Vlaenderen - A generalisation of classical electrodynamics for the prediction of scalar field effects (2004).pdf - Energy from the Vacuum - andreas50_54

A generalisation of classical electrodynamics for the prediction

of scalar eld eects

Koen J. van Vlaenderen

Institute for Basic Research

koenvanvlaenderen@wanadoo.nl

(July 9, 2004)

Abstract

Within the framework of Classical Electrodynamics (CED) it is common

practice to choose freely an arbitrary gauge condition with respect to a

gauge transformation of the electromagnetic potentials. The Lorenz gauge

condition allows for the derivation of the inhomogeneous potential wave

equations (IPWE), but this also means that scalar derivatives of the elec-

tromagnetic potentials are considered to be unphysical. However, these

scalar expressions might have the meaning of a new physical eld, S. If this

is the case, then a generalised CED is required such that scalar eld eects

are predicted and such that experiments can be performed in order to ver-

ify or falsify this generalised CED. The IPWE are viewed as a generalised

Gauss law and a generalised Ampere law, that also contain derivatives of

S, after reformulating the IPWE in terms of elds.

Since charge is conserved, scalar eld S satises the homogeneous wave

equation, thus one should expect primarily sources of dynamic scalar elds,

and not sources of static scalar elds. The collective tunneling of charges

might be an exception to this, since quantum tunneling is the quantum

equivalent of a classical local violation of charge continuity. Generalised

power/force theorems are derived that are useful in order to review historical

experiments since the beginning of electrical engineering, for instance Nikola

Tesla’s high voltage high frequency experiments. Longitudinal electro-scalar

vacuum waves, longitudinal forces that act on current elements, and applied

power by means of static charge and the S eld, are predicted by this theory.

The energy density and eld stress terms of scalar eld S are dened.

Some recent experiment show positive results that are in qualitative

agreement with the presented predictions of scalar eld eects, but further

quantitative tests are required in order to verify or falsify the presented

theory. The importance of Nikola Tesla’s pioneering research, with respect

to the predicted eects, cannot be overstated.

Classical Electrodynamics of Scalar Field Eects

I. INTRODUCTION

In general, the Maxwell/Heaviside equations, completed by the Lorentz force law, are

considered to be a complete theory for classical electrodynamics [9]. In dierential form

these quations are:

∇E

ǫ 0

Gauss law

(1)

∇×B−ǫ 0 0 ∂E

∂t

= 0 J

Ampere law

(2)

∂B

∂t

∇×E

= 0

Faraday law

(3)

∇B

(4)

The electromagnetic elds E

and B, and an extra scalar expression

S, can be dened in

terms of the electromagnetic potentials,

and

= ∇×A

(5)

=−∇ −

∂A

∂t

(6)

=−ǫ 0 0 ∂

∂t −∇A

(7)

In terms of the potentials and expressions S, the Gauss law and the Ampere law are:

ǫ 0 0 ∂ 2

∂S

∂t

ǫ 0

∂t 2 −∇ 2

(8)

ǫ 0 0 ∂ 2 A

∂t 2 −∇ 2 A

−∇S

= 0 J

(9)

The Maxwell/Heaviside equations are invariant with respect to a gauge transformation,

dened by a scalar function χ:

−→ ′ =

+ ∂χ

∂t

(10)

A−→A ′ = A−∇χ

(11)

B−→B ′

= B

(12)

E−→E ′

= E

(13)

S−→S ′

S−

ǫ 0 0

∂t 2 −∇ 2 χ

(14)

∂ 2 χ

Classical Electrodynamics of Scalar Field Eects

because the electromagnetic elds E and B are invariant with respect to this transforma-

tion, and the Maxwell/Heaviside equations do not contain partial derivatives of S. This

means that for each physical situation there is not a unique solution for the potentials

and A, because a particular solution for and A can be transformed into many other

solutions via an arbitrary scalar function χ. From the set of all equivalent electromagnetic

potential functions, one can choose freely a particular subset such that these potentials

satisfy an extra gauge condition, such as

(15)

which is known as the Lorenz condition [8]. For potentials that satisfy

0, equations

(8) and (9) become:

ǫ 0 0 ∂ 2

∂t 2 −∇ 2

ǫ 0

(16)

ǫ 0 0 ∂ 2 A

∂t 2 −∇ 2 A

= 0 J

(17)

which are the inhomogeneous potential wave equations (IPWE). Well known solutions

of these dierential equations are the retarded potentials, and in particular the Lienard-

Wiechert potentials [7] [21]. These solutions can be further evaluated and phenomena

like cyclotron radiation and synchrotron radiation can be explained by these evaluations

of the IPWE. It is necessary to prove that the retarded potentials satisfy the Lorenz con-

dition [19], and this is this case. However, other solutions than the retarded or advanced

potentials exist.

