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The Radius of the Electron

May 24, 2016

The Radius of the Electron www.vgelectron.net

Abstract

A lagrangian formulation for the vacuum gauge electron is developed bydrawing on analogies with classical theories of acoustic fields in continuousmedia. The theory has several notable underlying features including a secondrank vacuum stress tensor representing the deformed state of the vacuum inthe neighborhood of the charge. The vacuum tensor is characterized by a zerodeterminant and operates relative to a flat metric so that vacuum deformationscannot be directly associated with spacetime curvature.

The theory of the vacuum gauge electron is based solely on the requirementto enforce causality in the electromagnetic field. Like the fields of the particleitself, the associated lagrangian is a causal lagrangian having a naturally builtin symmetry for the propagation of vacuum waves. Among other things, themotion of the fields define a particle radius and provide stability relative to anarbitrary frame of reference.

Accelerated motions of the electron are easily accommodated by the vac-uum lagrangian with the addition of an acceleration strain tensor. The totallagrangian then generates a Noether current identical to the symmetric stresstensor derived in the conventional electromagnetic theory.

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“It is therefore impossible to get all the mass to be electromagnetic inthe way we had hoped. It is not a legal theory if we have nothing butelectrodynamics. Something else has to be added.”

R.P. Feynman, The Feynman Lectures, chapter 28

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Contents

1 Vacuum Gauge Electron as an Electromagnetic Theory 5

2 Deformation of the Vacuum by Electromagnetic Fields 92.1 Wave Propagation in Elastic Media . . . . . . . . . . . . . . . . . . . 92.2 Vacuum as a Deformable Medium . . . . . . . . . . . . . . . . . . . . 102.3 Propagation of Vacuum Waves . . . . . . . . . . . . . . . . . . . . . . 12

3 Lagrangian Formulation of the Vacuum Gauge Electron 143.1 Vacuum Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Properties of the Vacuum Lagrangian . . . . . . . . . . . . . . . . . . 183.3 Symmetric and Total Stress Tensors . . . . . . . . . . . . . . . . . . . 21

4 Stability of a Causal Electron: Vacuum Energy 254.1 Rest Frame Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Moving Frame Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3 Classical Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Accelerated Motions of the Electron 345.1 Theory of Acceleration Strain . . . . . . . . . . . . . . . . . . . . . . 355.2 Generalized Vacuum Lagrangian . . . . . . . . . . . . . . . . . . . . . 375.3 Symmetric and Total Stress Tensors . . . . . . . . . . . . . . . . . . . 39

A Lagrange Equations for Particle Accelerations 41

B Contractions of Vacuum Strain Tensors 42

C Derivatives of the Null Vector 43

List of Figures

1 Comparison of electromagnetic and vacuum lagrangians . . . . . . . . 192 Plot of vacuum lagrangian versus dilation . . . . . . . . . . . . . . . . 213 Rest frame step function with phase . . . . . . . . . . . . . . . . . . . 264 Spacetime diagram for particle radius . . . . . . . . . . . . . . . . . . 34

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1 Vacuum Gauge Electron as an Electromagnetic

Theory

Define an inertial reference frame in a flat spacetime using the Minkowski metric

gµν =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

(1.1)

Requiring a strict enforcement of causality, the the field strength tensor for a chargedparticle created at the origin at time ct = 0 is

F µν = F µνM · ϑ (1.2)

where F µνM is the conventional Maxwell-Lorentz field and ϑ = ϑ(ct − r) is a causal

light sphere beyond which no information is available about the particle. Takingthe divergence brings about a modification to the conventional notion of charge andcurrent density associated with the particle since

∂µFµν =

4π

cJ∗νe (1.3)

whereJ∗νe ≡ Jνe · ϑ+ JνN JνN ≡

c

4πF µν

M · ∂µϑ (1.4)

and where JνN may be referred to as the null current having a four-space norm ofzero.

Based on causal fields, a causal potential theory can be constructed from thehomogeneous Maxwell-Lorentz equations leading to

2Aν − ∂ν∂µAµ =4π

cJ∗νe (1.5)

However, unlike the conventional theory, the presence of the causal light sphere(causality step) mandates the application of the covariant vacuum gauge condi-tion,

∂νAνv ≡√

E2 − B2 (1.6)

Inserting the fields of the electron leads to a de-coupled theory of velocity and accel-eration potentials:

2Aνv − ∂ν∂µAµv =4π

cJ∗νe 2Aνa − ∂ν∂µAµa =

4π

cJνa (1.7)

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Solutions to these two equations can be combined to form the most general vacuumgauge solution for arbitrary motions of the charge:

Aν(rrr, t) =e(1− aλRλ)

ρ2Rν · ϑ (1.8)

For reference, the fields produced by Aνv and Aνa are

F µνv =

e

ρ3[Rµβν −Rνβµ] · ϑ (1.9a)

F µνa =

e

ρ2[Rµaν −Rνaµ − ξ(Rµβν −Rνβµ)] · ϑ (1.9b)

where ξ ≡ aλRλ/ρ. The structure of (1.8) also allows the potentials to be furtherdivided as

Aν = Aνe + Aν` + Aνa (1.10)

where

Aνe =e

ρβν · ϑ Aν` =

e

ρUν · ϑ Aνa = − e

ρ2(aλRλ)R

ν · ϑ (1.11)

Each of these potentials can be associated with its own vector current density, andeach current density points in the same direction as its associated potentials. Acovariant form for J∗νe follows as an integral over proper time while remaining currentdensities are

Jν` =ec

2πρ3Uν · ϑ Jνa = − ec

2πρ4χRν · ϑ (1.12)

Source wave equations for the potentials Aνe and Aν` can also be written as

2Aνe =4π

cJνe · ϑ 2Aν` =

4π

cJν` · ϑ (1.13)

Electromagnetic Lagrangian as a Vacuum Gauge Theory: The separation ofthe velocity and acceleration fields generated by the vacuum gauge condition allowsfor the possibility of constructing a classical lagrangian for a charged particle as afunction of independent field quantities. In consideration of notational simplicity itis useful to make the temporary replacements

Aνv → Vν Aνa → Aν ∂µAνv → Vµν ∂µAνa → Aµν (1.14)

so that Hamilton’s principle can be written

δS = δ

∫Lvg[V

µν ,Aµν ,Vν , Aν , xν ]d4x (1.15)

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The functional form of Lvg can be derived beginning with the classical electromagneticlagrangian

Lem = − 1

16π(∂µAν − ∂µAν)(∂µAν − ∂µAν)−

1

cJ∗νe Aν (1.16)

and making the replacements

Aν → Vν + Aν J∗νe → J∗νe + Jνa (1.17)

The vacuum gauge result is then

Lvg = − 1

8π[VµνVµν − VνµVµν + 2AµνVµν − 2AνµVµν + AµνAµν −AνµAµν ]

− 1

cJ∗νe Vν −

1

cJ∗νe Aν −

1

cJνa Vν −

1

cJνa Aν (1.18)

Obviously, this lagrangian differs from the conventional electromagnetic lagrangianbased on the requirement to include the acceleration current density, but this is notan issue since the additional terms containing Jνa have a functional value of zero. Infact, the functional value of the free field lagrangian is identical to the conventionaltheory since terms containing the field quantity Aµν also add to zero. Euler-Lagrangeequations for the independent field quantities now follow from Hamilton’s principleleading to equations of motion given by

∂µ∂Lvg

∂Vµν− ∂Lvg

∂Vν= 0 ∂µ

∂Lvg

∂Aµν

− ∂Lvg

∂Aν

= 0 (1.19)

Not surprisingly, each set of lagrange equations produces identical equations of motion

∂µFµνv =

4π

cJ∗νe ∂µF

µνa =

4π

cJνa (1.20)

This verifies the equivalence relation Lvg = Lem.

Symmetric Stress Tensor as a Vacuum Gauge Theory: Beginning with Lvg

the canonical stress tensor is

T µνvg =∂Lvg

∂VµλVνλ +

∂Lvg

∂Aµλ

Aνλ − gµνLvg (1.21)

It is convenient to write this equation as

T µνvg = Θµνvg + T µνD

where

Θµνvg =

∂Lvg

∂Vµλ(Vνλ − V ν

λ ) +∂Lvg

∂Aµλ

(Aνλ −A ν

λ )− gµνLvg (1.22)

T µνD =∂Lvg

∂VµλV νλ +

∂Lvg

∂Aµλ

A νλ (1.23)

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The tensor Θµνvg is the most general, symmetric and traceless, gauge invariant tensor.

