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arXiv:hep-th/9606171v4 16 Oct 1997

On the Existence of Undistorted Progressive Waves

(UPWs) of Arbitrary Speeds 0 ≤ v < ∞ in Nature

Waldyr A. Rodrigues, Jr.(a)and Jian-Yu Lu(b)

(a) Instituto de Matem´ atica, Estat´ ıstica e Computa¸ c˜ ao Cient´ ıfica

IMECC-UNICAMP; CP 6065, 13081-970, Campinas, SP, Brasil

e-mail: walrod@ime.unicamp.br

Biodynamics Research Unit, Department of Physiology and Biophysics

Mayo Clinic and Foundation, Rochester, MN55905, USA

e-mail: jian@us0.mayo.edu

(b)

Abstract

We present the theory, the experimental evidence and fundamental phys-

ical consequences concerning the existence of families of undistorted progres-

sive waves (UPWs) of arbitrary speeds 0 ≤ v < ∞, which are solutions of

the homogeneous wave equation, Maxwell equations, Dirac, Weyl and Klein-

Gordon equations.

PACS numbers: 41.10.Hv; 03.30.+p; 03.40Kf

1. Introduction

In this paper we present the theory, the experimental evidence, and the fun-

damental physical consequences concerning the existence of families of undistorted

progressive waves (UPWs)(∗)moving with arbitrary speeds(∗∗)0 ≤ v < ∞. We show

that the main equations of theoretical physics, namely: the scalar homogeneous

wave equation (HWE); the Klein-Gordon equation (KGE); the Maxwell equations,

the Dirac and Weyl equations have UPWs solutions in a homogeneous medium, in-

cluding the vacuum. By UPW, following Courant and Hilbert[1]we mean that the

UPW waves are distortion free, i.e. they are translationally invariant and thus do

not spread, or they reconstruct their original form after a certain period of time.

Explicit examples of how to construct the UPWs solutions for the HWE are found in

Appendix A. The UPWs solutions to any field equations have infinite energy. How-

ever, using the finite aperture approximation (FAA) for diffraction (Appendix A),

(∗)UPW is used for the singular, i.e., for undistorted progressive wave.

(∗∗)We use units where c = 1, c being the so called velocity of light in vacuum.

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we can project quasi undistorted progressive waves (QUPWs) for any field equation

which have finite energy and can then in principle be launched in physical space.

In section 2 we show results of a recent experiment proposed and realized by

us where the measurement of the speeds of a FAA to a subluminal(∗)Bessel pulse

[eq.(2.1)] and of the FAA to a superluminal X-wave [eq.(2.5)] are done. The results

are in excellent agreement with the theory.

In section 3 we discuss some examples of UPWs solutions of Maxwell equa-

tions; (i) subluminal solutions which are interesting concerning some recent attempts

appearing in the literature[2,3,4]of construction of purely electromagnetic particles

(PEP) and (ii) a superluminal UPW solution of Maxwell equations called the su-

perluminal electromagnetic X-wave[5](SEXW). We briefly discuss how to launch a

FAA to SEXW. In view of the experimental results presented in section 2 we are

confident that such electromagnetic waves will be produced in the next few years. In

section 4 we discuss the important question concerning the speed of propagation of

the energy carried by superluminal UPWs solutions of Maxwell equations, clearing

some misconceptions found in the literature. In section 5 we show that the experi-

mental production of a superluminal electromagnetic wave implies in a breakdown

of the Principle of Relativity. In section 6 we present our conclusions.

Appendix B presents a unified theory of how to construct UPWs of arbitrary

speeds 0 ≤ v < ∞ which are solutions of Maxwell, Dirac and Weyl equations. Our

unified theory is based on the Clifford bundle formalism[6,7,8,9,10]where all fields

quoted above are represented by objects of the same mathematical nature. We take

the care of translating all results in the standard mathematical formalisms used by

physicists in order for our work to be usefull for a larger audience.

Before starting the technical discussions it is worth to briefly recall the history of

the UPWs of arbitrary speeds 0 ≤ v < ∞, which are solutions of the main equations

of theoretical physics.

