Derivation Of The Photon Mass-energy Threshold

Derivation of the Photon Mass-Energy Threshold Riccardo C. Storti1, Todd J. Desiato

Abstract An analytical representation of the mass-energy threshold of a Photon is derived utilising finite reciprocal harmonics. The derived value is “< 5.75 x 10-17 (eV)” and is within 4.3(%) of the Eidelman et. al. value endorsed by the Particle Data Group (PDG) “< 6 x 10-17 (eV)”. The PDG value is an adjustment of theoretical predictions to fit physical observation. The derivation presented herein is without adjustment and may represent physical evidence of the existence of Euler’s Constant in nature at the quantum level.

1

[email protected], [email protected]: 1

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ITRODUCTIO

It shall be demonstrated that the Polarizable Vacuum (PV) model of gravitation, [1] complimenting General Relativity (GR) in the weak field, is capable of predicting the Photon massenergy threshold to within 4.3(%) of the Particle Data Group2 (PDG) prediction presented by Eidelman et. al. of “< 6 x 10-17 (eV)”. [2] The derived Photon mass-energy threshold “mγ”, based on the physical properties of the Electron, may be usefully described by a finite reciprocal harmonic series representation as the number of harmonic modes approaches infinity, producing the result mγ < 5.75 x 10-17 (eV). The proceeding section sets the foundation from which a complete construct may be formed based on practical modelling methods. The use of physical modelling techniques will be shown to be highly advantageous in the development of “mγ”. 2

MATHEMATICAL MODELLIG3

2.1

BUCKINGHAM “Π” THEORY

Commencing from first principles, we apply Buckingham's Π Theory (BPT). The underlying principle of BPT is the preservation of dynamic, kinematic and geometric similarity between a mathematical model and an Experimental Prototype (EP). When applying BPT, one selects a set of significant parameters avoiding repetition of dimensions as illustrated in table (1) in accordance with the application of BPT. [3-5] 2.2

FORMULATION OF “Π” GROUPINGS

The formulation of Π groupings begins with the determination of the number of groups to be formed. The difference between the number of significant parameters (a, B, E, ω, r and Q) and the number of dimensions (kg, m, s and C), represents the number of Π groups required (two). where, Significant Parameter

Units m/s2

Composition4 kg0 m1 s-2 C0

T

kg1 m0 s-1 C-1

V/m

kg1 m1 s-2 C-1

Magnitude of position vector

m

kg0 m1 s0 C0

Magnitude of electric charge Propagation frequency of field Table 1, significant parameters.

C Hz

kg0 m0 s0 C1 kg0 m0 s-1 C0

a

a ( B , E, ω , r , t )

Description Magnitude of acceleration vector

B

B( ω , r , t )

Magnitude of magnetic field vector

E

E( ω , r , t )

Magnitude of electric field vector

r r( x, y , z, t ) Q Q( r , t )

ω

We may write the general formulation of significant parameters as, x1

a K 0( X ) .B

2

x2

.E

.ω

x3 x x . 4. 5

r

Q

(1)

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). [2] 3 All mathematical modelling and output was formed using “MathCad 8 Professional” and appears in standard product notation. 4 The traditional representation of mass (M), length (L) and time (T), in BPT methodology has been replaced by dimensional representations familiar to most readers (kg, m and s). “C” denotes Coulombs, the MKSA units representing charge. 2

where, K0(X) represents an experimentally determined dimensionless relationship function and “X” denotes all variables, within the experimental environment that influences results and behaviour. This also includes all parameters that might otherwise be neglected, due to practical calculation limitations, in theoretical analysis. Hence, the general formulation may be expressed in terms of its dimensional composition as follows, 0

1

kg m s

2

0

C

1

0

K 0( X ) . kg m s

1

C

1

x1

. kg1 m1 s

2

C

1

x2

. kg0 m0 s

1

0

C

x3

. kg0 m1 s 0 C0

x4

. kg0 m0 s 0 C1

x5

(2)

Applying the indicial method [4] yields, x1

x2 0

x2

x4 1

x1 x1

2 .x 2 x2

x3

2

solve , x 2 , x 3 , x 4 , x 5

x1 x1

2 x1

1 0

x5 0

(3)

Substituting the expressions for xn into the general formulation and grouping terms yields, a r .ω

2

K 0( X ) .