Using the relations

F µνv = Vµν − Vνµ F µν

a = Aµν −Aνµ (1.24)

it can be written

Θµνvg ≡

1

4π

[F µλF ν

λ +1

4gµνFαλF

αλ

](1.25)

To determine T µνD simply write

T µνD = − 1

4π

(Vµλ − Vλµ + Aµλ −Aλµ

)V νλ −

1

4π

(Aµλ −Aλµ + Vµλ − Vλµ

)A νλ

= − 1

4π(F µλv + F µλ

a ) ∂λAνv −

1

4π(F µλa + F µλ

v ) ∂λAνa

= − 1

4πF µλ∂λA

ν (1.26)

This term can now be removed1 from (1.22) with the observation that ∂µTµνD = 0,

and leading to the source equation:

∂µΘµνvg =

1

cF λνJ∗e λ (1.27)

It is important to mention here that details of the derivation of (1.27) show no con-tribution from the acceleration current density which is something of a spectator inthe calculation. The equivalence of electromagnetic and vacuum gauge theories for acharged particle in arbitrary motion is therefore

Θµνvg ≡ Θµν

em = Θµν (1.28)

For later reference, the specific form of Θµν can be written

Θµν = Θµν1 +Θµν

2 +Θµν3 (1.29a)

with individual components given by

Θµν1 =

e2

4πρ4

[1

ρ(Rµβν +Rνβµ)− 1

ρ2RµRν − 1

2gµν]· ϑ (1.29b)

Θµν2 =

e2

4πρ4

[aµRν +Rµaν − ξ(Rµβν +Rνβµ) +

2ξ

ρRµRν

]· ϑ (1.29c)

Θµν3 = − e2

4πρ4(ξ2 + aλa

λ)RµRν · ϑ (1.29d)

1For details on symmetrizing the canonical stress tensor, see Jackson, Classical Electrodynamics,section 12.10; Landau and Lifshitz, Classical Theory of Fields, Section 94.

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2 Deformation of the Vacuum by Electromagnetic

Fields

Vacuum gauge theory is specifically designed to accommodate causality applied tothe electromagnetic field of an electron. However, the presence of the gauge fieldalso allows for an interpretation of vacuum gauge velocity potentials as a deformationof the surrounding medium in the neighborhood of the charge. The vacuum gaugecondition is a precise measure of this deformation which can be demonstrated bydrawing on analogies with the classical theory of elastic media.

2.1 Wave Propagation in Elastic Media

In three-dimensional space, the displacement of an elastic continuum from its equilib-rium configuration is represented by a vector field uuu = uuu(rrr). The displacement resultsin stresses and strains on the surrounding medium which are adequately described bya symmetric stress tensor typically written

Tij = −µ(∂ui∂xj

+∂ui∂xj

)− λ δij∇ · uuu (2.1)

where the positive constants µ and λ associated with the medium are measuredexperimentally. The elastic strain tensor is defined according to

εij ≡1

2

(∂ui∂xj

+∂ui∂xj

)and ε ≡

∑i

εii (2.2)

and allows the stress tensor to take the form

Tij = −2µεij − λ δijε (2.3)

The amount of stored potential energy V per unit volume associated with the de-formed configuration can be determined by combining the stress and strain tensorsas

V = −1

2

∑i,j

Tijεij (2.4)

Assuming no first order changes in the density ρ of the material, and noting thatthe velocity of a point in the medium is given by ∂uuu/∂t , an appropriate lagrangiandensity inclusive of an interacting source fff can be written

L = 12ρ |uuu|2 − µ

∑i,j

εijεij − 12λε2 + uuu · fff (2.5)

Lagrange’s equations are

∂

∂t

∂L∂(∂ui/∂t)

+∑j

∂

∂xj

∂L∂(∂ui/∂xj)

=∂L∂ui

(2.6)

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which can be used to derive the dynamical equation

ρ∂2uuu

∂t2= µ∇2uuu + (λ+ µ)∇∇ · uuu + fff (2.7)

If the deformation and the force density are written in terms of transverse and longi-tudinal components

uuu = uuut + uuul fff = ffft + fffl (2.8)

the dynamical equation de-couples with uuut and uuul independently satisfying identicalequations

ρ∂2uuut,l∂t2

= µ∇2uuut,l + (λ+ µ)∇∇ · uuut,l + ffft,l (2.9)

However, these equations can be simplified considerably since the solenoidal characterof uuut and the irrotational character of uuul imply the relations

∇ · uuut = 0 ∇× uuul = 0 (2.10)

Source wave equations for the separate components are then

2tuuut =

1

c2t

∂2uuut∂t2−∇2uuut = ffft (2.11a)

2luuul =

1

c2l

∂2uuul∂t2−∇2uuul = fffl (2.11b)

where the source terms have been re-defined accordingly, and where the transverseand longitudinal velocities of propagation are given by

ct =

√µ

ρcl =

√λ+ 2µ

ρ(2.12)

2.2 Vacuum as a Deformable Medium

For a charged particle, the four space analog of the dynamical equation in (2.7) willbe the Maxwell-Lorentz equations for the velocity potentials.

2Aνv − ∂ν∂µAµv =4π

cJ∗νe (2.13)

It has already been shown that the velocity equation can be divided into two separateequations with associated four-currents Jνe and Jν` . This allows the vacuum gauge

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source equations in (1.13) to follow the source equations in (2.11) according to

2Aνe =4π

cJνe ←→ 2

tuuut = ffft (2.14a)

2Aν` =4π

cJν` ←→ 2

luuul = fffl (2.14b)

Jνe (J )ν = 0 ←→ ffft · fffl = 0 (2.14c)

The potentials themselves map to the three-space deformation according to

Aνv = Aνe + Aν` ←→ u = uuut + uuul (2.15a)

∂νAνe = 0 ←→ ∇ · uuut = 0 (2.15b)

∂µAν` − ∂νAµ` = 0 ←→ ∇× uuul = 0 (2.15c)

Aν` = ∂νϕ ←→ uuul = −∇ϕ (2.15d)

Now suppose an elastic disturbance uuu(rrr, t) has a longitudinal component defined bythe wave vector kkk. The projection operation analogy takes the form

Aνe = Aνv + (AµvUµ)Uν ←→ uuut = uuu− (uuu · kkk)kkk (2.16a)

Aν` = −(AµvUµ)Uν ←→ uuul = (uuu · kkk)kkk (2.16b)

Each of the six previous equations for the potentials is also mirrored by the currentdensities:

Jµv = Jµe + Jµ` Jµe = Jµv + (JνvUν)Uµ ∂µJµe = 0

(2.17)

∂µJν` − ∂νJµ` = 0 Jµ` = −(JνvUν)Uµ Jµ` = ∂µψ

In words, the transverse and longitudinal deformations of the three-space continuumget replaced by timelike and spacelike potentials in the four-space theory of a chargedparticle; and these potentials are determined from associated timelike and spacelikecurrent densities. For an accelerating charge it could be argued that the analogywould fall apart because the total potential will acquire an acceleration term

Aν = Aνe + Aν` + Aνa (2.18)

This issue is resolved by asserting that the analogy only applies to the velocity portionof the total potential. This is reasonable because Aµa satisfies its own independentdifferential equation.

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Another correspondence is the connection

∂νAνv ←→ ∇ · uuu (2.19)

But since the transverse components do not contribute to the total divergence, then

∂νAνv = ∂νA

ν` = 2ϕ ←→ ∇ · uuu = ∇ · uuul = −∇2ϕ (2.20)

According to the theory of elastic media, the three-space divergence on the rightmeasures the net distortion, or volumetric dilatation, of the medium. This suggeststhat the vacuum gauge condition

∂νAνv =

e

ρ2(2.21)

is really a vacuum dilatation gauge condition—a measure of the amount by whichthe surrounding vacuum is deformed by the presence of the particle. The implicationhere is that a comprehensive description of the vacuum gauge electron will supportthe introduction of a radius re into the theory. This is a very desirable feature ofany classical theory of the electron to address—among other things—the well knownproblem of the divergent self energy. Moreover, a particle radius in the context of agauge theory necessarily avoids the introduction of form factors and other senselessconstructs based on macroscopic classical notions. The theory of vacuum dilatationcan also be compared with the Lienard Wiechert potentials where the application ofthe Lorentz gauge condition ∂µA

µe = 0 generates a dilatation of zero and is unmistak-

ably associated with a point-particle theory.

2.3 Propagation of Vacuum Waves

To initiate the transition to a theory of a particle radius it is important to discussthe coupling of causality to the velocity fields in equation (1.9a). Causality impliesmotion of the velocity fields; and a moving field will require both momentum andenergy for its propagation. The principle of energy conservation is therefore violatedin the causal theory, but the essential point here is that the propagation of the fieldsis not possible without the introduction of a particle radius.

The radius itself is easily demonstrated by writing the vector potential as

Av(R) = 4π σe(θ, φ)uuu(R) · ϑ(ctr) (2.22)

This equation defines a charge density and a radial function

σe(θ, φ) ≡ σeγ2(1− nnn ·βββ)2

uuu(R) ≡ re2

Rnnn (2.23)

It is suitable to refer to uuu(R) as a dilatation function describing a microscopicspherical cavity (pin-prick) in what would otherwise be a continuous vacuum. The

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radius re is simply a constant and it will be assumed that its value is an invariantproperty of the theory of vacuum dilatation. A differential element of surface at theboundary defined by the dilatation can be written

daaa = uuu(re)redΩ (2.24)

and the invariant charge e is easily determined through an integration over theDoppler charge density.

One could argue here that it might be possible to associate the factor of γ withthe dilatation and produce a theory of the electron with a contracting radius. Thelogistics of equations (2.22) and (2.23) do not seem to make a preference for eitherpossibility. On the other hand, it is easier to formulate a theory where the radius issimply a constant of nature like c or ~. As a compromise, the radius will always bewritten re, but with the understanding that it can mean re = re(γ) in a moving framefor a theory where it contracts.