To the best of our knowledge H. Bateman[11]in 1913 was the first person to

present a subluminal UPW solution of the HWE. This solution corresponds to what

we called the subluminal spherical Bessel beam in Appendix A [see eq.(A.31)]. Ap-

parently this solution has been rediscovered and used in diverse contexts many

times in the literature. It appears, e.g., in the papers of Mackinnon[12]of 1978 and of

Gueret and Vigier[13]and more recently in the papers of Barut and collaborators[14,15].

In particular in[14]Barut also shows that the HWE has superluminal solutions. In

(∗)In this experiment the waves are sound waves in water and, of course, the meaning of the

words subluminal, luminal and superluminal in this case is that the waves travel with speed less,

equal or greater than cs, the so called velocity of sound in water.

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1987 Durnin and collaborators rediscovered a subluminal UPW solution of the HWE

in cylindrical coordinates[16,17,18]. These are the Bessel beams of section A4 [see

eq.(A.41)]. We said rediscovered because these solutions are known at least since

1941, as they are explicitly written down in Stratton’s book[19]. The important point

here is that Durnin[16]and collaborators constructed an optical subluminal Bessel

beam. At that time they didn’t have the idea of measuring the speed of the beams,

since they were interested in the fact that the FAA to these beams were quasi UPWs

and could be very usefull for optical devices. Indeed they used the term “diffraction-

free beams” which has been adopted by some other authors later. Other authors

still use for UPWs the term non-dispersive beams. We quote also that Hsu and

collaborators[20]realized a FAA to the J0Bessel beam [eq.(A.41)] with a narrow band

PZT ultrasonic transducer of non-uniform poling. Lu and Greenleaf[21]produced the

first J0nondiffracting annular array transducers with PZT ceramic/polymer com-

posite and applied it to medical acoustic imaging and tissue characterization[22,23].

Also Campbell et al[24]used an annular array to realize a FAA to a J0Bessel beam

and compared the J0beam to the so called axicon beam[25]. For more on this topic

see[26].

Luminal solutions of a new kind for the HWE and Maxwell equations, also

known as focus wave mode [FWM] (see Appendix A), have been discovered by

Brittingham[27](1983) and his work inspired many interesting and important studies

as, e.g.,[29−40].

To our knowledge the first person to write about the possibility of a superluminal

UPW solution of HWE and, more important, of Maxwell equations was Band[41,42].

He constructed a superluminal electromagnetic UPW from the modified Bessel beam

[eq.(A.42)] which was used to generate in an appropriate way an electromagnetic

potential in the Lorentz gauge. He suggested that his solution could be used to

eventually launch a superluminal wave in the exterior of a conductor with cylin-

drical symmetry with appropriate charge density. We discuss more some of Band’s

statements in section 4.

In 1992 Lu and Greenleaf[43]presented the first superluminal UPW solution

of the HWE for acoustic waves which could be launched by a physical device[44].

They discovered the so called X-waves, a name due to their shape (see Fig. 3).

In the same year Donnelly and Ziolkowski[45]presented a thoughtfull method for

generating UPWs solutions of homogeneous partial equations. In particular they

studied also UPW solutions for the wave equation in a lossy infinite medium and to

the KGE. They clearly stated also how to use these solutions to obtain through the

Hertz potential method (see appendix B, section B3) UPWs solutions of Maxwell

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equations.

In 1993 Donnely and Ziolkowski[46]reinterpreted their study of[45]and obtained

subluminal, luminal and superluminal UPWs solutions of the HWE and of the KGE.

In Appendix A we make use of the methods of this important paper in order to obtain

some UPWs solutions. Also in 1992 Barut and Chandola[47]found superluminal

UPWs solutions of the HWE. In 1995 Rodrigues and Vaz[48]discovered in quite

an independent way(∗)subluminal and superluminal UPWs solutions of Maxwell

equations and the Weyl equation. At that time Lu and Greenleaf[5]proposed also

to launch a superluminal electromagnetic X-wave.(∗∗)

In September 1995 Professor Ziolkowski took knowledge of[48]and informed one

of the authors [WAR] of his publications and also of Lu’s contributions. Soon a

collaboration with Lu started which produced this paper. To end this introduction

we must call to the reader’s attention that in the last few years several important ex-

periments concerning the superluminal tunneling of electromagnetic waves appeared

in the literature[51,52]. Particularly interesting is Nimtz’s paper[53]announcing that

he transmitted Mozart’s Symphony # 40 at 4.7c through a retangular waveguide.

