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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
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