Returning to the invariant theory, re can be included as a phase in the step functionin equation (2.22) so that an expansion over Fourier components follows accordingto2

ϑ(ctr + re) =1

πi

∫ ∞−∞

eiω(tr+τe)

ωdω R ≤ ct+ re (2.25)

Pairing this with the dilatation function gives the Fourier transformation

uuu(R, t) =1

πi

∫ ∞−∞

uuuω(R)eiωt dω (2.26a)

uuuω(R) = aωe−iωR/c

ωRnnn where aω ≡ r2

e eiωτe (2.26b)

The form of (2.26) indicates that uuu(R, t) is a radiator of longitudinal travelling waves,having a polarization vector nnn, and radiating at all frequencies. Such waves might bereferred to as electromagnetic pressure waves or vacuum waves, and they radiatefrom the instantaneous retarded position of the particle. A Fourier component of thevector potential can be constructed as

Aω(R) = 4πaω · σe(θ, φ) · e−iωR/c

ωRnnn (2.27)

and this shows the role of the charge density as the directional component of thesignal amplitude.

2Certainly, the expansion over negative frequencies can be viewed as problematic. In truth, itshould be considered as a matter of convenience since the step can always be defined as a sum oversine waves.

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From here it is possible to develop a comprehensive theory of vacuum wave prop-agation using the method of Green functions. One is ultimately led to the infinitespace Green function

G(R,R′) =e−ik|R−R

′|

4π|R −R′|(2.28)

which characterizes the response of the vacuum to the creation of a charged particle.While the details of a Green function solution are important they can be addressedat a later time since they are not neccessary for the development of the lagrangianformulation presented here.

3 Lagrangian Formulation of the Vacuum Gauge

Electron

The correspondence between vacuum gauge velocity potentials and the theory ofelastic continua can be used as a guide for the development of a highly specializedlagrangian formulation for the classical electron. A strictly inflexible requirementfor a successful lagrangian must be the reproduction of all the usual gauge invariantresults of the conventional Maxwell-Lorentz electron theory. In addition—and basedon the work of the previous section—an appropriate formulation must also provide aframework to incorporate the proliferation of vacuum waves. All of these requirementscan be met beginning with the derivation of a four-space analog to the symmetricstress tensor Tij given in (2.1).

3.1 Vacuum Tensor

Derivation of the Vacuum Tensor: A firm theoretical foundation for the vacuumtensor can be established by working in the Maxwell limit and defining the thirdrank tensor Ψαµν composed of the four components of the vacuum gauge velocitypotentials3 along with two occurrences of the fourth rank, totally anti-symmetric,Levi-Civita symbol:

Ψαµν ≡ 12εαµστεστνκA

κ (3.1)

The field strength tensor follows from a divergence operation on the last index

∂νΨαµν = 12εαµστεστνκ ∂

νAκ = 12εαµστFD

στ = Fαµ (3.2)

Where FD

στ is the dual to the field strength tensor. Contractions on the third indexwith either βν or Uν also produces the field strength tensor

Fαµ = −1

ρΨαµνβ

ν =1

ρΨαµν Uν =

1

ρ[Aα, βµ] (3.3)

3For notational simplicity the subscript v on the velocity potentials and the velocity field strengthtensor will be removed from this point forward. No ambiguity with counterpart acceleration poten-tials and fields should arise.

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This derivation has the added benefit of showing the linear relationship between thefield strength tensor and the components of the potentials themselves. Now considera divergence operation on either of the two remaining indices. The inclusion of themetric is appropriate here to raise the last index with the result

∂αΨαµν = ∂α

[gνλΨαµλ

]= ∂νAµ − gµν∂αA

α (3.4)

The first term on the far right is the covariant derivative of the velocity potentials—exhibiting a single symmetric term dependent on accelerations of the particle whichcan be isolated from the velocity terms as

∂νAµ = ηνµ + Jµνa (3.5)

The symbol ηνµ = η†µν defines the components of a vacuum strain tensor whose

scalar contraction is exactly the vacuum gauge condition η = gµνηµν . Coupled with

the last term on the far right of (3.4) a concise representation of the vacuum tensorcan be written

∆†µν ≡ ηνµ − gµνη (3.6)

The peculiar inversion of the indices on the right side of this equation justifies theinclusion of the superscript (†) transpose operator in the definition, however this andthe untransposed version ∆µν will both be useful for the causal theory.

An explicit form of ∆µν can be determined by separating out the gauge field fromthe velocity potentials. Since Aν` is determined from the gradient of the scalar ϕ, then∆µν

` will be symmetric:∆µν

` = ∂µ∂νϕ− gµν2ϕ (3.7)

In terms of retarded coordinates, both tensors are

∆µν` =

e

ρ2

[2

ρ2RµRν − 2

ρ(Rµβν +Rνβµ) + βµβν

](3.8a)

∆µνe =

e

ρ2

[1

ρRµβν − βµβν

](3.8b)

Adding the two portions back together yields the final result

∆µν =e

ρ2

[2RµRν

ρ2− 2βµRν

ρ− Rµβν

ρ

](3.9)

with the observation that a term proportional to the metric is conspicuously absentfrom the equation.

15


Divergence of the Vacuum Tensor: An important contraction of the vacuumtensor is the divergence operation applied to the second index

∂ν∆µν = 0 (3.10)

This can be easily proved from the definition in equation (3.6) but it can also be provedby applying ∂ν directly to (3.9) where the divergence of each individual term can beshown to vanish. The divergence operator applied to the first index is a differentmatter. It is strictly only legitimate when applied to (3.5) yielding an additionalacceleration term

∂µ∆µν =

4π

c[J∗νe − Jνa ] (3.11)

The acceleration current density Jνa is identical to the current density determined bythe electromagnetic theory except now it should be linked to an independent vacuumtheory of acceleration strain. Keeping accelerations in the closet for now, the velocitytheory is then determined by

∂µ∆µλ =

4π

cJ∗λe (3.12)

This equation may be considered as the four-space analog of (2.7). It also represents alegitimate potential formulation for the velocity fields of the Maxwell-Lorentz electronderived from a completely different premise. The ultimate goal will be to show thatthe classical fields of the electron are indistinguishable from a theory of a deformedvacuum defined by vacuum gauge potentials. Whereas the field strength tensor is thefundamental field quantity of the electromagnetic theory, the counterpart quantity ofthe vacuum theory will be the vacuum strain tensor ηµν with the two theories linkedby the anti-symmetric combination

F µν = ∆µν −∆νµ (3.13)

This equation cannot be solved for either the vacuum tensor or the strain tensor interms of the field strength tensor.

Determinants of the Vacuum Tensor: A fundamental property of the vacuumtensor is its vanishing determinant

det∆µν =∑i,j,k,l

εijkl∆1i∆2j∆3k∆4l = 0 (3.14)

However, this is only part of the story because for each of 16 three-by-three minorsdefined by M[µν], one finds

detM[µν] =∑i,j,k

εijkM1iM2jM3k = 0 (3.15)

16


These determinants are interesting but nothing new since the field strength tensorexhibits the same properties. In fact, it can be shown that values for the nine sub-minors associated with each minor are proportional to those of F µν . Mathematically,

∆ij∆kl −∆kj∆il = −2(F ijF kl − F kjF il) (3.16)

On the other hand, by evaluating each detM[µν], a component of the velocity field(Ei or Bj) can be factored from the resulting homogeneous equation resulting in linearrelations among the components of ∆µν and F µν . Two classes of covariant equationsemerge from calculating along rows or columns yielding

∆µλFαν −∆αλF µν −∆νλFαµ = 0 (3.17a)

∆µλFαν −∆µαF λν −∆µνFαλ = 0 (3.17b)

The structure of each individual equation makes it possible to derive the covariantexpression for the homogeneous Maxwell-Lorentz equations simply by making thereplacement ∆αλ → ∂α where α is the unrepeated index in each occurence of the vac-uum tensor. Making the replacement ∆µν = F µν +∆νµ in both equations eliminatesthe vacuum tensor and leaves the covariant EEE ·BBB = 0 expression

F µλFαν + F µνF λα + F µαF νλ = 0 (3.18)

One can also eliminate the field strength tensor in (3.17) in favor of an expressionamong components of the vacuum tensor

∆αλ∆νµ −∆νλ∆αµ +∆µα∆λν −∆µν∆λα = 0 (3.19)

Other properties of equations (3.17) can be ascertained by considering individualcomponents. Of the several relations among vacuum gauge potentials and fields, themost interesting are the electric equations,

(∇ ·A)EEE−∇A ·EEE− ∂A/∂ct×BBB = 2ηEEE (3.20a)

(∇ ·A)EEE−EEE ·∇A + ∇A×BBB = 2ηEEE (3.20b)

and their magnetic counterparts

∇A ·BBB = ηBBB BBB ·∇A = ηBBB (3.20c)

The structure inherent in (3.20) compels a formal definition of the space-space com-ponents of the vacuum tensor which can be written

TTTΛ ≡ −∇A + 111 ∂νAν (3.21)

The magnetic equations are simply TTTΛ ·BBB = BBB · TTTΛ = 0 but more importantly is thefundamental expression determining the Poynting vector by subtracting equations(3.20b) and (3.20a):

SSS =c

4π[TTTΛ ·EEE−EEE · TTTΛ] (3.22)

To re-iterate, this is flux associated with velocity fields only.

