1

Derivation of Permittivity, Permeability, the Speed of Light, and Transit time Fluctuations due to Transient Dipoles in

the Vacuum.

Richard Bradford and Gordon Rogers

Abstract

Abstract (DRAFT)

Vacuum Permittivity, Permeability and the Speed, and Transit time fluctuations of light in the Vacuum are derived semi-classically directly from the electromagnetic influence of light on transient dipoles. The results are shown to be independent of mass of the transient dipoles for Permittivity and Permeability, while the speed of light and its Transit time fluctuations are independent of electrical charge and mass. There are no introduced experimental parameters in any of the cases. Given transient dipoles immersed in low–energy or slowly varying electromagnetic fields of light, two conservative field components contribute to the permittivity and permeability. One is the average intrinsic transient dipole moment and the other is an induced dipole moment. Also, the average total magnetic moment of the intrinsic transient dipole in the electromagnetic field is further decomposed into two components, which are the orbital motions and the coupled intrinsic spin from the dipoles’ oppositely charged constituent particles. It is assumed that there is unitary probability of absorption and emission of the photons of light by the transient dipoles given by the Thompson cross section. The cross section is shown to be proportional to the area of Compton wavelength squared.The cross section together with the aggregated mean-free path and the lifetime of the transient dipoles form the conduction path for light and determines its speed. Transit time fluctuations of light speed are derived from the distribution of the possible positions of the photons in lights’ wave packets. These characterizations are in accordance with the Heisenberg Uncertainty Relation, the Cauchy-Lorentz distribution and the standard Gamma exponential distribution. This provides an explicit mechanism for the permeability, permittivity, and the speed of light of typically seen as parameters in Maxwell’s classical Electromagnetic equations of light in the Vacuum.

Introduction

Mean values of four quantities from two curves’ graphs produces accurate results, 1% to 0.05%, for the vacuum

permittivity, permeability, and the speed of light using a semi-classical analysis. The algorithms for the analysis follow

those of references 1, 2, and 3 but differ from the referent algorithms in that there are no arbitrary parameters which

need to be set by comparison with measured quantities.

The derivations of the permittivity, permeability, speed of light, and transit time fluctuations due to the properties of transient dipoles with electric and magnetic moments in the vacuum are based on those of references 1, 2, and 3. On vacuum permittivity, the authors of references 1 and 2 use only the transient intrinsic electric dipole moment and its relative orientation to an external nearly constant weak electromagnetic field of light being carried while the author of reference 3 used only the induced electrical dipole moment. The proper analysis would utilize both of these contributions. As opposed to references 1 and 2, the vacuum permeability is determined by the relative orientation to the external field of the total magnetic moment of the transient dipole, which has contributions from both electrical currents in the charged dipole and the coupled spins of its constituent Fermi particles. The algorithm for the calculation of the speed of light due to the vacuum is similar to that of the authors in references 1 and 2. Deriving the transit time fluctuations initially follows from reference 2 but a different line of reasoning utilizing the Heisenberg Uncertainty Principle is added. There, an experiment has yet to be done. As opposed to references 1 and 2 there are no introduced parameters that need to be set by comparison with experimentally measured values. Four necessary mean values are established from the Cauchy-Lorentz and Exponential distributions. These values lead to nearly exact values for the vacuum permittivity, permeability, and the speed of light. The same values with uncertainties added are used in the formulation of the transit time fluctuation variance.

Permittivity of the vacuum

The electric dipoles are of the form di = eQiCƛCi,, where ƛCi = ħ/(mic), is the Compton wavelength of a fermion of mass, mi, in the dipole. C is a constant ≤ 1. If C > 1 the pair becomes real since the particles become distinguishable outside their Compton wavelengths. Qi is a fraction between -1 and 1 inclusive. Qie is the charge for a particular species of charged particles e.g. -1e, Qi = -1 for the electron, muon, tau or +2/3e, -1/3e for Qi = +2/3 and -1/3 for up and down quarks. The vacuum permittivity and permeability result from a contribution of all virtual particle-antiparticle dipoles. Needed in the derivations is the sum ∑iQi

