Derivation Of The Photon & Graviton Mass-energies & Radii

Derivation of the Photon & Graviton Mass-Energies & Radii Riccardo C. Storti1, Todd J. Desiato

Abstract The construct herein utilises the Photon mass-energy threshold “mγ”, as derived by Storti et. al., to facilitate the precise derivation of the mass-energies of a Photon and Graviton [mγγ = 3.2 x 10-45 (eV) and mgg = 6.4 x 10-45 (eV) respectively]. Moreover, recognising the wave-particle duality of the Photon, the root-mean-square (RMS) charge radii of a free Photon and Graviton [rγγ = 2.3 x 10-35 (m) and rgg = 3.1 x 10-35 (m) respectively] is derived to high computational precision. In addition, the RMS charge diameters of a Photon and Graviton (“φγγ” and “φgg” respectively) are shown to be in agreement with generalised Quantum Gravity (QG) models, implicitly supporting the limiting definition of the Planck length “λh”. The value of “φγγ” is illustrated to be “≈λh”, whilst the value of “φgg” is demonstrated to be “≈1.5λh”.

1

[email protected], [email protected]. 1

1

ITRODUCTIO

Storti et. al. demonstrated in [1], based on the physical properties of an Electron, that the Polarizable Vacuum (PV) model of gravitation, [2] complimenting General Relativity (GR) in the weak field, is capable of predicting the Photon mass-energy threshold “mγ” to within 4.3(%) of the Particle Data Group2 (PDG) prediction presented by Eidelman et. al. [3] The construct herein utilises Electro-Gravi-Magnetics3 (EGM) principles [2] to facilitate the precise derivation of the mass-energies of a Photon and Graviton [mγγ = 3.2 x 10-45 (eV) and mgg = 6.4 x 10-45 (eV) respectively]. Moreover, recognising the wave-particle duality of the Photon, the root-mean-square (RMS) charge radii of a free Photon and Graviton [rγγ = 2.3 x 10-35 (m) and rgg = 3.1 x 10-35 (m) respectively] is derived to high computational precision. In addition, the RMS charge diameters of a Photon and Graviton (“φγγ” and “φgg” respectively) are derived and shown to be in agreement with generalised Quantum Gravity (QG) models, implicitly supporting the limiting definition of the Planck4 length “λh”. [4] The value of “φγγ” is illustrated to be “≈λh”, whilst the value of “φgg” is demonstrated to be “≈1.5λh”. 2

THEORETICAL MODELLIG5

Assuming that “mγ”, as conjectured6 in [1], represents an exact boundary value in accordance with equation (1), a precise expression for “mγγ” may be derived as illustrated in the proceeding section. mγ

512. h . G. m e c . π .r e

2

n Ω r e, m e

.

ln 2 . n Ω r e , m e

γ

(1)

To initiate the derivation process, we require a definition of “mgg” from which to apply dynamic, kinematic and geometric similarity with respect to “mγγ”. where, Variable h G c me re nΩ γ

Description Planck's Constant Universal Gravitational Constant Velocity of light in a vacuum Electron rest mass Classical Electron radius Harmonic cut-off mode of PV Euler's Constant Table 1,

Units Js Nm2kg-2 ms-1 kg m None None

It was illustrated in [1] that only the odd modes of a finite reciprocal harmonic distribution contribute to the magnitude of gravitational acceleration “g” according to the distribution “nPV = nΩ, 2 - nΩ ... nΩ”, symmetrical about the 0th mode. Where, “nPV” represents the modes of space-time 2

A collaboration of leading Nuclear and Theoretical Particle physicists funded by the USDoE, CERN, INFN (Italy), US NSF, MEXT (Japan), MCYT (Spain), IHEP and RFBR (Russia). [3] 3 Electro-Gravi-Magnetics (EGM) is based on Buckingham’s Π Theory. [2] 4 Utilising the "Plain h" form where λh = 4.05131993288926 x 10-35 (m): Calculated from National Institute of Standards and Technology (NIST) 2002 values. [5] 5 All mathematical modelling and output was formed using “MathCad 8 Professional” and appears in standard product notation. 6 The reader should refer to [1] for details regarding the radiation of conjugate Photon pair populations and the Photon mass-energy threshold construct. 2

manifold in the PV model of gravitation and the terminating mode in the finite reciprocal harmonic distribution is denoted by “nΩ”. The PV spectrum is conjectured to be composed of mathematical wavefunctions, over the symmetrical frequency domain -ωΩ<ωPV<ωΩ, which physically manifest as conjugate Photon pair populations. Subsequently, we shall define the odd frequency modes to be representative of conjugate Photon pair populations constituting a population of Gravitons. Therefore, 1 Graviton shall be defined as 1 conjugate Photon pair according to the following relationship, mgg = 2mγγ 3

(2)

MATHEMATICAL MODELLIG

Recognising that the Photon energy “EΩ” [6] at the harmonic cut-off frequency “ωΩ” is proportional to the conjugate Photon pair population, we may determine the Photon population “Nγ” at the mass-energy threshold as follows, E Ω h .ω Ω r e , m e

Nγ

(3)

EΩ mγ

(4)

Performing the appropriate substitutions utilising equations defined in Appendix (A) yields, Nγ

2 3 . . . c .π .r e 2 c G me . . ln 2 . n Ω r e, m e 512. G. m e π .r e

γ

(5)

Hence, 3

m γγ

h . re

3

π .r e 512. G. m e . 2 2 . c . G. m e c .π

2

.

n Ω r e, m e ln 2 . n Ω r e , m e

γ

2

(6)

Evaluating yields, Nγ = 1.8 x 1028 [mγγ mgg] = [3.2 6.4] x 10-45 (eV) 4

(7) (8)

PHYSICAL MODELLIG

In accordance with the preceding definition of Photon and Graviton mass-energy, we may apply Buckingham Π Theory in terms of dynamic, kinematic and geometric similarity between two mass-energy systems defined at “ωΩ”. Subsequently, it follows that any two dimensionally similar systems may be represented by, ω Ω r 1, M 1

ω Ω r 2, M 2

(9)

Where, “r1,2” and “M1,2” denote arbitrary radii and mass values. Subsequently, utilising equations defined in Appendix (A) and performing the appropriate substitutions, the preceding equation may be simplified as follows, M1

2

M2

r1 r2

3

5

(10)

Let M1 = mγγ/c2, M2 = me, r1 = rγγ and r2 = re: solving for “rγγ” yields, 5

r γγ

r e.