17


Determinant of the Strain Tensor: Calculating the determinant of the straintensor is complicated by the presence of additional terms in the diagonal entries. Likethe vacuum tensor, det ηµν = 0. The simplest way to see this is in the rest framewhere all entries along the top row vanish. A more formal analysis shows that three-by-three minors are not zero, but the entire determinant can be written as a 4th orderpolynomial in η:

det ηµν = det∆µν+3∑

k=0

(−1)kM[kν] ·η+2(B2−E2)·η2−gαλ∆αλ ·η3−η4 = 0 (3.23)

In this equation the first two terms drop out as a result of properties of the vacuumtensor, while the last three terms cancel each other out.

3.2 Properties of the Vacuum Lagrangian

The details of an action integral S will be addressed in section 4.3. Until then it maybe supposed that Hamilton’s principle can be applied to a lagrangian Lvac producinga stationary value relative to potentials which differ only slightly from vacuum gaugevelocity potentials.

δS = δ

∫Lvac(∂

µAν , Aν , xν)d4x = 0 (3.24)

A specific form for Lvac could be postulated based on definitions of the vacuum tensor,the associated strain tensor, and knowledge of continuum mechanics; but a morerevealing approach is to begin with the classical electromagnetic lagrangian and insertequation (3.13). After re-arranging terms, and discarded a superfluous inverted nullterm, the form of an interacting vacuum lagrangian is

Lvac ≡ −1

8π∆µνηµν −

1

cJ∗νe Aν (3.25)

The functional value of the free term is available by insertion of the velocity potentials

Lvac = − 1

8πη2 (3.26)

Although the sign is negative4 its absolute value has a physical interpretation as storedenergy in the vacuum in the near neighborhood of the particle. It is a direct result ofthe vacuum gauge condition which can be re-formulated in terms of the lagrangianitself by writing

|η| ≡√−8πLvac (3.27)

4The sign of the vacuum lagrangian is equal and opposite to the sign of the conventional electro-magnetic free field lagrangian. The sign difference is not simply a matter of taste either since it willbe required for the inclusion of accelerated motions of the particle.

18


This equation may be coined as the genuine vacuum gauge condition, defining arelativistic theory of potentials, well suited for the description of a classical chargedparticle. The legitimacy of Lvac can be immediately verified by applying the Euler-Lagrange equations resulting in the Maxwell-Lorentz theory written as

∂µ∆µν =

4π

cJ∗νe (3.28)

One might consider the vacuum lagrangian and its equation of motion as a symmetryproperty of classical electron theory—resulting in a re-interpretation of both fields andpotentials. The essential difference is that the potentials are the physically meaningfulquantities of the vacuum theory while the fields fade into the background. This is insharp contradistinction to the conventional theory which begins with the fields anddetermines ‘unphysical’ potentials from an appropriate gauge condition.

An important comparison between the two theories is illustrated in figure 1. Theextra ‘diagonal’ field quantities utilized by the vacuum theory are a novel additionwhich serve to bolster its computational power. To gain some additional perspective,

Figure 1: Comparison of electromagnetic and vacuum lagrangians

it will be convenient to make use of equations (3.8) and separate the individualcomponents of the vacuum lagrangian as:

Lvac = − 1

8π[ ∆µν

` +∆µνe ] · [(η`)µν + (ηe)µν ] (3.29)

= − 1

8π∆µν

` (η`)µν −1

8π∆µνe (ηe)µν (3.30)

Assume the first term on the right can be viewed as a spacelike dilatation stressdensity in contrast to the second term which represents a timelike distortional stress.Determination of functional values for each term shows that the dilatation portion isnegative and takes out twice what the distortional portion puts in. Interaction termscan be added by considering the null current and the point source separately:

Lint =1

cJνe · ϑ(Ae)ν +

1

cJνN(A`)ν (3.31)

19


Both interaction terms are delta functions and independent lagrangians follow as

Ldis = − 1

8π∆µνe (ηe)µν −

1

cJνe · ϑ(Ae)ν (3.32a)

Ldil = − 1

8π∆µν

` (η`)µν −1

cJνN(A`)ν (3.32b)

Applying Euler-Lagrange equations produces the separate equations

∂µ∆µνe =

4π

cJνe · ϑ ∂µ∆

µν` =

4π

cJνN (3.33)

These results are exactly equations (1.13).

Lagrangian for a Moving Vacuum: The tensor defined by equation (3.6) is afour-space stress tensor. Of particular interest is the term containing the metric whichhas three positive definite space-space diagonal components. These terms representthe admissibility property of the stress tensor allowing the vacuum continuum to self-generate. The lagrangian in equation (3.25) can only accommodate this feature withan additional term. It must be linear in field quantities and can have no effect on theequation of motion. The simplest possibility is to write

Lvac −→ −1

8π[η − 2πσe]

2 (3.34)

Expanding the right side and applying the vacuum gauge condition allows Lvac to bewritten

Lvac −→ −1

8π∆µνηµν +

1

2σe η (3.35)

A more meaningful approach begins with the observation that the vacuum lagrangianposesses an internal symmetry under the variation

∆µν −→ ∆µν − 4πσe gµν (3.36)

which also results in (3.35). The additional term will require a modification to theaction integral—an issue which will be addressed in section 4.3. For now it will sufficeto simply quote an appropriate interacting velocity lagrangian specifically designedto accommodate the causality principle applied to the fields:

Lvac ≡ −1

8π∆µνηµν +

1

2σe η −

1

cJ∗µe Aµ (3.37)

It is important to observe that both free terms can be considered gauge invariant eventhough they are constructed from vacuum gauge potentials. The linear term shouldbe properly referred to as a vacuum dilatation producing dilatation stresses which—as already indicated—function to propagate the vacuum away from the source. Its

20


Figure 2: Plot of Lvac versus the scalar dilatation η.

presence is reminiscent of a term added to a lagrangian in point particle mechanicsto represent forces of constraint. The analogy has some grey area but σe assumesthe role of a Lagrange multiplier. A good understanding of the linear term is bestdemonstrated by first calculating the associated stress tensor.

3.3 Symmetric and Total Stress Tensors

The Noether current generated from invariance under the infinitesimal translationgroup xµ → xµ + εµ is the canonical stress tensor

T µνvac =∂Lvac

∂ηµληνλ − gµνLvac (3.38)

To facilitate the construction of an appropriate stress tensor based on (3.37) it isbeneficial to begin by considering only terms quadratic in field quantites for which

T µνvac =1

4π

[12gµν∆αληαλ −∆µληνλ

](3.39)

The velocity portion of a symmetric stress tensor can be extracted from this expressionby writing

T µνvac = Θµνvac + T µνD (3.40)

where the two individual tensors are given by

Θµνvac =

1

4π

[12gµνη2 − ηµληνλ

]T µνD =

1

4πηηνµ (3.41)

21


As with the conventional theory the extra term T µνD can be eliminated by showingthat ∂µT

µνD = 0 which leaves only the symmmetric tensor.

Applying equation (3.38) to the propagation term of the lagrangian is rather trivialhere, providing an additional stress

Λµν ≡ σe2∆

†µν (3.42)

which stands on its own without need of any symmetrization. The Total VacuumStress Tensor then follows as

Tµν ≡ 1

4π


]+ Λµν (3.43)

In the space surrounding the charge the conservation law implied by Noether’s the-orem is ∂µT

µνvac = 0. It can be verified explicitly for the vacuum theory by considering

only the second term on the right in (3.39) and differentiating:

∂µ(∆µληνλ) = ∆µλ · ∂µηνλ = ∆µλ · ∂νηµλ = 12∂ν(∆µληµλ) (3.44)

If the point source interaction term is to be inserted into the theory it is only requiredto re-visit the previous calculation and include the current density from the divergenceof ∆µλ. Since ∂µΛµν = 0, the differential law for the total vacuum stress tensorbecomes

∂µTµν = −1

cJ∗λe η

νλ (3.45)

The source term here is exactly the Lorentz force density associated with a pointelectron interacting with its own fields, and this implies an equivalence relation linkingthe velocity portions of the electromagnetic and vacuum gauge theories

Θµνvac ≡ Θµν

em (3.46)

An argument can be made for a slight disparity with the electromagnetic theory bycalculating the divergence on the second index generating an extra delta functionfrom the vacuum tensor.

Canonical Momentum in the Electromagnetic Field: For the vacuum gaugeelectron, the motion of the vacuum is understood in terms of a stress tensor with anadmissibility property. Based on equivalence with the electromagnetic theory, thisproperty must be associated with momentum propagating thru the velocity fields.

Terms corresponding to time-space and space-time components of the vacuumtensor are

−∂Lvac

∂ηi0=

1

4π∆i0 = πi0 −∂Lvac

∂η0i

=1

4π∆0i = π0i (3.47)

22


If these terms are to be associated with momentum flux it will be necessary to multiplythem by a constant having units of surface charge density. Choosing 4πσe with thecorrect sign gives5:

πππV ≡ −σe∇A πππA ≡ σe1

c

∂A

∂t(3.48)

A similar analysis for an electromagnetic lagrangian gives momentum componentsdefined by

∂Lem

∂(∂0A)=

1

4πEEE −→ πππE (3.49)

Unfortunately, if Lem is the conventional electromagnetic lagrangian, this definition isfundamentally flawed because the vector EEE in the conventional theory includes bothvelocity and acceleration fields. For a causality based theory a more reasonable choicefor an electromagnetic lagrangian is therefore the lagrangian in equation (1.18) whichyields (3.49) for a velocity field only.

