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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
Page 16
We are not going to write explicitly the expression for F<corresponding to π<
because it is very long and will not be used in what follows.
We end this section with the following observations: (i) In general for sublu-
minal solutions of ME (SSME) the propagation vector satisfies an equation like
eq.(3.30). (ii) As can be easily verified, for a SSME the field invariants are non-
null. (iii) A SSME is not a transverse wave.
eq.(3.21). Conditions (i), (ii) and (iii) are in contrast with the case of the PWS of
ME. In[49,50]Rodrigues and Vaz showed that for free electromagnetic fields (∂F = 0)
such that F2?= 0, there exists a Dirac-Hestenes equation (see section A.8) for
ψ ∈ sec(?0(M) +?2(M) +?4(M)) ⊂ secCℓ(M) where F = ψγ1γ2?ψ. This was the
Maxwell equations (and also of Weyl equation)[48]which solve the Dirac-Hestenes
equation [eq.(B.40)].
This can be seen explicitly from
reason why Rodrigues and Vaz discovered subluminal and superluminal solutions of
3.3 The Superluminal Electromagnetic X-Wave (SEXW)
To simplify the matter in what follows we now suppose that the functions ΦXn
[eq.(A.52)] and ΦXBBn[eq.(A.53)] which are superluminal solutions of the scalar
wave equation are 0-forms sections of the complexified Clifford bundle CℓC(M) =
I C ⊗ Cℓ(M) (see section B4). We rewrite eqs.(A.52) and (A.53) as(∗)
?∞
and choosing B(k) = a0, we have
ΦXn(t,? x) = einθ
0
B(k)Jn(kρsinη)e−k[a0−i(z cosη−t)]dk
(3.33)
ΦXBBn(t,? x) =
a0(ρsinη)neinθ
√M(τ +√M)n
τ = [a0− i(z cosη − t)].
(3.34)
M = (ρsinη)2+ τ2; (3.35)
As in section 2, when a finite broadband X-wave is obtained from eq.(3.31) with
B(k) given by the Blackman spectral function [eq.(2.8)] we denote the resulting X-
wave by ΦXBLn(BL means band limited wave). The finite aperture approximation
(FAA) obtained with eq.(A.28) to ΦXBLnwill be denoted FAAΦXBLnand the FAA
to ΦXBBnwill be denoted by FAAΦXBBn. We use the same nomenclature for the
electromagnetic fields derived from these functions. Further, we suppose now that
(∗)In what follows n = 0,1,2,...
16
Page 17
the Hertz potential π, the vector potential A and the corresponding electromagnetic
field F are appropriate sections of CℓC(M). We take
π = Φγ1γ2∈ sec I C ⊗?2(M) ⊂ secCℓC(M),
where Φ can be ΦXn,ΦXBBn,ΦXBLn, FAA ΦXBBnor FAAΦXBLn. Let us start by
giving the explicit form of the FXBBn, i.e., the SEXWs. In this case eq.(B.81) gives
π = ? πmand
? πm= ΦXBBnz
(3.36)
(3.37)
where z is the versor of the z-axis. Also, let ρ, θ be respectively the versors of the
ρ and θ directions where (ρ,θ,z) are the usual cylindrical coordinates. Writing
FXBBn=?EXBBn+ γ5?BXBBn
(3.38)
we obtain from equations (A.53) and (B.25):
?EXBBn= −ρ
ρ
∂2
∂t∂θΦXBBn+ θ
∂2
∂t∂ρΦXBBn; (3.39)
?BXBBn= ρ
∂2
∂ρ∂zΦXBBn+ θ1
ρ
∂2
∂θ∂zΦXBBn+ z
?
∂2
∂z2ΦXBBn−∂2
∂t2ΦXBBn
?
; (3.40)
Explicitly we get for the components in cylindrical coordinates:
(?EXBBn)ρ= −1
(?EXBBn)θ=1
ρnM3
M6
√MM2
√MΦXBBn; (3.41a)
ρi
ΦXBBn;(3.41b)
(?BXBBn)ρ= cosη(?EXBBn)θ;
(?BXBBn)θ= −cosη(?EXBBn)ρ;
(?BXBBn)z= −sin2ηM7
(3.41c)
(3.41d)
√MΦXBBn. (3.41e)
The functions Mi, (i = 2,...,7) in (3.41) are:
M2= τ +√M;
1
√Mτ;
3
√Mτ;
(3.42a)
M3= n +(3.42b)
M4= 2n +(3.42c)
17
Page 18
M5= τ + n√M;
M6= (ρ2sin2ηM4
(3.42d)
M− nM3)M2+ nρ2M5
1
√M+ 3n1
Msin2η;
√M3τ2.
(3.42e)
M7= (n2− 1)
We immediately see from eqs.(3.41) that the FXBBnare indeed superluminal
UPWs solutions of ME, propagating with speed 1/cosη in the z-direction. That
FXBBnare UPWs is trivial and that they propagate with speed c1= 1/cosη follows
because FXBBndepends only on the combination of variables (z − c1t) and any
derivatives of ΦXBBnwill keep the (z − c1t) dependence structure.