2

m γγ m e .c

2

(11)

where, “rγγ” may be expressed7 in terms of Compton and Planck characteristics as follows, r γγ

Variable λh λCN λCP ωCN ωCP ωh mn mp mh

λ CN c ω CP h .mp . γ . λ h. . λ CP ω h ω CN c m h m n

Description Planck Length Neutron Compton Wavelength Proton Compton Wavelength Neutron Compton Frequency Proton Compton Frequency Planck Frequency Neutron rest mass Proton rest mass Planck mass Table 2,

(12) Units m m m Hz Hz Hz kg kg kg

Hence, r gg

5

4 . r γγ

(13)

Therefore, the Photon and Graviton RMS charge diameters may be expressed as multiples of the Planck length as follows, 2 .

r γγ

λ h r gg

1.1529 1.5213

[φγγ φgg] ≈ [1 1.5] λh 5

(14) (15)

COCLUSIOS

The construct herein derives the mass-energies and RMS charge diameters of a Photon and Graviton. The results agree with generalised Quantum Gravity (QG) models, implicitly supporting the limiting definition of Planck length “λh” according to “φγγ ≈ λh” and “φgg ≈ 1.5λh”.

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To within 5x10-3(%) of the precise numerical result. 4

References [1] R. C. Storti, T. J. Desiato, “Derivation of the Photon mass-energy threshold”, The Nature of Light: What Is a Photon?, edited by C. Roychoudhuri, K. Creath, A. Kracklauer, Proceedings of SPIE Vol. 5866 (SPIE, Bellingham, WA, 2005) [pg. 207 - 213]. [2] R. C. Storti, T. J. Desiato, “Electro-Gravi-Magnetics (EGM) - Practical modelling methods of the polarizable vacuum – I”, Physics Essays: Vol. 19, No. 1: March 2006. [3] Particle Data Group: http://pdg.lbl.gov/index.html Citation: S. Eidelman et al., Phys. Lett. B 592, 1 (2004). [4] Wolfram Research: http://scienceworld.wolfram.com/physics/PlanckLength.html [5] NIST: http://physics.nist.gov/cuu/ [6] Wolfram Research: http://scienceworld.wolfram.com/physics/Photon.html

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APPEDIX A ω Ω ( r , M ) n Ω ( r , M ) .ω PV( 1 , r , M ) n Ω ( r, M )

Ω ( r, M )

4

12

Ω ( r, M )

(A1)

1

(A2)

3

108.

Ω ( r, M )

U m( r , M ) U ω ( r, M )

U ω ( r, M )

(A3)

2 3.M .c

U m( r , M )

3 4.π .r

(A4)

U ω ( r, M ) .

U ω n PV, r , M

2

U m( r , M )

12. 768 81.

n PV

2

4

4

n PV

(A5)

4 h . U ω ( r, M ) ω PV( 1 , r , M ) 3 2.c

(A6) G .M

n PV 3 2 . c . G. M . .e r π .r

ω PV n PV, r , M

2 r .c

(A7)

G .M 2 r .c

e

1

(A8)

U m( r , M ) 3 . r2 . c4 3 π . r . U ω ( r , M ) 4 . h . G 2 . c . G. M 3

Ω ( r , M ) 3.c .

(A9)

3 2 6.r .c . π .r h . G 2 . c . G. M

(A10)

3

3 2 Ω ( r , M ) c . 6.r .c . π .r . . 12 4 h G 2 c . G. M

n Ω ( r, M )

(A11) 1

3

1 3

2 3 π .r c . 6.r .c . 3 3 2 3 3 2 . 4 h G 2 . c . G. M . 2 . c . G. M c . 6 . r . c . π . r . 2 . c . G. M c . 6 . r . c . π .r ω Ω ( r, M ) . . . . . . . . . . 4r 4 r h G 2 c . G. M r h G 2 c GM π r π r

1 2 c . 6.r .c ω Ω ( r, M ) 4.r h . G

3

. . . . 2 c GM π .r

3

.

π .r 2 . c . G. M

1

1

13

3.

3.

9

ω Ω ( r, M ) 3 h

2

1

1

1

2

.π

9

9

2 c . 12. r . c . M 4.r π .h

14

5

2

1

3

.

π .r 2 . c . G. M

Utilising c ω h

2

. G. h

.

3

3 ω Ω ( r, M ) c . . 2

ω h 4.π .h

2

2

ω Ω r 1, M 1

M1

ω Ω r 2, M 2

M2

6

1

3.

9. 9

12 2

.M

5

c

14

1

. . . . 2 c GM π .r 2

.r 9 .M 9 .h

c

1

1

3.

9.

G

3

(A12) 2

π

9

(A13) 1

9

. M

2

5

r

9

(A14) (A15)

2

5

r

14

4

13 2 2 .π .G

yields:

9

1

1

3

3

9

h 5

1.

1

1

.c 9 .r 9 .M 9 .G

9

3

(A16)

5

9

.

r2 r1

9

(A17)