Canonical momentum vectors for that lagrangian can be constructed from thedefinition Aπ ≡ σeA. When combined with equation (3.47) electric and magneticflux vectors are

πππE =γ

ρ(Aπ − Aπβββ) πππB =

γ

ρ(βββ ×Aπ) (3.50)

which can also be written

πππE =eσe(θ, φ)

R2·[nnn−βββ

1− nnn ·βββ

]· ϑ πππB =

eσe(θ, φ)

R2·[βββ × nnn

1− nnn ·βββ

]· ϑ (3.51)

While electric field momentum points radially outward from the present position ofthe source, the direction with which magnetic field momentum circles the particlewill depend on the frame of reference. An important symmetry operation is thetransformation

A −→ Ae + A` A −→ Ae + A` (3.52)

which leaves both flux vectors in equation (3.50) invariant. This gives new meaningto the concept of a gauge transformation and allows the velocity fields to be writtenas linear functions of components of the gauge field itself. This property is specific tothe velocity fields and will not work when applied to acceleration fields. Recordingthe force density Jνπ ≡ σeJ

νe now re-defines the Maxwell-Lorentz source equations for

the velocity fields:

∇ ·πππE = 4πρπ ∇×πππB =4π

cJJJπ +

1

c

∂πππE

∂t(3.53a)

∇×πππE = −1

c

∂πππB

∂t∇ ·πππB = 0 (3.53b)

5The subscript V appended to πππV has been borrowed from the conventional theory and onlyserves as a means of distinction from the momentum πππA.

23


The covariant connection between the theories is established by defining the anti-symmetric electromagnetic momentum flux tensor

πµν ≡

0 −πEx −πEy −πEz

πEx 0 −πBz πBy

πEy πBz 0 −πBx

πEz −πBy πBx 0

v

(3.54)

which easily follows from an anti-symmetric combination of Λµν . Using the transfor-mations (3.52), πµν can also be determined by a bracket among potentials only:

aeπµν = [Aµe , A

ν` ] where ae = 4πr2

e (3.55)

Finally, the vacuum lagrangian in equation (3.37) knows all about canonical momen-tum since the propagation term can be written

LΛ = 12σe η = 1

2πππE · nnn (3.56)

For arbitrary motions of the electron, a well known representation of the symmetricstress tensor is

Θµν =

U 1cSSS

1cSSS −TTT

(3.57)

The elements of this tensor are the energy density U , the energy flux vector (or Poynt-

ing vector) SSS, and the Maxwell stress tensor TTT. Source differential equations relatingindividual components are determined by application of the divergence operator toeither index of Θµν :

∂U

∂t+ ∇ ·SSS = −JJJe ·EEE ∇ · TTT− 1

c2

∂SSS

∂t= ρeEEE + JJJe ×BBB (3.58)

A matrix equation similar to (3.57) can also be constructed for Λµν . Componentsof this tensor can be grouped according to

UΛ ≡ −1

2σe∇ ·A

1

cSSSV ≡ −

1

2σe∇A

(3.59)

1

cSSSA ≡

σe2

∂A

∂tTij

Λ ≡σe2

[−∂Ai∂xj

+ δij∂αAα

]and this allows for

Λµν =

UΛ12πππV

12πππA TTTΛ

=

UΛ1cSSSV

1cSSSA TTTΛ

(3.60)

The elements of this tensor are the canonical work-energy density UΛ, the six “compo-nents” of the canonical energy flux vectors SSSV and SSSA, and the canonical stress-straintensor TTTΛ.

24


4 Stability of a Causal Electron: Vacuum Energy

The most straight forward application of the new formalism is in the rest framewhere the tensor Λµν provides a simple solution for a stable particle of radius re. Thissolution is easily extended to a moving frame, but equally important is the naturalemergence of a theory of inertia.

4.1 Rest Frame Solution

In the rest frame most of the momentum components are zero and one has

Tµν =

U 0

0 −TTT

+

UΛ1cSSSV

0 TTTΛ

(4.1)

For reference, the expanded form of the symmetric stress tensor is

Θµν =1

8π

E2 0 0 0

0 −E2x + E2

y + E2z −2ExEy −2ExEz

0 −2ExEy −E2y + E2

x + E2z −2EyEz

0 −2ExEz −2EyEz −E2z + E2

x + E2y

· ϑ (4.2)

For the vacuum tensor, the strain portion of TΛTΛTΛ can be symmetrized since

∂Ai∂xj− ∂Aj∂xi

= 0 β = 0 (4.3)

The time dependence of the scalar potential also vanishes and it is then appropriateto write

TijΛ =

σe2

[−1

2

(∂Ai∂xj

+∂Aj∂xi

)+ δij∇ ·A

]β = 0 (4.4)

The strain portion of TΛ has a restoring character as is evidenced by the minus sign,while the diagonal term has an admissibility character. If necessary, this equationcan be written interms of the dilatation function using (2.22) which shows that

TijΛ = πo

[−1

2

(∂ui∂xj

+∂uj∂xi

)+ δij∇ · uuu

]β = 0 (4.5)

where πo = 2πσ2e is a single vacuum constant similar to the Lame coefficients µ and

λ of the three-space theory—and having units of a momentum flux density (N/m2)equivalent to a pressure force.

25


Spacelike Integration of Components: Using (cτ, rrr) as rest frame coordinates,the integrated total stress-energy Eµν(τ) at a time τ > re/c is

Eµν(τ) =

∫Tµν · ϑ(cτ − r + re)d

3r =

∫Θµν · ϑ d3r +

∫Λµν · ϑ d3r (4.6)

The phase re has been included in the causality step to faciliate the calculation andrequires a separate interpretation for each integral on the right side of the equation.In the first integral, the coordinate r must have a lower limit to prevent the integralfrom diverging. The implication is a lower bound on proper time which is assumedto be cτ ≥ 0. If this condition is not met then the first integrand vanishes. Nosuch requirement is needed for the second integral where the time coordinate can beextended to cτ = −re allowing the field to propagate from r = 0. Figure 3 is includedfor reference and shows an upper limit of integration cτ + re for both integrals.

Figure 3: Spacetime diagram indicating the expansion of the radial step ϑ(cτ − r+ re)in the rest frame. The inclusion of the phase re defines two regions inside and outsidethe radius of the electron.

Integrated components are as follows:

T ττ :***************************************************************

T ττ = Θττ + Λττ

=1

8πE2 − 1

2σe∇ ·A =

e2

8π

[1

r4− 1

r2e r

2

]

26


Integrating as required by (4.6) produces

E ττ (τ) =

∫T ττ · ϑ r2 drdΩ =

e2

2

[1

re− 1

cτ + re

]− 2πeσe(cτ + re)→ −

1

2%cτ

where % = 4πσe e > 0 is the total momentum per unit time radiated.

T τx,T τy,T τz:***************************************************************There is no contribution from Θτi here and it is appropriate to use the vector

1

cSSSV = Λtjeeej = −1

2σe∇A

The integral is1

c

∫SSSV · ϑ d3r = 000

which means that E τi = 0, for i = x, y, z. These terms must be zero since the flow ofcanonical energy is symmetric about the source.

T xx, T yy, T zz:***************************************************************

T xx = Θxx + Λxx

=1

8π(−E2

x + E2y + E2

z) +1

2σe

[∂αA

α − ∂Ax∂x

]

=e2

8π

[(−x2 + y2 + z2)

r6+

1

r2r2e

+(x2 − y2 − z2)

r4r2e

]

The electromagnetic stress on the third line is removed by the vacuum strain at r = re.Integrating produces

E xx(τ) =

∫Txx · ϑ d3r =

1

3

e2

2

[1

re− 1

cτ + re

]+

1

34πeσe(cτ + re)

→ mc2 +1

3%cτ

where use has been made of e2/2re = mc2. Similarly E yy = E zz = E xx.

27


T xy, T xz, T yz:***************************************************************

T xy = Θxy + Λxy = − 1

4πExEy −

1

4σe

[∂Ax∂y

+∂Ay∂x

]=e2

4π

[−xyr6

+xy

r2e r

4

]Here again the (symmetrized) vacuum strain removes the electromagnetic stress atr = re. It is easy to see that

E xy =

∫T xy · ϑ d3r = 0

and from similar calculations E xz = E yz = 0.