2 over all species of charged particles in the Standard Model. One

generation of particles is e-, electron, +2/3e, up quark, and -1/3e, down quark. The sum is ( -1)2 + 3X((+2/3)

2 + (-

2

1/3)2) = 8/3. The three is for the color charges of the quarks. Since there are three generations, then ΣiQi

2 is 8/3 X 3 =

8. The sum of the mass of the various dipole species turns out to not be required since the mass terms cancel out.

In a small volume of dipoles, a slowly varying external electromagnetic field of light is approximately constant. There are two components contributing to the vacuum permittivity from transient electric dipoles immersed in the field. One is the average intrinsic electric dipole moment and the other is an induced dipole moment. The orientations of intrinsic transient electrical dipoles in the vacuum without an external electromagnetic field are random and the net vacuum dipole moment is zero. Given an external electromagnetic field, E, the net dipole moment in the vacuum is non-zero. The energy of orientation of the electrical transient dipole, d, to the electric field is d*E or d*E*cosθ, where θ is the

angle between the dipole vector and the vector of the external electric field.

The Heisenberg Uncertainty Relation for energy-time of a transient dipole is Ƭ = ħ/2ε, where ε is the energy of the dipole and Ƭ is the time of its existence. Adding orientation energy, the uncertainty in the time of existence of the dipole is modified to Ƭ(θ) = ħ/2(ε – d*E). The time of existence of the transient dipole is then asymmetric with the

result that longer life transient dipoles dominate over ones with shorter life and a net electric dipole is established in the vacuum. The transient time Ƭ(θ)= ħ/2(ε – d*E) = ħ/2(1 – ηcosθ), where η = dE/ε. The average dipole aligned with the external field E is determined by an integration weighted by the transient time as a function of orientation angle τ(θ) as,

Di = ∫0πdcosθƬ(θ)2πsinθdθ/∫0

πƬ(θ)2πsinθdθ. To lowest order it equals Di = (d

2/3ε)E ≠ 0.

The calculated permittivity is ε’ and the polarization P = ε’E, which also equals Ni*D, where Ni is the density of dipoles in the vacuum. Then, ε’ = ND/E = Nidi

2/3ε.

The induced polarization onto the vacuum transient electrical dipoles from the external electromagnetic field of light field is given by p = eQx, where p is the induced dipole moment of the dipole, eQ is the charge of the dipole, and x is the amount of displacement of the charges in the dipole. The transient dipole is considered a harmonic oscillator described by mω0

2x = eQE, where m is the mass, ω0 = energy/ħ, is the resonant frequency. The resonance

frequency is determined by energy associated with a quantum transition from a ground state with energy 2mic2. The

maximum quantum transition, Egapmax, is the energy transition that creates two transient dipoles at rest, 4mic2, from

the ground state. The value of ω02 is determined from a transition from an energy state determined below to 4mic

2

and Egapmax.

Then, p = [e2Q

2/(mω0

2)]E. The vacuum polarization, P = e

2Q

2Ni/(mω0

2), where Ni is the dipole density in the vacuum.

The quantities needed are: Ni, di, ε, ω02

These are determined by using two statistical distributions. One is the Cauchy-Lorentz (CL) distribution and the other a special case of the Gamma distribution or Exponential (e) distribution. The CL-distribution is used to determine the quantities ε, the dipole energy and ω0

2, the square of the resonance frequency, and the e-distribution is used to

determine the size of the dipole or equivalently the constant C in CƛC. Given CƛC the dipole density N = [CƛC]-3

follows.

A CL-distribution neglecting the 1/π factor is pictured in Figure 1 on page 3. The energy ε ranges from the rest energy of the dipole, 2mic

2 to a maximum of 4mic

2 where two dipoles at rest may be created by the Heisenberg

Indetermination of vacuum energy, ∆ε ≤ ħ/2∆Ƭ. ∆ε is then 2mic2. The maximum width at half height is set to the

maximum dipole energy of 4mic2 and is equal to Γ. The equation describing the curve is f(E) = Γ/2/(E

2 + (Γ/2)

2) and

without the mic2 factors is f(E) = 2/(E

2 + 4). The maximum height of the curve at E = 0 is f(E) = ½. E = 2 corresponds

to 2mic2 at half height, f(E) = ¼. E = 1 corresponds to mic

2 and f(E) = 2/5. Note: The actual total value of energy is

twice that of 2mic2 and mic

2. The mean value of f(E) = (f(2) + f(1))/2 = (1/4 + 2/5)/2 = 0.325. This value of F(E)

corresponds to an E given by 0.325 = 2/(E2 + 4) or E = 1.46mic

2 with the energy factor inserted. The total energy

width is then 2(1.46mic2) = 2.92mic

2 = ε.