B.ω .r

x1

E

(4)

Note that the variable for electric charge has dropped out of the general formulation. This implies that the acceleration derived is not to be associated with the Lorentz force. 2.3

WAVEFUNCTION PRECIPITATION

Storti et. al. [1] illustrated that the preceding equation may be precipitated into a number of useful forms by the application of limits and boundary values. A particular form of importance, known as the wavefunction precipitation (x1 → 1, B/E → 1/c), may be generated utilising the preceding equation and defined, such that a ≡ aPV and ω ≡ ωPV as follows5, 3 2 ω PV . r a PV K 0 ω PV, r , E, B, X . c

(5)

Where, “aPV” and “ωPV” denote the magnitude of the acceleration vector and harmonic frequency modes of the PV respectively. It was illustrated in [1] that the experimental relationship function “K0(ωPV,r,E,B,X)”, formed by the method of incorporation may be shown to be, K0(ωPV,r,E,B,X) = K0(X)

(6)

By application of the Equivalence Principle6, we may determine the value of K0(X) of a homogenous solid spherical mass by using the weak field approximation to the gravitational potential [1,6] such that for an observer at infinity, 3.

K 0 ω PV, r , E, B, X

5

e

G .M 2 r .c

(7)

For investigations involving transverse plane wave solutions in a vacuum, by this we mean the PV background field, Maxwell’s equations require E/B = c, when r → λ/2π in the frequency range 0<ω<∞, 6 Indicating that an accelerated reference frame is equivalent to a uniform gravitational field. 3

where, Variable G M c

2.4

Description Universal Gravitational Constant Rest mass of solid spherical object Velocity of light in a vacuum Table 2, variable definitions.

Units Nm2kg-2 kg ms-1

CONSTANT ACCELERATION

A constant function may be expressed as a summation of trigonometric terms [1,7] over the longest period in a spectrum of frequencies. It is convenient to model a gravitational field utilizing modified Complex Fourier Series, according to the harmonic distribution “nPV = -nΩ, 2 - nΩ ... nΩ”, where “nΩ” is a terminating odd number harmonic in the PV model of gravitation. Hence, the magnitude of the gravitational acceleration vector “g” may be usefully represented by equation (8) as7 |nPV| → ∞, g( r, M )

G. M . 2

r

n PV

2 . i . π .n PV .ω e π . n PV

.. PV( 1 , r , M ) t i

(8)

Hence, the amplitude spectrum [7] “CPV” may be written as, G. M .

C PV n PV, r , M

2

r

2 π . n PV

(9)

such that, Variable Description ωPV(1,r,M) Fundamental field harmonic of PV t Time Table 3, variable definitions. 2.5

Units Hz s

FREQUENCY SPECTRUM

In accordance with the harmonic representation of “g” illustrated by equation (8), “K0(ωPV,r,E,B,X)” is a frequency dependent experimental function approximating unity. Hence, an expression for the frequency spectrum may be derived in terms of harmonic frequency mode. This may be achieved by applying the Equivalence Principle and assuming that an arbitrary acceleration described by equation (5) “aPV” is dynamically, kinematically and geometrically similar to the amplitude of the 1st harmonic “CPV(1,r,M)” described by equation (9) as follows, aPV ≡ CPV(1,r,M)

(10)

Therefore, utilising equation (5), (7) and (10), it follows that all frequency modes may be represented by, G .M

ω PV n PV, r , M

7

n PV

3

. . . . 2 c G M .e r π .r

2 r .c

Representing the magnitude of a periodic square wave solution with constant amplitude. 4

(11)

2.6

ENERGY DENSITY

The gravitational field surrounding a homogenous solid spherical mass may be characterised by its energy density. Assuming that the magnitude of the field is directly proportional to the massenergy density of the object, then the energy density “Uω” may be evaluated over the difference between successive odd frequency modes as follows, U ω ( r, M ) .

U ω n PV, r , M

n PV

2

4

4

n PV

(12)

Where, U ω ( r, M )

Variable Uω(nPV,r,M) h

2.7

4 h . ω PV( 1 , r , M ) 3 2.c

(13)

Description Energy density per change in odd harmonic mode Planck’s Constant Table 4, variable definitions.