The solutions of Maxwell equations in a waveguide lead to solutions of Maxwell

equations that propagate with subluminal or superluminal speeds. These solutions

can be obtained with the methods discussed in this paper and will be discussed in

another publication.

2.

Finite Aperture Bessel Pulses and X-Waves.

Experimental Determination of the Speeds of Acoustic

In appendix A we show the existence of several UPWs solutions to the HWE,

in particular the subluminal UPWs Bessel beams [eq.(A.36)] and the superluminal

UPWs X-waves [eq.(A.52)]. Theoretically the UPWs X-waves, both the broad-band

and band limited [see eq.(2.4)] travel with speed v = cs/cosη > 1. Since only FAA

to these X-waves can be launched with appropriate devices, the question arises if

these FAA X-waves travel also with speed greater than cs, what can be answered

only by experiment. Here we present the results of measurements of the speeds of a

(∗)Rodrigues and Vaz are interested in obtaining solutions of Maxwell equations characterized

by non-null field invariants, since solutions of this kind are[49,50]necessary in proving a surprising

relationship between Maxwell and Dirac equations.

(∗∗)A version of [5] was submitted to IEEE Trans. Antennas Propag. in 1991. See reference

40 of[43].

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FAA to a broad band Bessel beam, called a Bessel pulse (see below) and of a FAA

to a band limited X-wave, both moving in water. We write the formulas for these

beams inserting into the HWE the parameter csknown as the speed of sound in

water. In this way the dispersion relation [eq.(A.37)] must read

ω2

c2

s

− k2= α2. (2.1)

Then we write for the Bessel beams

Φ<

Jn(t,? x) = Jn(αρ)ei(kz−ωt+nθ), n = 0,1,2,...(2.2)

Bessel pulses are obtained from eq.(2.2) by weighting it with a transmitting transfer

function, T(ω) and then linearly superposing the result over angular frequency ω,

i.e., we have

Φ<

JBBn(t,? x) = 2πeinθJn(αρ)F−1[T(ω)eikz], (2.3)

where F−1is the inverse Fourier transform. The FAA to Φ<

FAAΦ<

We recall that the X-waves are given by eq.(A.52), i.e.,

?∞

JBBnwill be denoted by

JBBn(or Φ<

FAJn).

Φ>

Xn(t,? x) = einθ

0

B(k)Jn(kρsinη)e−k[a0−i(z cosη−cst)]dk ,

(2.4)

where k = k/cosη, k = ω/cs. By choosing B(k) = a0we have the infinite aperture

broad bandwidth X-wave [eq.(A.53)] given by

Φ>

XBBn(t,? x) =

a0(ρsinη)neinθ

√M(τ +√M)n,

(2.5)

M = (ρsinη)2+ τ2, τ = [a0− i(z cosη − cst)].

XBBnwill be denoted by FAAΦ>

from a constant, e.g., if B(k) is the Blackman window function we denote the X-

wave by Φ>

FAAΦXBLn. Also when T(ω) in eq.(2.3) is the Blackman window function we denote

the respective wave by ΦJBLn.