The value of ε is 2.92mic2

. This is shown in Figure 1 on Page 3.

The CL-distribution shown in figure 2 on page 4 is used to set the value of ω02. The ground state of the transient

dipole is equal to 2mic2 and its maximum excited state is 4mic

2 or Egapmax, which is equal to 4mic

2 – 2mic

2 = 2mic

2.

The full width at half height is (Egapmax)2 = 4mi

2c

4 and is equal to Γ

2. Egap with ε taken from above is 4mic

2 – 2.92mic

2 =

1.1mic2 and Egap

2 = 1.2mi

2c

4. The equation describing the curve without the mi

2c

4 factor is f(E

2) = (Γ

2/2)/((E

2)2 –

(Γ2/2)

2) = 2/((E

2)2 – 4). The maximum height at E = 0 is again f(E

2) = 1/2. One-half height that gives Γ

2 = 4mi

2c

4 is f(E

2)

= 1/4, which occurs at E2 = 2 or 2mi

2c

4. The E

2 that corresponds to 1.2mi

2c

4 is (1.2 mi

2c

4)/2 = 0.6mi

2c

4 or without the

energy square factor is 0.6 giving f(E2) = 0.46. The mean value of f(E

2) is (1/4 + 0.46)/2 = 0.36. The E

2 that

corresponds to f(E2) = 0.36 is 1.25mi

2c

4. Then the full width is 2(1.25mi

2c

4) = 2.5mi

2c

4. Then, ω0

2 = 2.5mi

2c

4.

3

As shown in figure 3 on page 5, the exponential distribution, which is a special case of the Γ-distribution, is used to

set the value of C in CƛCi, which determines the size of the dipole and the density of dipoles in the vacuum. The

exponential distribution curve with parameter 1/ƛC is described by the equation f(r) = (1/ƛCi)exp(-r/ƛCi) where ƛCi =

ħ/mic. The value of f(r) at r = 0 is 1/ƛCi and its value at r = ∞ is 0. The mean value of f(r) is (f(0) + f(∞))/2 = (1/ƛC – 0)/2

= 1/2ƛCi. Then (1/ƛCi)exp(-r/ƛCi) = 1/2ƛCi and r = 0.69ƛCi giving C the value of 0.69. The dipole di= eQ0.69ƛCi. The

density of dipoles, Ni, is then given by Ni≈1/(0.69ƛCi)3 = 3/ƛCi

3 = 3.0mi

3c

3/ħ

3.

To summarize: ε = 2.92mic2, ω0

2 = 2.5mi

2c

4/ħ

2, di = eQ0.69ƛCi and Ni = 3.0/ƛCi.

The Pauli Exclusion Principle excludes two dipoles, each having a constituent particle with the same quantum numbers from forming at the same place at one time. The authors of reference 2 using the analysis of Fermi energies in a solid state determine that the separation of the identical particles is ∆xi = 2πƛCi/(KW – 1)

1/2. KW originated as a

parameter in determining the transient time of the dipole Ƭi = ħ/KW4mic2. KW = 31.9 was determined by comparison

with the experimentally determined value of the vacuum permeability. The value was ∆x = 0.2ƛCi and so the density of dipoles is Ni = 1/∆xi

3 = 130/ƛCi

3. The authors in reference 1 state that the minimum distance of separation of two

identical particles due to the Pauli Principle is ƛCi/2 or Ni = 8/ƛCi3. The density found in the derivation here is 3/ƛCi

3,

which is without conflict.

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The calculated permittivity can now be done.