Units Pa Js

CUT-OFF MODE AND FREQUENCY

Utilizing the approximate rest mass-energy density of a homogenous solid spherical object “Um”, the terminating harmonic mode of the PV, “nΩ” may be derived as follows, U m( r , M )

3.M .c

2

3 4.π .r

(14)

Storti et. al. conjectured in [1] that experimental investigations into the characteristics of the PV may be conducted at a specific frequency mode provided dynamic, kinematic and geometric similarity is preserved between a mathematical model and an EP. Therefore, assuming that the total rest mass-energy density may be characterised by one change in harmonic mode “|Um(r,M)| = |Uω(nPV,r,M)|”, equation (12) may be solved for the maximum value of “|nPV|”. This is termed the harmonic cut-off mode “nΩ” and may be applied to equation (8) as the terminating mode in the finite reciprocal harmonic Fourier Series as follows, n Ω ( r, M )

Ω ( r, M )

4

12

Ω ( r, M )

1

(15)

Where, “Ω(r,M)” is termed the harmonic cut-off function, 3

Ω ( r, M )

108.

U m( r , M )

12. 768 81.

U ω ( r, M )

U m( r , M ) U ω ( r, M )

2

(16)

Consequently, the upper boundary of the PV frequency spectrum “ωΩ”, termed the harmonic cut-off frequency, may be calculated as follows, ω Ω ( r , M ) n Ω ( r , M ) .ω PV( 1 , r , M )

(17)

The derivation of equations (15-17) is based on the compression of energy density to one change in odd harmonic mode whilst preserving dynamic, kinematic and geometric similarity in accordance with BPT. The subsequent application of these results to equation (8) acts to decompress the energy density over the Fourier domain yielding a highly precise reciprocal harmonic representation of “g”.

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3

PHYSICAL MODELLIG

3.1

CONJUGATE PHOTON PAIR POPULATIONS

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. It shall be illustrated that the pending definition leads to a solution for the mass-energy threshold of a population of Photons based on the physical properties of an Electron as defined by, i. The geometry of a free Electron at rest is usefully approximated to spherical. ii. Electrons radiate a spectrum of conjugate Photon pairs through their spherical geometric boundary. iii. The term “conjugate Photon pair” denotes a theoretical particle population involving energy transfer resulting in the magnitude of the local acceleration vector “g”. The existence of conjugate Photon pair populations requires experimental validation and is conjectured to be equivalent to the Polarization electric field, “4πP” of a polarized dielectric medium coupled to the source field. iv. The modes of the PV spectrum contributing to gravitational effects exist as odd harmonics over the domain “nPV = -nΩ, 2 - nΩ ... nΩ”, symmetrical about the 0th mode. The even modes (Imaginary component) of the complex Fourier function are disregarded due to null summation for all “|nPV|”. v. The amplitude spectrum of the Fourier distribution is proportional to the conjugate Photon pair population. We shall continue the construct by establishing a useful mathematical operator for subsequent use. It takes the form of the average value at each harmonic mode utilising the summation operand defined by equation (8) and may be generated as follows, 1 n PV. ω PV( 1 , r , M ) n PV. ω PV( 1 , r , M ) .

2 . i . π .n PV .ω e π . n PV

.. PV( 1 , r , M ) t i

0.( s )

dt

4 2 n PV. π

(18)

By considering an Electron at rest as a solid spherical particle with uniform surface and homogenous mass-energy distribution, we may determine the magnitude of the average power at each odd harmonic mode. An important aspect to this, assuming an Electron radiates a spectrum of conjugate Photon pairs through its geometric boundary, is the proportional rest mass-energy power flow “cUm(re,me)” through the surface “4πre2”. Hence the mass-energy power flow at each mode “Ste” may be formed as follows, St e n PV

2 4.π .r e . c .U m r e , m e

4

. 2

π . n PV

(19)

Subsequently, the magnitude of the average energy per odd harmonic period on either side of the PV spectrum “Stg” is defined by, St g n PV

St e n PV n PV . ω PV 1 , r e , m e

(20)

Recognising that the PV spectrum is symmetrical about the 0th mode, we may formulate an expression for the mass-energy of the odd harmonic conjugate Photon pair population “mg”. Assuming that |nPV| = nΩ at the spherical boundary of an Electron, an upper limit for “mg” may be defined as follows, 6

mg N g

2 . St g n Ω r e , m e Ng

(21)

where, Variable Ng re me

Description Photon pair population Classical Electron radius Rest mass of an Electron Table 5, variable definitions.