As discussed in Appendix A and detailed in[26,44]to produce a FAA to a given

beam the aperture of the transducer used must be finite. In this case the beams

produced, in our case FAAΦJBL0and FAAΦXBB0, have a finite depth of field[26]

A FAA to Φ>

XBBn. When B(k) in eq.(2.4) is different

XBLn, where BL means band limited. A FAA to Φ>

XBLnwill be denoted

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(DF)(∗)and can be approximately produced by truncating the infinite aperture

beams ΦJBL0and ΦXBB0(or ΦXBL0) at the transducer surface (z = 0). Broad band

pulses for z > 0 can be obtained by first calculating the fields at all frequencies with

eq.(A.28), i.e.,

??ΦFA(ω,? x) =

1

iλ

?a

0

?π

−πρ′dρ′dθ′??Φ(ω,? x′)eikR

?π

R2z

(2.6)

+

1

2π

?a

0

−πρ′dρ′dθ′??Φ(ω,? x′)eikR

R3z,

where the aperture weighting function??Φ(ω,? x′) is obtained from the temporal Fourier

the depth of field of the FAAΦJBL0pulse, denoted BZmaxand the depth of field of

the FAAΦXBB0or FAA ΦXBL0denoted by XZmaxare given by[26]

transform of eqs.(2.3) and (2.4). If the aperture is circular of radius a [as in eq.(2.6)],

BZmax= a

??ω0

csα

?2

− 1; XZmax= acotη.(2.7)

For the FAAΦJBL0pulse we choose T(ω) as the Blackman window function[54]

that is peaked at the central frequency f0= 2.5MHz with a relative bandwidth of

about 81% (−6dB bandwidth divided by the central frequency). We have

The “scaling factor” in the experiment is α = 1202.45m−1and the weighting function

??ΦJBB0(ω,? x) in eq.(2.6) is approximated with stepwise functions. Practically this is

The diameter of the array is 50mm. Fig. 1(∗∗)shows the block diagram for the

production of FAA Φ>

FAA Bessel pulse has been done by comparing the speed with which the peak of the

FAA Bessel pulse travels with the speed of the peak of a pulse produced by a small

B(k) =

a0

?

0.42 − 0.5πk

0 otherwise.

k0

+ 0.08cos2πk

k0

?

, 0 ≤ k ≤ 2k0;

(2.8)

done with the 10-element annular array transfer built by Lu and Greenleaf[26,44].

XBL0and FAAΦ<

JBL0. The measurement of the speed of the

(∗)DF is the distance where the field maximum drops to half the value at the surface of the

transducer.

(∗∗)Reprinted with permission from fig. 2 of[44].

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circular element of the array (about 4mm or 6.67λ in diameter, where λ is 0.6mm

in water). This pulse travels with speed cs= 1.5mm/µs. The distance between

the peaks and the surface of the transducer are 104.33(9)mm and 103.70(5)mm for

the single-element wave and the Bessel pulse, respectively, at the same instant t of

measurement. The results can be seen in the pictures taken from of the experiment

in Fig. 2. As predicted by the theory developed in Appendix A the speed of the

Bessel pulse is 0.611(3)% slower than the speed csof the usual sound wave produced

by the single element.

The measurement of the speed of the central peak of the FAA Φ>

obtained from eq.(2.4) with a Blackman window function [eq.(2.8)] has been done in

the same way as for the Bessel pulse. The FAAΦXBL0wave has been produced by

the 10-element array transducer of 50mm of diameter with the techniques developed

by Lu and Greenleaf[26,44]. The distances traveled at the same instant t by the single

element wave and the X-wave are respectively 173.48(9)mm and 173.77(3)mm. Fig.

3 shows the pictures taken from the experiment. In this experiment the axicon angle

is η = 40. The theoretical speed of the infinite aperture X-wave is predicted to be

0.2242% greater then cs. We found that the FAAΦXBB0wave traveled with speed

0.267(6)% greater then cs!

These results, which we believe are the first experimental determination of the

speeds of subluminal and superluminal quasi-UPWs FAAΦ>

tions of the HWE, together with the fact that, as already quoted, Durnin[16]produced

subluminal optical Bessel beams, give us confidence that electromagnetic subluminal

and superluminal waves may be physically launched with appropriate devices. In

the next section we study in particular the superluminal electromagnetic X-wave

(SEXW).