ε’ = (Nd2)/(3ε) + (Ne

2Q

2)/(mω0

2), which the total contribution from the intrinsic and induced electric dipoles in an

external field, respectively. Using the values found and the analysis above the calculated permittivity is:

ε’ = Σi[3.0mi3c

3(0.69)

2e

2Qi

2ħ

2]/[ħ

3mi

2c

2*3*2.92mic

2] + [3.0mi

3c

3e

2Qi

2ħ

2]/[ħ

3mi*2.5mi

2c

4] =

(1.3 + 9.6)e2/ħc = 10.9*8.12X10

-13F/m = 8.85X10

-12F/m. The measured value is ε0 = 8.8542X10

-12F/m. The calculated

and measured value are the same to three significant figures.

Permeability of the vacuum.

The simplest assumption is that a transient dipole is created from one transient photon of sufficient energy and annihilates into one transient photon. One transient photon, which has a zero net electric charge, can create two charges of opposite electrical polarity forming a transient dipole. Conservation laws of momentum and energy are violated due to a short time period, which allows the one transient photon process whereas this process is forbidden on the mass shell where the conservation laws hold over a long duration of time.

In Figure 4 on page 6, it is shown in a semi-classical way the creation point of a transient dipole where an e- that represents a charged particle and an e+, that represents the oppositely charged anti-particle, are brought into existence. The opposing point on the orbit is the annihilation point of the transient dipole. The e+ is depicted to move

5

on one side of the orbit while the e- moves on the other. Their angular momentum vectors Le- + Le+ then sum to zero.

Even though the opposite electrical charges move with opposite angular momentum, their electric currents move in the same direction around the orbit and produces a total magnetic moment 2μB, where μB is the Bohr magneton = eħ/2m. The orbital magnetic moment couples to an external nearly constant electromagnetic field, which has an energy associated with the orientation of the total orbital magnetic moment to the magnetic field of -2μBBcosθ.

Figure 3.

The intrinsic spins of the two components of the dipole are oriented in the opposite directions and due to their opposite charges contribute 2μB to the total magnetic moment. The coupling of the magnetic moment from orbital and intrinsic spin to the external magnetic field then has a total orientation energy of -4μBBcosθ. The Bohr magneton is also the natural constant for the orbital and intrinsic magnetic moment of spin ½ fermions.

A spin one of magnitude ħ transient photon creates a transient dipole. Then the transient dipole must assume a spin one status. The total angular momentum of the particle and anti-particle according to the analysis in Figure 4 is zero. Thus, the spin coupling in the transient dipole must have size ħ or spin one. The spin state that is spin equal one and

where the particle and anti-particle have opposite spins is Then the transient photon creates a transient dipole .<0 1 ا

with spin state and decays to a transient photon with spin 1. The charge conjugation of the e-, e+ dipole is <0 1 ا

equivalent to an exchange of spin labels giving a sign change of -(-1)S and parity along with interchanging the

electrical charges. The eigenvalues of charge conjugation are (-1)l+S

. The dipole with total angular momentum l = 0 and total spin s = 1 has odd charge conjugation parity or (-1)

0+1 = -1. The interchanging of the positive and negative

charges reverses the direction of polarization of the electric field so the charge conjugation of a photon is -1. A system of n photons has charge conjugation parity of (-1)

n. With one photon, n = 1, the odd charge conjugation parity

of the transient photons is conserved during creation and upon annihilation of the transient dipole. Charge conjugation and spin one is conserved.

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As with the analysis for the permittivity, the transient time of existence of the dipole is modified by the total magnetic moment magnetic field coupling of -4μBcosθ to give τ(θ) = ħ/[ε - 4μBBcosθ]. Again there are dipoles with a longer transient time and some shorter. There is then an average vacuum magnetic moment ≠ 0 of the dipole given by <m> = ∫0

π4μBcosθτ(θ)2πsinθdθ/∫0

πτ(θ)2πsinθdθ.