T xτ ,T yτ ,T zτ :***************************************************************For the last three components, use the vector

1

cSSSA = Λjteeej =

1

2σe∂A

∂cτ= 000

These terms represent canonical energy flux based on explicit changes of the vacuumdeformation with time and they vanish for the stationary electron.***************************************************************

In a matrix format the previously integrated terms are

Eµν(τ) =

−12%cτ 0 0 0

0 mc2 + 13%cτ 0 0

0 0 mc2 + 13%cτ 0

0 0 0 mc2 + 13%cτ

(4.7)

The energy component should have yielded a term mc2 from an integral of the elec-tromagnetic energy density but this term was obliterated by the presence of vacuumstresses inside the particle radius. In addition to this, mc2 terms in the remainingdiagonal components are an admixture of electromagnetic and vacuum stresses. Butthe dilemma created by the integration can be easily disentangled with the inclusionof an integration constant added to the total energy tensor

Eµν −→ Eµν + gµνmc2 ≡ Eµνpart + Eµν

vac (4.8)

where

Eµνpart =

mc2 0 0 0

0 0 0 00 0 0 00 0 0 0

Eµνvac =

−1

2%cτ 0 0 00 1

3%cτ 0 0

0 0 13%cτ 0

0 0 0 13%cτ

(4.9)

28


On the left, Eµνpart is determined by a structureless constant—completely free of all

electromagnetic stresses. On the right, the tensor Eµνvac represents an unstable con-

tinuum, with properties defined by the particle radius. Of particular interest is thenegative sign appearing in the energy component. In the language of deformable me-dia it represents the total canonical stress imparted to the vacuum in a time cτ by thecanonical momentum. Equivalently, it represents the total work done on the vacuumby the admissibility force constant λ = πo. Nevertheless, this should be interpretedas positive energy density surrounding the particle which is necessarily one-half themomentum density times a factor of c—and requiring a quantum of the theory tosatisfy

u = 12|ppp|c (4.10)

It is assumed that the factor of 1/2 reflects that the associated longitudinal wavehas only one polarization state—as opposed to two polarization states for transversewaves.

Now define Pin as the inertial power and having a numerical value of

Pin = −∂Eττvac

∂τ= 2πσeec = 1.74× 1010 watts (4.11)

This is quite a large number and is obviously related to the large value for the speedof light. On the other hand, as a first approximation, it may be supposed thatcomponents of Eµν

vac have no mass equivalent and bear no functional relationship toany other established law of physics. This doesn’t seem unreasonable since the overtviolation of energy conservation is determined by the presence of the gauge field Aν` .

Cast in a different light, the local deformation of the vacuum, the radius of theparticle, and the generation of canonical field energy are all intimately connectedproperties in the abstract space provided by the vacuum gauge potentials. Sincepotentials are of no tangible substance, it can only be concluded that all of theseproperties lie beyond the possibility of any experimental verification. The classicalvacuum gauge electron therefore bears a strong similarity with its quantum mechan-ical counterpart in the sense that there exists well defined limitations on measurableinformation which can be obtained about the particle.

Timelike Integration: Integrating over causal space is not the only way to arriveat the tensors Eµν

part and Eµνvac in the rest frame. The spacelike three-volume element

dVs = r2e dΩcdτ (4.12)

represents a differential surface element of the spacetime cylinder in figure 3. Re-stricting the proper time interval to [0, τe], the integral over components of Tµν is

Eµν =

∫τe

TµνdVs (4.13)

29


This is a flux integral equivalent to an evaluation of the total vacuum energy containedwithin the particle radius. The resulting tensor reads

Eµν =

0 0 0 00 mc2 0 00 0 mc2 00 0 0 mc2

(4.14)

and re-produces Eµνpart with the inclusion of the integration constant. The quantity

Eµνvac results from replacing the integrand in equation (4.13) with Λµν and extending

to an arbirtrary time τ .

4.2 Moving Frame Solution

A covariant solution to the stability problem can be established by first generalizingthe total stress tensor in equation (3.43) to include acceleration components

Tµν = Θµν1 +Θµν

2 +Θµν3 + Λµν (4.15)

While this equation has not been adequately justified yet it is easy to show thatmagnitudes of the components of Θµν

1 and Λµν are the same in the neighborhoodρ ∼ re. On the other hand, the ratios of individual terms of the symmetric tensor are∣∣∣Θµν

2

Θµν1

∣∣∣ ∼ aνRν

c2

∣∣∣Θµν3

Θµν1

∣∣∣ ∼ [aνRν

c2

]2

(4.16)

Near the radius of the particle, these terms are utterly neglible unless accelerationsare on the order of 1031m/s2. This means that the stability of the particle will bedetermined exclusively by the velocity tensors only.

Covariant Theory of Stability: To address the stability problem first define

Gµν ≡ 1

ρ(Rµβν +Rνβµ)− 1

ρ2RµRν − 1

2gµν (4.17)

Πµν ≡ −1

2σeη

[gµν − Rνβµ

ρ

]= −1

2σeη ∂

νRµ (4.18)

This allows the symmetric stress tensor and the vacuum tensor to be written

Θµν1 =

1

4πη2Gµν Λµν = −σeη Gµν + Πµν (4.19)

and the total stress tensor is

Tµν =

[1

4πη2 − σeη

]Gµν + Πµν (4.20)

30


But the last term on the right represents radiated stress from canonical energy flux.Stability is therefore determined by the condition ρ = re for which the first term onthe right vanishes. Now form the covariant flux integral

dEµ = −∫ρ=re

TµνUνρ2dΩ cdτ = −∫ρ=re

ΠµνUνρ2dΩ cdτ (4.21)

Dividing both sides by the proper time element leads to the four-vector vacuumradiation rate

dEµ

dτ= Pinβ

µ (4.22)

a result which is proportional to the energy-momentum four-vector of the particle.To recover the original vacuum lagrangian, form the contraction

Lvac = −βµTµνnewβν (4.23)

and then apply the vacuum gauge condition of equation (3.27).

Vacuum Radiation in a Moving Frame: A Lorentz transformation of Eµνvac in

equation (4.9) can be implemented by considering the timelike energy term and space-like pressure terms separately. Witholding subscripts, appropriate tensors are

Eµν = −12%cτ

1 0 0 00 0 0 00 0 0 00 0 0 0

P µν = 13%τ

0 0 0 00 1 0 00 0 1 00 0 0 1

(4.24)

The general Lorentz transformation is X ′µν = Lµα X αβ L νβ which leads to the trans-

formed quantities

E′µν = −1

2%cτβµβν P

′µν = 13%τ · (βµβν − gµν) (4.25)

One can also arrive at these results by integrating the vacuum tensor over an appro-priate surface in the moving frame. Begin with the integral

E′µν + P

′µνc =1

2

∫∫σe(θ, φ)∆

†µνρ2dΩ cdτ (4.26)

where all angles (θ, φ) are defined relative to the retarded position of the charge. Thevacuum tensor can be expanded as in equations (3.8),

∆†µν = ∆

†µν` +∆

†µνe (4.27)

but the integration over ∆†µνe is zero while the remaining symmetric timelike and

spacelike terms can be separated as

E′µν = −e

2

∫∫σe(θ, φ)βµβνdΩ cdτ = −1

2%cτβµβν (4.28a)

P′µν =

e

c

∫∫σe(θ, φ)UµUνdΩ cdτ = 1

3%τ(βµβν − gµν) (4.28b)

31


Contracting both terms with the metric tensor then derives the general scalar relation

E = 12

Pc (4.29)

where both E and P are seen to be scalar invariants. It is important to understandthat ∆

†µνe plays no role in the integration implying that both tensors in equation

(4.28) are determined exclusively from the gauge field Aµ` . The gauge field is thepropagator of vacuum energy in the causal theory.

The sum of the two contributions in (4.28) define a quantity resembling the rela-tivistic perfect fluid stress tensor

T µνvac ≡ −1

2%cτβµβν + 1

3%cτ (βµβν − gµν) (4.30)

The comparison seems to have limitations though since (4.30) is a total energy tensorwhich is a linear function of proper time. Moreover, the four velocity βν is the motionof the source while the actual ‘fluid’ is radiating outward at speed c. Keeping theenergy and momentum terms separate seems to be a more useful approach. As anexample, a divergence operation applied to E ′µν with a minus sign leads to the four-vector inertial power formula already derived in (4.22). A similar operation on P ′µν

leads to a rate of momentum change four-vector

F ν = (γβββ · FFF , γFFF ) where FFF = 13ρβββ (4.31)

Both calculations here assume that βν is not a function of proper time. Even ifaccelerations are considered however it may not be appropriate to differentiate thefour-velocity since extra terms must be added by the acceleration theory.

4.3 Classical Action

To calculate the action for the vacuum gauge particle assume that a differential changein the vacuum strain dηµν , is accompanied by a differential change in the action dSthrough the relation

dS = − 1

4πc

∫∆µνdηµν d

4x (4.32)

It is straightforward to show that the integrand can be written as a total differential

∆µνdηµν =1

2d [∆µνηµν ] =

1

2

d

dR[∆µνηµν ] dR (4.33)

where R is a variable representing intermediate states of the electron radius as itexpands towards its value of re. The action follows as

S = − 1

8πc

∫ ∫ re

0

d

dR[∆µνηµν ] dR d

4x (4.34)

32


But the integration over the radius is trivial so that

S = − 1

8π

∫re

∆µνηµν d4x (4.35)

The propagation of the field follows as in section 3.3 from the symmetry operation

∆µν −→ ∆µν − 4πσe gµν (4.36)

and the transformed action becomes S = S1 + S2 with individual contributions givenby

S1 = − 1

8πc

∫re

∆µνηµν d4x S2 =

1

2c

∫σeη d

4x (4.37)

The term S1 is a Lorentz scalar since the integrand and the volume element are bothLorentz scalars. The value of the integral can be determined in the rest frame wherea lower integration limit at the classical electron radius leads to

dS1 = −mc2 · dτ (4.38)

The upper integration limit is governed by causality, but it can be assumed here thatthe integrand diminishes fast enough to extend the limit to infinity without fueling adebate.