Now, the Poynting vector?PXBBnand the energy density uXBBnfor FXBBnare
obtained by considering the real parts of?EXBBnand?BXBBn. We have
Mτ + 3
1
(3.42f)
(?PXBBn)ρ= −Re{(?EXBBn)θ}Re{(?BXBBn)z};
(?PXBBn)θ= Re{(?EXBBn)ρ}Re{(?BXBBn)z};
(?PXBBn)z= cosη
(3.43a)
(3.43b)
(3.43c)
?
|Re{(?EXBBn)ρ}|2+ |Re{(?EXBBn)θ}|2?
|Re{(?EXBBn)ρ}|2+ |Re{(?EXBBn)θ}|2?
The total energy of FXBBnis then
;
uXBBn= (1 + cos2η)
?
+ |Re{(?BXBBn)z}|2.
(3.44)
εXBBn=
?π
−πdθ
?+∞
−∞
dz
?∞
0
ρdρuXBBn
(3.45)
Since as z → ∞,?EXBBndecreases as 1/|z − tcosη|1/2, what occurs for the X-
branches of FXBBn, εXBBnmay not be finite. Nevertheless, as in the case of the
acoustic X-waves discussed in section 2, we are quite sure that a FAAFXBLncan
be launched over a large distance. Obviously in this case the total energy of the
FAAFXBLnis finite.
We now restrict our attention to FXBB0. In this case from eq.(3.40) and eqs.(3.43)
we see that (?EXBB0)ρ = (?BXBB0)θ = (?PXBB0)θ = 0.
amplitudes of Re{ΦXBB0} [4(1)], Re{(?EXBB0)θ} [4(2)], Re{(?BXBB0)ρ} [4(3)] and
Re{(?BXBB0)z} [4(4)]. Fig. 5 shows respectively (?PXBB0)ρ [5(1)], (?PXBB0)z [5(2)]
and uXBB0[5(3)]. The size of each panel in Figures 4 and 5 is 4m (ρ-direction) ×
(∗)Figures 4, 5 and 6 were reprinted with permission from[5].
In Fig. 4(∗)we see the
18
Page 19
2mm (z-direction) and the maxima and minima of the images in Figures 4 and 5
(before scaling) are shown in Table 1, in MKSA units(∗∗).
Re{ΦXBB0}
1.0
0.0
Re{(?EXBB0)θ}
9.5 × 106
−9.5 × 106
Re{(?BXBB0)ρ}
2.5 × 104
−2.5 × 104
Re{(?BXBB0)z}
6.1
−1.5
max
min
(?PXBB0)ρ
2.4 × 107
−2.4 × 107
(?PXBB0)z
2.4 × 1011
0.0
UXBB0
1.6 × 103
0.0
max
min
Table 1: Maxima and Minima of the zeroth-order nondiffracting
electromagnetic X waves (units: MKSA).
Fig. 6 shows the beam plots of FXBB0in Fig. 4 along one of the X-branches
(from left to right). Fig. 6(1) represents the beam plots of Re{ΦXBB0} (full line),
Re{(?EXBB0)θ} (dotted line), Re{(?BXBB0)ρ} (dashed line) and Re{(?BXBB0)z} (long
dashed line). Fig. 6(2) represents the beam plots of (?PXBB0)ρ(full line), (?PXBB0)z
(dotted line) and uXBB0(dashed line).
3.4 Finite Aperture Approximation to FXBB0and FXBL0
From eqs.(3.40), (3.43) and (3.44) we see that?EXBB0,?BXBB0,?PXBB0and uXBB0
are related to the scalar field ΦXBB0. It follows that the depth of the field[5](or non
diffracting distance — see section 2) of the FAAFXBB0and of the FAAFXBL0, which
of course are to be produced by a finite aperture radiator, are equal and given by
Zmax= D/2cotη, (3.46)
where D is the diameter of the radiator and η is the axicon angle. It can be proved
also[5]that for ΦXBL0(and more generally for ΦXBLn), that Zmaxis independent of
the central frequency of the spectrum B(k) in eq.(3.1). Then if we want, e.g., that
FXBB0or FXBL0travel 115 km with a 20 m diameter radiator, we need η = 0.005o.
Figure 7 shows the envelope of Re{FAAΦXBB0} obtained with the finite aperture
approximation (FAA) given by eq.(A.28), with D = 20 m, a0 = 0.05 mm and
η = 0.005o, for distances z = 10 km [6(1)] and z = 100 km [6(2)], respectively,
(∗∗)Reprinted with permission from Table I of[5].