To lowest order <m> = [16μB2/3ε]B. The magnetic moment per unit volume is M = <m>N. The calculated permeability,

μ’, is 1/μ’ = M/B = Σi[3.0mi3c

3*16ħ

2e

2Qi

2]/[ħ

3*4mi

2*3*2.92mic

2] = 10.96e

2c/ħ = 10.96*7.314X10

4. μ’ =1/

[10.96*7.314X104] = 12.5X10

-7N/A

2 The measured value is μ0 = 12.57X10

-7N/A

2.

Figure 4.

Speed of light, c.

The speed of light following references 1 and 2 is developed as follows. The Thomson cross section, σ, was used to describe absorption and emission of photons (hυ < < 2mc

2) by transient dipoles in the vacuum without altering the

photon beam structure. A red long lasting laser stream or white light from the sun streaming photons is obviously preserved when propagating a long distance.

In the Thomson cross section, σThom = (8π/3)α2ƛCi2, one α is the probability for absorption and the other α is the

probability for emission as stated in references 1 and 2. Since the dipole is transient, its absorbed photon is emitted with probability of α = 1 upon annihilation. References 1 and 2 obtains a Q

2 as a result of α

2 in σThom one Qi for each

α. Since the probability of emission is one, i.e. one α = 1, this implies that all dipoles regardless of their charge emits a photon upon its disappearance. Then, the emission of the photon is independent of the dipole charge and one Qi is eliminated. The probability of absorption, the other α must depend on the other Qi. One Qi summed over a generation of charged particles species is ΣiQi = [-1 + 3(2/3 – 1/3)*3] = 0. Based on the derivation of references 1 and 2, the speed of light is cave = 1/[ΣiσiNi/2]. Since σThom = (8π/3)α

2ƛCi

2, the authors obtain σi = (8π/3)αQi

2ƛCi

2 because one α =

1. Now one of the Q factors was argued away in the emission probability as being independent of it. The result is σi = (8π/3)αQiƛCi

2. The summation on Qi = 0 that renders c = 1/0 = ∞. The viable solution is that a photon appearing at a

dipole during its transient existence will absorb the photon with a probability of the other α = 1. Since an appearing dipole will absorb the photon with probability α = 1 it is also independent of its associated charge and, thus, no Q factors appear. The cross section is then proportional to the area of the dipole only.

With α2 = 1, σThom = (8π/3)ƛCi

2. The mean free path between interactions is Λ =1/(σN). Travelling a distance L the

average number of stops is Nstopave = L/Λ. The photon will possibly encounter a transient dipole during its average

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time of its existence or Ƭ/2. The total mean time to cross the distance L is Tave = NstopaveƬ/2. Then the average photon speed is cave =L/Ƭave = 1/(σiNiƬ/2). Using the above determined values

Cave = 1/[(8π0.692ħ

2*3mi

3c

3ħ)/(3.0mi

2c

2ħ

3*4*2.92mic

2)] = 1/[(8π(0.69)

2*3)/(3*4*2.92c)]. There is no sum since all mass

terms cancel and there are no Qi terms.

Then cave = 0.99c.

The accuracies of these three calculated values suggest that this approach closely approximates the physical mechanisms in the vacuum.

Transit time fluctuations.

The transit time fluctuations is a variance of propagation time σƬ of a photon over a distance L. The main influence is due to the leading term consisting of transient electron-positron dipoles.

As quoted σt2 = σNstop

2(τ/2)

2 + Nstopaveστ

2. στ

2 = τ

2/12 and σNstop

2 = Nstop. Nstop is variance of the number of interactions

and σT2 = (1/3)Nstopaveτ

2 = (L/3)σNτ

2 = (2L/3)(σNτ/2)τ = (2L/3)(1/c)τ.

The variance in the number NStop is given by fixed quantities and no differences in σ or N.

The variance in the transit time is equal to the variance in 1/c together with the spread in possible positions of the photon during its propagation from emission to absorption. The spread of positions of where the photon is located in the span of the mean free path is represented by a Gaussian distribution with the standard deviation equal to the change in the mean free path, ∆Λ = σΛ. At one standard deviation the total width of the spread of positions is +-σΛ or 2σΛ = 2∆Λ. It makes sense to define a local low sample variance in 1/c and 2∆Λ, since there are approximately 10 mean free paths in a distance of ƛC or approximately 1000 in a cubic volume of ƛC

3. This implies that a small region

can have a local density, some deviations in cross sections, transient time fluctuations, and possible positions of the photon in 2∆Λ. A natural multiplier to quantify the number of these small volumes traversed with distance, L, is the non-unit ratio L/ƛC. The accumulation of the transit time fluctuations then occur with an increasing L.