The term S2 is also a Lorentz scalar but unlike S1, it will diverge unless causalityforces an upper limit to the integral. Furthermore, it will be mandatory to extendthe lower integration limit to include R = 0. Distinct portions of S2 corresponding tovolumes inside an outside the ‘hot spot’ of the electron are

S2 =1

2c

∫V<

σeη d4x+

1

2c

∫V>

σeη d4x = S< + S> (4.39)

The relevant volumes are shown in figure 4 in an abitrary Lorentz frame. Evaluatingthese integrals in the rest frame and withholding the time integral shows that

dS< = mc2 · dτ dS> =1

2%cτ · dτ (4.40)

Individual conributions to the action can now be added together. For consistency anintegration constant can be included in the form dS0 = −mc2dτ to show that thetotal rate at which the action changes with respect to proper time is given by

dS

dτ= −H (4.41)

where H is the energy component associated with (4.9). Of particular interest is aslice of the action from the radiation term

δS> =dS>

dτ· δτe = mc2τ (4.42)

33


Figure 4: Two dimensional spacetime diagram showing four-volumes V< and V> as-sociated with the causal light cone.

where δτe is the time required for light to traverse the radius of the particle. A Diracplane wave in moving frame coordinates is easily constructed from

ψ(ct, rrr) = ψoei δS>/~ = ψoe

i(Et−p·rrr)/~ (4.43)

For accelerated motions of the particle it might be expected that the analysis givenhere will require modification. However, vacuum gauge theory will show that moregeneral motions of the particle have no effect on the action integral.

5 Accelerated Motions of the Electron

It is well known that an accelerating charge emits transverse electromagnetic wavesover a range of frequencies. For the vacuum gauge particle, such waves can be inter-preted as resulting from induced variations of the surface charge density, in additionto the velocity configuration σe(θ, φ). A description of these variations will naturallyrequire the inclusion of an acceleration strain tensor εµν . One essential property ofεµν is immediately available by requiring that it may not be a source of additionalvacuum dilatation so that ε = 0 only. This requirement is consistent with classicalradiation theory which utilizes only transverse potentials for the description of propa-gating photons. In other words, classical radiation theory de-couples from the vacuum

34


theory of the electron.6

Naturally, εµν is expected to be functionally dependent on the four-acceleration ofthe particle, and as a preliminary calculation it will be useful to define the quantityaν⊥ orthogonal to the spacelike vector Uν and given by

aν⊥ ≡ aν + (aλUλ)Uν (5.1)

A link between aν⊥ and the velocity theory can be immediately established from thecontractions

∆µνa⊥ν = ∆µνa⊥

µ = 0 (5.2)

which are the same as the two eigenvalue equations:

ηµνa⊥ν = ηaµ⊥ ηµνa⊥

µ = ηaν⊥ (5.3)

It is also important to observe that aν⊥ is orthogonal to the four-vectors Rν and βν inaddition to its defined orthogonality with Uν .

5.1 Theory of Acceleration Strain

The vacuum gauge condition allows the velocity and acceleration fields of the electronto be addressed as independent theories. An initial conjecture for the form of anacceleration strain tensor might therefore be to simply follow the velocity theory anddifferentiate the acceleration potentials. Employing equation (3.5) allows this tensorto be written

εµν ≡ ∂µAνa = −χ(ηµν + Jµνa )− ∂µχ · Aν (5.4)

A problem already presents itself though because the covariant derivative in this equa-tion contains a term proportional to the velocity strain. Nevertheless, an accelerationlagrangian can still be written as

La =1

8π

[εµνεµν − ε2

]+

1

cJνaA

aν (5.5)

for which correct equations of motion are easily verified. However, it is evident thatboth ε and La both have the non-zero values

ε = −2χη La = χ2ηµνηµν (5.6)

and this violates the premise that acceleration strain may not dilate the vacuum.A clue to resolving the problem can be found from a review of equation (3.11)

which makes no specification as to what theory of acceleration strain is used to pro-duce the current density Jνa . This important observation is suggestive of a strategy to‘cleanse’ εµν of unwanted terms leaving an appropriate tensor εµν with the required

6This section relies heavily on various contractions of the tensors ηµν and εµν . These contractionshave been evaluated explicitly in separate tables in Appendix B serving as a useful calculational tool.

35


property ε = 0. As an initial guess, suppose all symmetric terms are separated fromεµν by

εµν = sµν − ενµ (5.7)

This relation can be inserted into La producing the quadratic form

La =1

8π

[sµνsµν − 2εµνsνµ + εµνεµν − s2 + 2sε− ε2

]+

1

cJνaA

aν (5.8)

Viewing this equation in terms independent field quantities sµν and εµν , and associ-ating the current Jνa with εµν , a legimate set of Euler-Lagrange equations is:

∂µεµν − ∂µ(sνµ − gµνs) =

4π

cJνa

∂µ(sµν − gµνs)− ∂µενµ = 0 (5.9)

The tensor sµν is now easily eliminated resulting in the equation

∂µ[εµν − ενµ] =4π

cJνa (5.10)

The task of removing the symmetric terms is tricky business and depends criticallyon the expansion of the acceleration four-vector in terms of aν⊥. The suprisingly simpleresult is

εµν ≡ Aµaν⊥ (5.11)

There are many interesting properties of this bi-linear quantity including ε = 0 andvanishing determinants of all individual two-by-two minors which implies det εµν = 0.Divergences on each index of εµν are

∂µενµ =

4π

c(jνa − Jνa ) ∂µε

µν =4π

cjνa (5.12)

and the second equation is the acceleration analog of equation (3.12) producing thecurrent density jνa given by

jνa ≡1

4πcη aν⊥ (5.13)

Properties of the Acceleration Current: A Comparison with equation (5.3)shows that jνa results from an interaction of aν⊥ with the vacuum strain. It can beevaluated at the classical radius taking the form

jνa = σe(θ, φ)βν⊥ at R = re (5.14)

This equation can be interpreted as a modulation of σe(θ, φ) during accelerated mo-tions and functioning to generate acceleration strain waves.

36


Although jνa is not a conserved current it is not associated with any additionalcharge density either as can be seen in the instantaneous rest frame where its timecomponent vanishes. Two important covariant integrals of the strain current are

Prad = − e

c2

∫jνa a

⊥ν ρ

2 dΩ

∫jνa Uνρ2dΩ dτ = 0 (5.15)

The power formula withoholds the integration over proper time and both integralsare valid at any radius.

5.2 Generalized Vacuum Lagrangian

While the acceleration potentials have no direct connection with an acceleration straintensor, they are still useful—and necessary—for the development of a generalizedvacuum lagrangian which accommodates accelerations of the particle. Hamilton’sprinciple for the velocity fields is

δS = δ

∫Lvac(∂

µAν , Aν , xν)d4x = 0 (5.16)

and the correct potentials Aν which satisfy this condition follow by considering vari-ations of the form

Aν −→ Aν + αξν (5.17)

where ξν is an arbitrary function of the coordinates and α is a scalar. However,comparison of (5.17) with the general form of the vacuum gauge potentials in equation(1.8) suggests that the acceleration potentials are to be used as variable functions withthe replacements

α −→ e ξν −→ −aλRλ

ρ2Rν (5.18)

In other words, the velocity theory is a stationary value relative to all possible accel-erations of the particle and the lagrange equations (with a slight change of notation)are then derivable from the condition

dS

de=

∫ [∂Lvac

∂Aλ

∂Aλ∂e

+∂Lvac

∂Aλ, µ

∂Aλ, µ∂e

]d4x = 0 (5.19)

There is no injustice here in choosing e as a vanishing scalar parameter since previouscalculations require the charge to be associated with velocity fields only. Performingan integration by parts gives

dS

de=

∫ [∂Lvac

∂Aλ− d

dxµ

∂Lvac

∂Aλ, µ

]∂Aλ∂e

d4x+

∫d

dxµ

[∂Lvac

∂Aλ, µ

∂Aλ∂e

]d4x = 0 (5.20)

37


The second integral might be shown to be zero for acceleration potentials whichvanish at endpoints xν1 and xν2; however no such requirement is necessary based onthe divergence calculation

∂µ[∆µλξλ

]= 0 = 1

2∂µJ

µa (5.21)

This is not the whole story though because both terms in (5.20) allow for the inclu-sion of additional terms in the integrand without changing the stationary condition.Specifically, any vector field which is orthogonal to the null vector Rλ is fair gameleading to possible current densities

J λ = h1 · aλ⊥ + h2 ·Rλ (5.22)

Both current densities in equation (5.12) are exactly of this form.Suppose therefore that the explicit form of the velocity lagrangian7 is written

Lvac = − 1

8π

[∂µAν∂µAν − (∂νA

ν)2]− 1

cj∗νe Aν (5.23)

and assume that accelerated motions of the electron follow from a first order correctionto the strain tensor accompanied by the causal appearance of an acceleration four-current:

∂µAν −→ ∂µAν − Aµaν⊥ j∗νe −→ j∗νe − jνa (5.24)

Inserting these variations and keeping only terms first order in accelerations resultsin

Lvac = − 1

8π

[∂µAν∂µAν − 2∂µAνAµa

⊥ν − (∂νA

ν)2 + 2∂νAνAµa⊥

µ

]− 1

c(j∗νe − jνa)Aν

(5.25)The form of this lagrangian motivates (briefly) a definition for an acceleration stresstensor

∆µνa ≡ Aµaν⊥ − gµνAλa

λ⊥ (5.26)

The Lagrange equations will then be determined from

∂Lvac

∂(∂µAν)= − 1

4π[∆µν −∆µν

a ]∂Lvac

∂Aν=

1

4π∆µνa⊥

µ −1

c(j∗νe − jνa) (5.27)

Making the connection ∆µνa → εµν , equations of motion are now easily identified with

the help of (5.2) rendering

∂µ∆µν =

4π

cj∗νe ∂µε

µν =4π

cjνa (5.28)

One can argue that a proper generalized lagrangian should lead to both strain equa-tions in (5.12). The inclusion of the conjugate tensor is not difficult and properlyaccounted for in Appendix A.