It is important to observe here the following crucial points: (i) The FAA ΦXBBnis

produced by the source (transducer) in a short period of time ∆t. However, different

parts of the transducer are activated at different times, from 0 to ∆t, calculated from

eqs.(A.9) and (A.28). As a result the wave is born as an integral object for time

∆t and propagates with the same speed as the peak. This is exactly what has been

seen in the experiments and is corroborated by the computer simulations we did for

the superluminal electromagnetic waves (see section 3). (ii) One can find in almost

all textbooks that the velocity of transport of energy for waves obeying the scalar

wave equation

?1

c2

XBL0wave

JBL0and FAAΦ<

JBB0solu-

∂2

∂t2− ∇2

?

φ = 0(2.9)

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is given by

? vε=

?S

u,

(2.10)

where?S is the flux of momentum and u is the energy density, given by

?S = ∇φ∂φ

∂t,

u =1

2

(∇φ)2+1

c2

?∂φ

∂t

?2

, (2.11)

from which it follows that

vε=|?S|

u

≤ cs. (2.12)

Our acoustic experiment shows that for the X-waves the speed of transport of

energy is cs/cosη, since it is the energy of the wave that activates the detector

(hydro-phone). This shows explicitly that the definition of vεis meaningless. This

fundamental experimental result must be kept in mind when we discuss the meaning

of the velocity of transport of electromagnetic waves in section 4.

Figure 1: Block diagram of acoustic production of Bessel pulse and X-Waves.

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Figure 2: Propagation speed of the peak of Bessel pulse and its comparison with that

of a pulse produced by a small circular element (about 4 mm or 6.67 λ in diameter,

where λ is 0.6 mm in water). The Bessel pulse was produced by a 50 mm diameter

transducer. The distances between the peaks and the surface of the transducer

are 104.339 mm and 103.705 mm for the single-element wave and the Bessel pulse,

respectively. The time used by these pulses is the same. Therefore, the speed of the

peak of the Bessel pulse is 0.611(3)% slower than that of the single-element wave.

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Figure 3: Propagation speed of peak of X-wave and its comparison with that of

a pulse produced by small circular element (about 4 mm or 6.67 λ, where λ is

0.6 mm in water). The X-wave was produced by a 50 mm diameter transducer.

The distance between the peaks and the surface of the transducer are 173.489 mm

and 173.773 mm for the single-element wave and the X-wave, respectively. The time

used by these pulses is the same. Therefore, the speed of the peak of the X-wave

is 0.2441(8)% faster than that of the single-element wave. The theoretical ratio for

X-waves and the speed of sound is(cs/cosη − cs)

cs

= 0.2442% for η = 4o.

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3. Subluminal and Superluminal UPWs Solutions of Maxwell

Equations(ME)

In this section we make full use of the Clifford bundle formalism (CBF) resumed

in Appendix B, but we give translation of all the main results in the standard vec-

tor formalism used by physicists. We start by reanalyzing in section 3.1 the plane

wave solutions (PWS) of ME with the CBF. We clarify some misconceptions and ex-

plain the fundamental role of the duality operator γ5and the meaning of i =√−1 in

standard formulations of electromagnetic theory. Next in section 3.2 we discuss sub-

luminal UPWs solutions of ME and an unexpected relation between these solutions

and the possible existence of purely electromagnetic particles (PEPs) envisaged by

Einstein[55], Poincar´ e[56], Ehrenfest[57]and recently discussed by Waite, Barut and

Zeni[2,3]. In section 3.3 we discuss in detail the theory of superluminal electromag-

netic X-waves (SEXWs) and how to produce these waves by appropriate physical

devices.

3.1 Plane Wave Solutions of Maxwell Equations

We recall that Maxwell equations in vacuum can be written as [eq.(B.6)]

∂F = 0, (3.1)

where F sec?2(M) ⊂ secCℓ(M). The well known PWS of eq.(3.1) are obtained

as follows. We write in a given Lorentzian chart ?xµ? of the maximal atlas of M

(section B2) a PWS moving in the z-direction

F = feγ5kx, (3.2)

k = kµγµ, k1= k2= 0, x = xµγµ, (3.3)

where k, x ∈ sec?1(M) ⊂ secCℓ(M) and where f is a constant 2-form. From

eqs.(3.1) and (3.2) we obtain

kF = 0 (3.4)

Multiplying eq.(3.4) by k we get

k2F = 0 (3.5)

and since k ∈ sec?1(M) ⊂ secCℓ(M) then

k2= 0 ↔ k0= ±|?k| = ±k3, (3.6)

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i.e., the propagation vector is light-like. Also

F2= F. F + F ∧ F = 0 (3.7)

as can be easily seen by multiplying both members of eq.(3.4) by F and taking into

account that k ?= 0. Eq(3.7) says that the field invariants are null.