Then σT2 = (L/ƛC)*(σ1/cσΛ)

2. The mean values in determining the variance of 1/c together with ∆Λ are used from

above.

The uncertainty between the minimum and maximum values of the dipole is ∆ε = 2mc2. Then the uncertainty in the

time of existence of the dipole is ∆Ƭ = ħ/2∆ε = ħ/(2*2mc2) = 0.25ħ/mc

2. The change in the average time of existence

of the dipole using values above is ∆Ƭ/2 = Ƭ2/2– Ƭ1/2, where Ƭ2/2 = ħ/[4(2.92 – 0.25)mc2] = 0.094ħ/mc

2 and Ƭ1/2 =

ħ/[4(2.92 + 0.25)mc2] = 0.079ħ/mc

2. Then ∆Ƭ = 0.015ħ/mc

2.

From the relativistic energy-momentum relation, the minimum value of the momentum of the dipole is (2mc2)2 =

(2mc2)2 + p

2c

2 that leads to p = 0 or a dipole created at rest. The maximum value of the momentum is governed by

the maximum energy before creating two dipoles at rest or (4mc2)2 = (2mc

2)2 + p

2c

2 where p = √12mc. Then ∆p =

√12mc and ∆x = ħ/2√12mc = 0.14ħ/mc = 0.14ƛC.

Based on ∆x, the change in the cross section is ∆σ = σ2 – σ1, where σ2 = (8πƛC2/3)[(0.69 + 0.14)

2 = (8πƛC

2/3)(0.69)

and σ1 = (8πƛC2/3)(0.69 – 0.14)

2 = (8πƛC

2/3)(0.3). Then, ∆σ = (8πƛC

2/3)*0.39. Also based on ∆x, ∆N = N2 – N1. N2 =

1/[(0.69 – 0.14)3ƛC

3] = 6.0/ƛC

3 and N1 = 1/[(0.69 + 0.14)

3ƛC

3] = 1.7/ƛC

3. ∆N = 4.3/ƛC

3.

The Gaussian distribution having a standard deviation of σΛ = ∆Λ is given by (1/∆σ)(1/∆N). The total spread, 2∆Λ, is 2(1/∆σ)(1/∆N) = 2*(3/8π)1/0.39*1/4.3ƛC = 0.14*ƛC.

The standard deviation of 1/c and ∆Λ together is viewed as the possible values in a 4-dimensional abstract volume composed of ∆σ, ∆N, and ∆Ƭ/2, and ∆Λ.

All of the values of 1/c and 2∆Λ are contained in the 4-volume if each of its components values fall within the range of ∆σ, ∆N, ∆τ/2, and 2∆Λ governed by the uncertainties determined by the minimum and maximum energy of the dipole, and momentum, i.e. 2mc

2 for one dipole at rest and 4mc

2 for the creation of two dipoles at rest, √12mc for

momentum.

Then (σ1/cσΛ) = 2*∫0.30.69

∫1.76.0

∫0.0790.094

∫00.071

dσdN(dτ/2)dΛ. = 2*0.3ا0.69

σ1.7ا6.0

N0.079ا0.094

τ/20ا0.071

Λ

Where “ا” is simply the evaluation of the integrals at the noted endpoints.

=[(8πƛC2/3)*0.39][4.3/ƛC

3][0.015ħ/mc

2][2*3/8π)*0.60ħ/mc]=2*[(8π/3*3/8π)*0.39*4.3*0.015*0.60][1/c][ƛC]=

[2*0.39*4.3*0.015*0.60]ƛC/c = 0.030ƛC/c. The variance is (σ1/cσΛ)2 = 0.00090ƛC

2/c

2.

