clear

of this. Therefore, the same argument could be used in favor of a nuclear system[12].

2. Nuclear Binding Energy Approximation from Black Hole Formalism

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

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32

2

2/32/101

8821

212)(

+

+

=

AAAAAEAf , (20)

where, )(Af is the energy for the free falling particle. It seems in some way natural to

identify the mass number A with a multiple integer (quantum) of the Schwarzschild

radius (A=r/m); since the height of a free fall particle is directly related to its energy.A=r/m could be identified with energy packets or discrete mass particles. With this

analogy )(Af is the binding energy of a nucleon bound to a nucleus of mass number A.

Other physical interpretation is obtained by considering this partition r/m, of the energy

as energy quanta which become particles or nuclei fragments after a very energeticcollision. In analogy to a free fall object attracted by a massive body.

It must also be included in the equation for the binding energy per nucleon theelectromagnetic interaction and the quantum effects. The deviation of the curve due to a

quantum interaction for instance, for He is discussed with detail by J. Sakurai [7]. In thenext equation the last term and the first are part of the electromagnetic force. The

potential energy was calculated for Z particles in the nucleus interacting with the freefalling nucleon with Z=1.

Finally, one does not need any numerical expansion to determine the free fall energy ofan object in the neighborhood of a black hole. Because the semi-empirical mass formula

of Bethe and Bacher, in p. 183 [2] is quadratic in Z, there is an atomic number Z which

minimizes M which is called the nuclear charge of the most stable isobar.The Weizsckers semi-empirical formula for the total energy of a nucleus is slightlysimplified by the authors of Ref 2. It is given by

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23/22 )/(5/3/)( +++++= AZreAAZNAZMNMM pn . (21)


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

0)(2

1

5

3)(2

2/10

2

=+

=

np MM

Ar

Ze

A

ZN

Z

M . (22)

By substituting A=200 in eq. (20) it is found that the Z value that minimizes M, which is

the nuclear charge of the most stable isobar, denoted Z0=80.The most stable nucleus of atomic weight 200 has the nuclear charge 80, correspond to

Hg200

.All the elements from Z=1 to Z=80 at least have one stable nuclide. The first 82 from

hydrogen to lead except for technetium (Z=43) and promethium (Z=61). Elements withZ>82 only have radioactive isotopes.

Therefore if we take as a probability energy distribution function f(A,Z)=f(A)/A or M/A

AAZreAAZNAZMNMAMZAf 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 O16

and Hg200

where, constAM / . Therefore

200

11),(

200

0

200

0

==>== kkdAdAZAf (24)

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

2/1

2

1

32

2

2/52/3

0 882121

2200

),(

+

+

=

AAAAAA

EZAf (25)


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Figure 1. This figure depicts the experimental data for stable nuclides (with Z82).


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Figure 2. The Figure shows the Binding Energy per Nucleon obtained from eq. (22) no

Coulomb 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

Coulomb interaction. And compare it with the experimental data of the Binding Energyper Nucleon


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Figure 4. This figure shows all the three curves for the Binding Energy per Nucleondisplayed above.


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

Nucleon 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 ofthe unification of the two forces, gravity and nuclear forces. Quantization imposes a

constraint in a nucleus, equation (19) in this paper.What we are suggesting is 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 prevents the occurrence of this event [10]. Hot superconductivity may beexplained by superconductivity states in nuclei.

Acknowledgments

My most sincere acknowledgment goes for Roger M. Pecina for his sharp suggestions

during the preparation of this manuscript.


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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.NY p.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

haven National 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.

[12] A. Bohr, B.R. Motelson, Nuclear Structure Vol I,

Amsterdam: North-Holand 1970.

could nucleons fall as newton's apples do

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