It is interesting to understand the fundamental role of the volume element γ5

(duality operator) in electromagnetic theory. In particular since eγ5kx= coskx +

γ5sinkx, γ5≡ i, writing F =?E + i?B (see eq.(B.17)), f = ? e1+ i? e2, we see that

?E + i?B = ? e1coskx −? e2sinkx + i(? e1sinkx +? e2coskx).

From this equation, using ∂F = 0, it follows that ? e1.? e2= 0,?k.? e1=?k.? e2= 0 and

then

?E.?B = 0.

(3.8)

(3.9)

This equation is important because it shows that we must take care with the i =

√−1 that appears in usual formulations of Maxwell Theory using complex electric

and magnetic fields. The i =√−1 in many cases unfolds a secret that can only be

known through eq.(3.8). It also follows that?k.?E =?k.?B = 0, i.e., PWS of ME are

transverse waves. We can rewrite eq.(3.4) as

kγ0γ0Fγ0= 0 (3.10)

and since kγ0= k0+?k, γ0Fγ0= −?E + i?B we have

?kf = k0f. (3.11)

Now, we recall that in Cℓ+(M) (where, as we say in Appendix B, the typical

fiber is isomorphic to the Pauli algebra Cℓ3,0) we can introduce the operator of space

conjugation denoted by ∗ such that writing f = ? e + i?b we have

f∗= −? e + i?b ; k∗

We can now interpret the two solutions of k2= 0, i.e. k0= |?k| and k0= −|?k| as

corresponding to the solutions k0f =?kf and k0f∗= −?kf∗; f and f∗correspond

in quantum theory to “photons” of positive or negative helicities. We can interpret

k0= |?k| as a particle and k0= −|?k| as an antiparticle.

Summarizing we have the following important facts concerning PWS of ME: (i)

the propagation vector is light-like, k2= 0; (ii) the field invariants are null, F2= 0;

0= k0 ;?k∗= −?k.

(3.12)

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(iii) the PWS are transverse waves, i.e.,?k.?E =?k.?B = 0.

3.2 Subluminal Solutions of Maxwell Equations and Purely Electromag-

netic Particles.

We take Φ ∈ sec(?0(M)⊕?4(M)) ⊂ secCℓ(M) and consider the following Hertz

potential π ∈ sec?2(M) ⊂ secCℓ(M) [eq.(B.25)]

π = Φγ1γ2. (3.13)

We now write

Φ(t,? x) = φ(? x)eγ5Ωt. (3.14)

Since π satisfies the wave equation, we have

∇2φ(? x) + Ω2φ(? x) = 0. (3.15)

Solutions of eq.(3.15) (the Helmholtz equation) are well known. Here we consider

the simplest solution in spherical coordinates,

φ(? x) = CsinΩr

r

, r =

?

x2+ y2+ z2, (3.16)

where C is an arbitrary real constant. From the results of Appendix B we obtain

the following stationary electromagnetic field, which is at rest in the reference frame

Z where ?xµ? are naturally adapted coordinates (section B2):

F0 =

C

r3[sinΩt(αΩr sinθsinϕ − β sinθcosθcosϕ)γ0γ1

− sinΩt(αΩr sinθcosϕ + β sinθcosθsinϕ)γ0γ2

+ sinΩt(β sin2θ − 2α)γ0γ3+ cosΩt(β sin2θ − 2α)γ1γ2

+ cosΩt(β sinθcosθsinϕ + αΩrsinθcosϕ)γ1γ3

+ cosΩt(−β sinθcosθcosϕ + αΩrsinθsinϕ)γ2γ3]

with α = Ωr cosΩr−sinΩr and β = 3α+Ω2r2sinΩr. Observe that F0is regular at

the origin and vanishes at infinity. Let us rewrite the solution using the Pauli-algebra

in Cℓ+(M). Writing (i ≡ γ5)

F0=?E0+ i?B0

(3.17)

(3.18)

we get

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?E0=?W sinΩt,

?B0=?W cosΩt,(3.19)

with

?W = −C

?αΩy

r3

−βxz

r5,−αΩx

r3

−βyz

r5,β(x2+ y2)

r5

−2α

r3

?