7Use lowercase j∗νe now for the point current density.

38


5.3 Symmetric and Total Stress Tensors

The most general symmetric stress tensor for accelerated motions of the electron isderivable from two different approaches.

Stress Tensor from the Velocity Theory: The velocity portion of the symmetricstress tensor has already been derived in section 3.3:

Θµνvac =

1

4π


](5.29)

The generalization of this tensor follows immediately by inserting the variation

ηµν −→ ηµν − εµν (5.30)

The resulting stress tensor is then

Θµνvac =

1

4π

[12gµνη2 − ηµληνλ + ηµλενλ − εµλενλ + εµληνλ

](5.31)

which is exactly equation (1.29a). Unfortunately, this simple procedure is not com-pletely valid. Specifically, a legitimate variation of the velocity strain must includeboth the tensor εµν and its conjugate. If the variation is required to be a symmetrictensor then the appropriate replacement should be

ηµν −→ ηµν − εµν − ενµ ≡ ζµν (5.32)

It is easy to verify that all extra terms resulting from the presence of the conjugatetensor are zero so that equation (5.31) remains unchanged.

Stress Tensor Using the Total Strain: The generalized stress tensor can also bedetermined by using the quantity ζµν as the field quantity and jν as an appropriatelychosen total current density, which casts the lagrangian into the compact form

Lvac =1

8π[ζ2 − ζµνζµν ]−

1

cjνAν (5.33)

This lagrangian produces both acceleration strain equations of motion in addition tothe velocity equation. Invoking translational invariance as usual leads to the Noethercurrent

T µνvac =∂Lvac

∂ζµλζνλ − gµνLvac =

1

4π

[12gµνζ2 − ζµλζνλ + ζζνµ

](5.34)

To extract the symmetric stress tensor one can simply write

T µνvac = Θµνvac + ζζνµ (5.35)

where

Θµνvac =

1

4π

[12gµνζ2 − ζµλζνλ

](5.36)

39


After crossing off superfluous null terms this tensor re-produces (5.31). If the tensorRµν is defined by

Rµν ≡ 1

4πζµλζνλ (5.37)

then a concise representation of the total stress tensor, inclusive of the requirementto propagate the velocity field, is

Tµν = 14gµνR − Rµν + Λµν (5.38)

Unfortunately, this derivation neglects the offending term which can be written

ζζνµ =1

4πηηνµ −Θµν

2 (5.39)

The problem of removing the velocity term has already been addressed. The presenceof Θµν

2 is assumed to be a consequence of neglecting acceleration terms to formulatethe velocity theory. More specifically, a divergence operation on equation (5.29) isprohibited from carrying acceleration terms which rightfully belong to the accelerationtheory. Relaxing this requirement gives rise to a small but significant modification tothe velocity theory

∂µΘµν1 −→ ∂µΘ

µν1 +

1

cηJνa (5.40)

Enforcement of a zero divergence for all regions of space excluding the location of theparticle will then require the appearance of an interaction term satisfying

∂µΘµνint = −1

cηJνa (5.41)

To solve this equation for Θµνint write

4π · ∂µΘµνint = −η∂µ(εµν − ενµ)

= ∂µ(ηεµν + ηενµ) = 4π · ∂µΘµν2 (5.42)

This argument adequately accounts for the interaction term leading to equations(5.36) and (5.38). A final calculation determines the general source equation bysubstituting ζµν into equation (3.45) leaving

∂µTµν = −1

cj∗λe ζ

νλ = −1

cjλe(ηνλ − ε

νλ

)(5.43)

On the far right the asterisk has been removed from the point charge density sincethe null current does not interact with either the velocity strain or the accelerationstrain. Except for the inclusion of the propagation term, (5.43) can be shown to becompletely equivalent to the electromagnetic theory.

40


A Lagrange Equations for Particle Accelerations

Beginning with Aν , the interacting charged particle velocity lagrangian is:

Lvac = − 1

8π


ν)2]− 1

cj∗νe Aν (A.1)

Under conditions where accelerated motions occur assume that ∂µAν receives a smallanti-symmetric perturbation first order in the quantity aν⊥ with the appearance of anacceleration four-current:

∂µAν −→ ∂µAν − Aµaν⊥ + Aνaµ⊥ j∗νe −→ j∗νe − Jνa (A.2)

Still keeping only first order corrections in aν⊥ the modified velocity theory can bewritten

Lvac = − 1

8π


ν)2]− 1

cj∗νe Aν

+1

4π

[∂µAνAµa

⊥ν − ∂µAνAνa⊥

µ

]+

1

cJνaAν (A.3)

Derivatives with respect to the field quantity and its derivative are

∂Lvac

∂(∂µAλ)= − 1

4π

[∂µAλ − gµλ∂νA

ν − Aµaλ⊥ + Aλaµ⊥]

(A.4)

∂Lvac

∂Aλ=

1

4π

[∂λAµ − ∂µAλ

]a⊥µ −

1

c

[j∗λe − Jλa

](A.5)

Appealing to equation (5.3), the second equation produces all the correct currentdensities necessary to write

∂µ[∂µAλ − gµλ∂νA

ν]

=4π

cj∗λe = ∂µ∆

µλ (A.6a)

∂µ[Aµaλ⊥

]=

4π

cjλa = ∂µε

µλ (A.6b)

∂µ[Aλaµ⊥

]=

4π

c

[jλa − Jλa

]= ∂µε

λµ (A.6c)

These results may also be obtained beginning with the symmetric variation

∂µAν −→ ∂µAν − Aµaν⊥ − Aνaµ⊥ (A.7)

In this case however it is necessary to use acceleration current densities derived inequations (A.6) for interaction terms instead of Jνa .

41


B Contractions of Vacuum Strain Tensors

ηµν =e

ρ2gµν − eRµβν

ρ3− 2eRνβµ

ρ3+

2eRµRν

ρ4(B.1)

εµν =e

ρ2Rµaν − e

ρ3χRµβν +

e

ρ4χRµRν (B.2)

ηµλη νλ =

e2

ρ4gµν − e2

ρ5Rµβν ηµληνλ =

e2

ρ4gµν − e2

ρ5(Rµβν +Rνβµ) +

e2

ρ6RµRν

ηλµηνλ =e2

ρ4gµν − e2

ρ5βµRν ηλµη ν

λ =e2

ρ4gµν − e

ρ3(Rµβν + βµRν)

Table 1: Contractions of ηµν .

ηµλεν λ = ηενµ εµλην λ = ηεµν εµλεν λ = −Θµν3

ηµλε νλ = 0 εµλη ν

λ = ηεµν εµλε νλ = 0

ηλµεν λ = ηενµ ελµηνλ = 0 ελµενλ = 0

ηλµε νλ = −ηεµν ελµη ν

λ = −ηενµ ελµε νλ = 0

Table 2: Contractions of ηµν and εµν .

42


C Derivatives of the Null Vector

The covariant derivative of Rν is

∂µRν = gµν − Rµβν

ρ(C.1)

The trace of the resulting matrix gives the 4-divergence ∂νRν = 3. In terms of

individual components–and with the inclusion of a sign–a useful construction is:

− ∂µRν =

−∂R

∂t−∂R∂t

∇R ∇R

(C.2)

where individual components are given by

∂R

∂t= 1− γR

ρ

∂R

∂t=−γRβββρ

(C.3)

∇R =γR

ρ∇R = 111 +

γRβββ

ρ(C.4)

The determinant of (C.2) can be written det[∂µRν ] = 0. The divergence of R followsfrom Tr[∇R] and has a value

∇ ·R = 3 +γR ·βββρ

(C.5)

Let www(ctr) be the retarded position of a charged particle at time ctr. The lightcone condition is defined by

R ≡ rrr −www(ctr) R ≡ ct− ctr (C.6)

Suppose that retarded coordinates are viewed collectively as xµr = (ctr,R). A trans-formation to present time coordinates xµ = (ct, rrr) is then xνr = xνr(x

µ) and it followsthat

dxνr =∂xνr∂xµ

dxµ (C.7)

The matrix generated by this transformation can be written

∂xνr∂xµ

=

∂ctr∂t

∂R

∂t

−∇ctr −∇R

Derivatives of R and R have already been evaluated while derivatives of the retardedtime are

∂tr∂t

=γR

ρ∇ctr =

−γRρ

(C.8)

43

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