A very dierent philosophy is to regard the IPWE as generalised Gauss and Ampere

laws. In the spirit of J.C. Maxwell, who added the famous displacement term to the

Ampere law, one can add derivatives of expression S to the Maxwell/Heaviside equations:

∇E− ∂S

∂t

ǫ 0

(18)

+ ∇S−ǫ 0 0 ∂E

∂t

∇×B

= 0 J

(19)

∂B

∂t

∇×E

= 0

(20)

∇B

(21)

Classical Electrodynamics of Scalar Field Eects

When these extra derivatives of S are likewise added to equations (8) and (9), this yield au-

tomatically the IPWE without the need of an extra gauge condition. These eld equations

are a generalisation of classical electrodynamics, since the special case S = 0 results into

the usual Maxwell/Heaviside equations, and they are variant with respect to an arbitrary

scalar gauge transformation χ, see Eq. (14), unless χ is a solution of the homogeneous

wave equation. The expression S now has the meaning of a physical and observable scalar

eld. This scalar eld interacts with the vector elds E and B, as described by the gen-

eralised eld equations. The question: ”Is classical electrodynamics a complete classical

eld theory, with respect to scalar expression S?”, can not be answered within the context

of the standard classical electrodynamics, since this theory treats S as a non-observable

non-physical function, and this is premature. The usual gauge freedom and gauge condi-

tion S = 0 are based on the presumption that partial derivatives of S are not part of the

standard Maxwell eld equations in the rst place, which implies that S is disregarded as

a physical eld even before the theoretical development of the gauge transformation. In

other words, the assumed gauge freedom and free choice of gauge conditions are part of

a sequence of circular arguments, that seem to ”prove” that S has no physical relevance.

Oliver Heaviside did not like the abstract electromagnetic potentials and he preferred the

concept of observable elds. Therefore it is also in the spirit of Heaviside to assume that

S can cause observable eld eects as required by a testable theory, and such that the

Lorenz condition has only meaning as a special physical condition similar to: ’the electric

eld is zero’.

Next, the induction of scalar elds is discussed, followed by the derivation of gener-

alised force/power theorems in order to predict the type of observable phenomena at-

tributable to the presence of scalar elds.

II. THE INDUCTION OF SCALAR FIELDS

Considering the denition of S (S =−ǫ 0 0 ∂ ∂t −∇A), one might design an electrical

device such that factor ∂ / ∂t or factor ∇A is optimised, and such that these two scalar

factors do not cancel each other. With ∂ / ∂t we can associate systems of high voltage

and high frequency, such as pulsed power systems. With ∇A we can associate a source

of divergent/convergent currents, which is similar to the induction of a magnetic eld

Classical Electrodynamics of Scalar Field Eects

by rotating currents, B = ∇×A. For instance, a spherical or cylindrical capacitor can

show currents with non-zero divergence/convergence. If the capacity is high, then we can

expect a high ∇A, since strong currents need to charge/discharge the capacitor. If the

capacity is low, then a higher factor ∂ / ∂t can be expected, since then it takes less time

to charge and discharge the capacitor to high voltages.

Electromagnetic elds are of static or dynamic type. Considering the inhomogeneous

eld wave equations:

ǫ 0 0 ∂ 2 E

− ∂ J

∇ρ

ǫ 0 0

∂t 2 −∇ 2 E

= 0

∂t −

(22)

ǫ 0 0

∂t 2 −∇ 2 B

= 0 ( ∇× J)

(23)

−∇ J− ∂ρ

∂t

ǫ 0 0 ∂ 2 S

∂t 2 −∇ 2 S

= 0

(24)

that are deduced from the generalised Maxwell/Heaviside eld equations, we can expect

primarily dynamic scalar elds, because of the conservation of charge. This is the reason

why the discovery of scalar eld S is not as easy as the discovery of the electromagnetic

elds via simple static eld type experiments. Quantum tunneling of electrons can be

understood on the classical level as a local violation of charge conservation, for instance

at Josephson junctions. Hence, collective quantum tunneling devices might induce a

new type of classical eld: a static scalar eld. A dynamic scalar eld is induced by a

charge/current density wave: set E = 0

and

B = 0, then Eq.(18) and Eq.(19) become:

−

∂S

∂t

ǫ 0

(25)

∇S

= 0 J

(26)

Since S satises wave Eq.(24), also the charge density ρ and current density J are wave

solutions, however these wave solutions also have speed c. There are also wave solutions

of charge/current density with speed less than c, in case the electric eld (and/or the

magnetic eld) and scalar eld are not zero. Conclusion, a scalar eld

can be induced

by a dynamic charge/current distribution.

∂ 2 B

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