1

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 rstorti@deltagroupengineering.com, tdesiato@deltagroupengineering.com.

2

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.

m γ

512 h. G. me.

c π r e. 2.

n Ω r e me,

ln 2 n Ω r e me,. 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 Description Units

h Planck's Constant Js

G Universal Gravitational Constant Nm2kg

-2

c Velocity of light in a vacuum ms-1

me Electron rest mass kg

re Classical Electron radius m

nΩ Harmonic cut-off mode of PV None

γ Euler's Constant None

Table 1,

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.

3

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 (2)

3 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, EΩ h ω Ω r e me,.

(3)

N γ

EΩ

m γ (4)

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

N γ

c π2. r e

.

512 G. me.

32 c. G. me

.

π r e.

. ln 2 n Ω r e me,. γ.

(5)

Hence,

m γγh

r e3

3π r e

.

2 c. G. me.

.512 G. me

.

c π2.

2

.n Ω r e me,

ln 2 n Ω r e me,. γ2

.

(6)

Evaluating yields,

Nγ = 1.8 x 1028 (7)

[mγγ mgg] = [3.2 6.4] x 10

-45 (eV) (8)

4 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,

M 1

M 2

2r 1

r 2

5

(10)

4

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

r γγ r e

5

m γγ

me c2.

2

.

(11)

where, “rγγ” may be expressed

7 in terms of Compton and Planck characteristics as follows,

r γγ γ λ h

λ CN

λ CP

. c

ω h

ω CP

ω CN

. h

c mh.

mp

mn

..

(12)

Variable Description Units

λh Planck Length m

λCN Neutron Compton Wavelength m

λCP Proton Compton Wavelength m

ωCN Neutron Compton Frequency Hz

ωCP Proton Compton Frequency Hz

ωh Planck Frequency Hz

mn Neutron rest mass kg

mp Proton rest mass kg

mh Planck mass kg

Table 2,

Hence,

r gg54 r γγ

. (13)

Therefore, the Photon and Graviton RMS charge diameters may be expressed as multiples of the

Planck length as follows,

2

λ h

r γγ

r gg

.1.1529

1.5213 (14)

[φγγ φgg] ≈ [1 1.5] λh (15)

5 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”.

7 To within 5x10

-3(%) of the precise numerical result.

5

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

6

APPEDIX A

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

(A1)

n Ω r M,( )Ω r M,( )

12

4

Ω r M,( )1

(A2)

Ω r M,( )

3

108Um r M,( )

U ω r M,( )

. 12 768 81Um r M,( )

U ω r M,( )

2

..

(A3)

Um r M,( )3 M. c

2.

4 π. r3. (A4)

U ω n PV r, M, U ω r M,( ) n PV 24

n PV4. (A5)

U ω r M,( )h

2 c3.

ω PV 1 r, M,( )4.

(A6)

ω PV n PV r, M,n PV

r

32 c. G. M.

π r.. e

G M.

r c2.

.

(A7)

e

G M.

r c2.

1 (A8) Um r M,( )

U ω r M,( )

3 r2. c

4.

4 h. G.

3π r.

2 c. G. M..

(A9)

Ω r M,( ) 3 c.

3

6 r2. c.

h G.

3π r.

2 c. G. M...

(A10)

n Ω r M,( )Ω r M,( )

12

c

4

3

6 r2. c.

h G.

3π r.

2 c. G. M...

(A11)

ω Ω r M,( )

c

4

3

6 r2. c.

h G.

3π r.

2 c. G. M...

r

32 c. G. M.

π r.. c

4 r.

3

6 r2. c.

h G.

3π r.

2 c. G. M...

32 c. G. M.

π r.. c

4 r.

6 r2. c.

h G.

π r.

2 c. G. M.

1

3

.

1

3

. 2 c. G. M.

π r.

1

3

.

(A12)

ω Ω r M,( )c

4 r.

6 r2. c.

h G.

1

3

. 2 c. G. M.

π r.

1

3

. π r.

2 c. G. M.

1

9

. c

4 r.

12 r. c2. M.

π h.

1

3

. π r.

2 c. G. M.

1

9

. 1

412

1

3. 2

1

9. c

14

9. r

5

9. M

2

9. h

1

3. G

1

9. π

2

9.

(A13)

ω Ω r M,( ) 3

1

3h

1

3. 2

13

9. π

2

9. c

14

9. r

5

9. M

2

9. G

1

9. 3

h

1

3c14

213

π2. G.

1

9

. M2

r5

1

9

.

(A14)

Utilising c5

ω h2G. h. yields: (A15)

ω Ω r M,( ) c

33

2

.

9

ω h

4 π. h.

2

M2

r5

..

(A16)

ω Ω r 1 M 1,

ω Ω r 2 M 2,

M 1

M 2

2

9r 2

r 1

5

9

.

(A17)