8

The variance in transit time, σT2 is equal to the variance of 1/c and ∆Λ together, (σ1/cσΛ)

2 multiplied by the natural non-

unit number L/ƛC. The derivation of the speed of light and of the transit time fluctuations assumes that the photon travels on a straight path due to an overall conservation of linear momentum when traveling a large distance L. Traveling in a deviant random non-straight path after each emission sets a random walk situation where the probability is low that the photon will cross the base line straight path as the number of absorptions and emissions increase with time with the result that the photon will overwhelmingly may not reach the target as is not observed physically.

Then, σT2 = 0.00090(L/ƛC)(ƛC

2/c

2) = 0.00090LƛC/c

2. And σT = √(0.00090)√L√ƛC/c = 0.030√L*2.1fs/m

1/2 =0.063fs/m

1/2.

Astronomical events at distances on the order of 1026

to 1020

m having a short durations of brightness, 4ms and 1μs, respectfully, have placed upper limits on the transit time fluctuations of 0.3fs/m

1/2 and 0.2fs/m

1/2.

The authors in reference 2 propose an experiment with an initial 9fs pulse being reflected up and back travelling a total distance of 3km and broadening to a 13fs pulse having a difference of 4fs from start to finish. See reference 2 for references. The authors result for σT was 0.05fs/m

1/2. Then 0.05fs/m

1/2√3000m ≈ 2.7fs. With the value calculated in

this paper the result is 0.063fs/m1/2

√3000m = 3.5fs. The authors in reference 2 are gathering funds to perform the experiment and the result will be determined.

We note that the expansion of the universe needs to be taken into account when correlating received pulse width from the emitted width when dealing with large distance astrophysical objects. With z =0.9 and a distance of 2X10

26m

and an initial pulse of 4ms at the source, the pulse duration added to 4ms is approximately 3.6ms. As example, the initial pulse length is 4ms*c = 1.2X10

6m. Then the back of the pulse as a z value with respect to the front of z =

0.9*1.2X106m/2X10

26m = 5.4X10

-21. The transit time of the pulse from the source to the receiver is t = 2X10

26m/c =

6.7X1017

s. The relative speed of the back of the pulse away from the front is u = 5.4X10-21*

c = 1.6X10-12

m/s. The lengthening of the pulse 6.7X10

17s*1.62X10

-12m/s = 1.1X10

6m. Then the duration added is 1.1X10

6m/c = 3.6ms.

Then the initial pulse at the source 0.4ms. This would require analysis of what type of event and whether the brightness and size would support such a time of occurrence. For the Crab pulsar at distance approximately 1X10

20m

the duration added to the 1μs pulse is approximately 4X10-13

s. Then the initial pulse is 1μs, which is supported.

References

1. Does the Speed of Light Depend upon the Vacuum? Urban, Couchot, Sarazin. Arxiv.org 1106.3996v1 2. The Quantum Vacuum as the Origin of the Speed of Light. Marcel Urban, Francois Couchot, Xavier Sarazin,

Arache Djannati-Atai. Eur. Phy. J D (2013) DOI 10.1140/epjd/e2013-30578-7 3. The Quantum Vacuum as the Foundations of Classical Electrodynamics. G. Leuchs, A.S. Villar, L.L.

Sanchez-Soto App1 Phys B (2010) 100: 9-13 DOI 10.1007/s00340-010-4069-8 4. Positronium: Review of Symmetry, Conserved Quantities and Decay for the Radiological Physicist. Micheal

Harpen Dept of Radiology, 2451 Fillingim St, Mobile, AL 36617 5. J.D. Jackson Classical Electrodynamics 3

rd edition, Wiley

6. The Gaussian and Normal Probability Density Function. John Cimbala Penn State University Sept 2013.

Richard Bradford completed his Master’s degree in physics at UC-Davis, after receiving a departmental

citation there for his Bachelor’s degree in physics, as well as the Saxon-Patten Award for Physics.

Gordon Rogers BA Mathematics U. C. Berkeley, Mathematics where he concentrated in Algebras,

Optical Physics, Geometric Optics, Opto-electronic Computer Architecture, under EECS Dean John

Whinnery, Electro Dynamics under theoretical physicist Abraham Taub: Honors Courses in the above as

well as geometries and algebras, physics.