. (3.20)

We verify that div?W = 0, div?E0= div?B0= 0, rot?E0+∂?B0/∂t = 0, rot?B0−∂?E0/∂t =

0, and

rot?W = Ω?W.

(3.21)

Now, from eq.(B.88) we know that T0=1

energy density and the Poynting vector. It follows that?E0×?B0= 0, i.e., the solution

has zero angular momentum. The energy density u = S00is given by

2

?Fγ0F is the 1-form representing the

u =1

r6[sin2θ(Ω2r2α2+ β2cos2θ) + (β sin2θ − 2α)2] (3.22)

Then

constructed by considering “wave packets” with a distribution of intrinsic frequencies

F(Ω) satisfying appropriate conditions. Many possibilities exist, but they will not

be discussed here. Instead, we prefer to direct our attention to eq.(3.21). As it

is well known, this is a very important equation (called the force free equation[2])

that appears e.g. in hydrodynamics and in several different situations in plasma

physics[58]. The following considerations are more important.

Einstein[55]among others (see[3]for a review) studied the possibility of construct-

ing purely electromagnetic particles (PEPs). He started from Maxwell equations for

a PEP configuration described by an electromagnetic field Fpand a current density

Jp, where

∂Fp= Jp

? ??

I R3 udv = ∞. As explained in section A.6 a finite energy solution can be

(3.23)

and rightly concluded that the condition for existence of PEPs is

Jp.Fp= 0. (3.24)

This condition implies in vector notation

ρp?Ep= 0, ?jp.?Ep= 0, ?jp×?Bp= 0.(3.25)

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From eq.(3.24) Einstein concluded that the only possible solution of eq.(3.22) with

the subsidiary condition given by eq.(3.23) is Jp= 0. However, this conclusion is

correct, as pointed in[2,3], only if J2

However, if we suppose that Jpcan be spacelike, i.e., J2

frame where ρp= 0 and a possible solution of eq.(3.24) is

p> 0, i.e., if Jpis a time-like current density.

p< 0, there exists a reference

ρp= 0,

?Ep.?Bp= 0, ?jp= KC?Bp, (3.26)

where K = ±1 is called the chirality of the solution and C is a real constant. In[2,3]

static solutions of eqs.(3.22) and (3.23) are exhibited where?Ep= 0. In this case we

can verify that?Bpsatisfies

∇ ×?Bp= KC?Bp.

Now, if we choose F0∈ sec?2(M) ⊂ secCℓ(M) such that

F0=?E0+ i?B0,

?E0=?BpcosΩt,

(3.27)

?B0=?BpsinΩt

(3.28)

and Ω = KC > 0, we immediately realize that

∂F0= 0. (3.29)

This is an amazing result, since it means that the free Maxwell equations may

have stationary solutions that may be used to model PEPs. In such solutions the

structure of the field F0is such that we can write

F0= F

′

p+ F = i?W cosΩt −?W sinΩt,

∂F

′

p= −∂F = J

′

p,

(3.30)

i.e., ∂F0= 0 is equivalent to a field plus a current. This fact opens several interesting

possibilities for modeling PEPs (see also[4]) and we discuss more this issue in another

publication.

We observe that moving subluminal solutions of ME can be easily obtained

choosing as Hertz potential, e.g.,

π<(t,? x) = CsinΩξ<

ξ<

exp[γ5(ω<t − k<z)]γ1γ2, (3.31)

ω2

<− k2

<= Ω2

<(z − v<t)2],

, v<= dω</dk<.

<;

ξ<= [x2+ y2+ γ2

(3.32)

γ<=

1

?

1 − v2

<

15