Units None m kg

Evaluating equation (21) assuming that the population of conjugate Photon pairs is mode normalised to unity (Ng = 1) yields, mg ≈ 1.2 x 10-15 (eV) (22) 3.2

PHOTON MASS-ENERGY THRESHOLD

To predict the mass-energy threshold of a Photon “mγ”, we shall utilise the conjugate Photon pair population principles defined above. Firstly, we shall establish some useful mathematical relationships that facilitate the concise representation of “mγ”. It has been illustrated that the summation of the odd harmonic modes are representative of the magnitude of the acceleration vector “g”. [1] Therefore, summing the spectrum over the odd modes across both sides of the spectrum leads to the following representation with vanishing error, [8] proportional to the sum of all modes on the positive side of the spectrum as |nPV| → nΩ and nΩ >> 1, n Ω ( r, M ) 1 n PV

1

ln( 2 )

n PV

n PV = 1

ln 2 . n Ω ( r , M )

n PV

γ

(23) where, i. The LHS8 of the preceding equation denotes the summation of all odd modes across the entire spectrum, symmetrical about the 0th mode, following the sequence “nPV = -nΩ, 2 - nΩ ... nΩ”. ii. The middle expression of the preceding equation represents the summation of all odd and even modes on the RHS side of the spectrum following the sequence “nPV = 1, 2 … nΩ”. iii. “γ” On the RHS of the preceding equation denotes Euler’s Constant. There are half as many odd modes as there are “odd + even” modes when |nPV| → nΩ. Hence, we may deduce “mγ” by the following ratio, mg 1 > . ln 2 . n Ω r e , m e mγ 2

γ

(24)

Performing the appropriate substitutions and recognising that the preceding equation may be further reduced by usefully approximating the exponential term in equation (11) to unity, yields the Photon mass-energy threshold to be, 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

(25)

Evaluating yields, mγ < 5.75 x 10-17 (eV)

8

LHS = Left hand side, RHS = Right hand side. 7

(26)

By comparing the value of “mγ” derived to the value for the Photon mass-energy threshold by Eidelman et. al. and endorsed by the PDG “< 6 x 10-17 (eV)”, [2] it is apparent that “mγ” compares favourably. 4

COCLUSIOS

It has been illustrated that the PV model of gravity based on the existence of a spectrum of frequencies makes the following predictions, i. The Photon mass-energy threshold for a mode normalised population of Photons is believed to be “< 5.75 x 10-17 (eV)”, based on the physical properties of an Electron. ii. Experimental validation of the Photon mass-energy boundary predicted herein may be natural evidence of Euler’s Constant at a quantum level. References [1] 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. [2] Particle Data Group: http://pdg.lbl.gov/index.html; Citation: S. Eidelman et al., Phys. Lett. B 592, 1 (2004). [3] B.S. Massey, “Mechanics of Fluids sixth edition”, Van Nostrand Reinhold (International), 1989, Ch. 9. [4] Rogers & Mayhew, “Engineering Thermodynamics Work & Heat Transfer third edition”, Longman Scientific & Technical, 1980, Part IV, Ch. 22. [5] Douglas, Gasiorek, Swaffield, “Fluid Mechanics second edition”, Longman Scientific & Technical, 1987, Part VII, Ch. 25. [6] H. E. Puthoff, et. al., “Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar Flight”, JBIS, Vol. 55, pp.137-144, http://xxx.lanl.gov/abs/astro-ph/0107316, v1, Jul. 2001. [7] Erwin Kreyszig, “Advanced Engineering Mathematics Seventh Edition”, John Wiley & Sons, 1993, Ch. 10. [8] Lennart Rade, Bertil Westergren, “Beta Mathematics Handbook Second Edition”, ChartwellBratt Ltd, 1990, Page 175.

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