19
Page 20
from the radiator which is located at the plane z = 0. Figures 7(3) and 7(4) show
the envelope of Re{FAAΦXBL0} for the same distances and the same parameters
(D,a0and η) where B(k) is the following Blackman window function, peaked at the
frequency f0= 700 GHz with a 6 dB bandwidth about 576 GHz:
B(k) =
?
a0[0.42 − 0.5cosπk
0 otherwise;
k0+ 0.08cos2πk
k0], 0 ≤ k ≤ 2k0;
(3.47)
where k0= 2πf0/c (c = 300,000km/s). From eq.(3.46) it follows that for the above
choice of D, a0and η
Zmax= 115km(3.48)
Figs. 8(1) and 8(2) show the lateral beam plots and Figs. 8(3) and 8(4) show the
axial beam plots respectively for Re{FAAΦXBB0} and for Re{FAAΦXBL0} used to
calculate FXBB0and FXBL0. The full and dotted lines represent X-waves at distances
z = 10 km and z = 100 km. Fig. 9 shows the peak values of Re{FAAΦXBB0}
(full line) and Re{FAAΦXBL0} (dotted line) along the z-axis from z = 3.45 km to
z = 230 km. The dashed line represents the result of the exact ΦXBB0solution. The
6 dB lateral and axial beam widths of ΦXBB0, which can be measured in Fig 7(1)
and 7(2), are about 1.96 m and 0.17 mm respectively, and those of the FAAΦXBL0
are about 2.5 m and 0.48 mm as can be measured from 7(3) and 7(4). For ΦXBB0
we can calculate[43,26]the theoretical values of the 6 dB lateral (BWL) and axial
(BWA) beam widths, which are given by
BWL=2√3a0
|sinη|;
With the values of D, a0and η given above, we have BWL= 1.98 m and BWA=
0.17 mm. These are to be compared with the values of these quantities for the
FAAΦXBL0.
We remark also that eq.(3.46) says that Zmaxdoes not depend on a0. Then we
can choose an arbitrarily small a0to increase the localization (reduced BWLand
BWA) of the X-wave without altering Zmax. Smaller a0requires that the FAAΦXBL0
be transmitted with broader bandwidth. The depths of field of ΦXBB0and of ΦXBL0
that we can measure in Fig. 9 are approximately 109 km and 110 km, very close to
the value given by eq.(3.46) which is 115 km.
We conclude this section with the following observations.
BWA=2√3a0
|cosη|. (3.49)
(i) In general both subluminal and superluminal UPWs solutions of ME have non
null field invariants and are not transverse waves. In particular our solutions
20
Page 21
have a longitudinal component along the z-axis.
because it shows that, contrary to the speculations of Evans[59], we do not
need an electromagnetic theory with a non zero photon-mass, i.e., with F
satisfying Proca equation in order to have an electromagnetic wave with a
longitudinal component. Since Evans presents evidence[59]of the existence on
longitudinal magnetic fields in many different physical situations, we conclude
that the theoretical and experimental study of subluminal and superluminal
UPW solutions of ME must be continued.
This result is important
(ii) We recall that in microwave and optics, as it is well known, the electromag-
netic intensity is approximately represented by the magnitude of a scalar field
solution of the HWE. We already quoted in the introduction that Durnin[16]
produced an optical J0-beam, which as seen from eq.(3.1) is related to ΦXBB0
(ΦXBL0). If we take into account this fact together with the results of the
acoustic experiments described in section 2, we arrive at the conclusion that
subluminal electromagnetic pulses J0and also superluminal X-waves can be
launched with appropriate antennas using present technology.
(iii) If we take a look at the structure of e.g.
eq.(A.28) we see that it is a “packet” of wavelets, each one traveling with
speed c. Nevertheless, the electromagnetic X-wave wave that is an interfer-
ence pattern is such that its peak travels with speed c/cosη > 1. (This
indeed happens in the acoustic experiment with c ?→ cs, see section 2). Since
as discussed above we can project an experiment to launch the peak of the
FAAΦXBB0from a point z1to a point z2, the question arises: Is the existence
of superluminal electromagnetic waves in conflict with Einstein’s Special Rel-
ativity? We give our answer to this fundamental issue in section 5, but first
we discuss in section 4 the speed of propagation of the energy associated with
a superluminal electromagnetic wave.
the FAAΦXBB0[eq.(3.40)] plus
21
Page 22
Figure 4: Real part of field components of the exact solution superluminal electro-
magnetic X-wave at distance z = ct/cosη (η = 0.005o, a0= 0.05 mm, n = 0).
22
Page 23
Figure 5: Poynting flux and energy density of the exact solution superluminal elec-
tromagnetic X-wave at distance z = ct/cosη, (η = 0.005o, a0= 0.05 mm, n = 0).
23
Page 24
Figure 6: (6.1) Beam plots along the X-branches of FXBB0for Re{ΦXBB0} or Hertz
potential, Re{(?EXBB0)θ}, Re{(?BXBB0)ρ}, and Re{(?BXBB0)z}. (6.2) Beam plots for
(?PXBB0)ρ(full line), (?PXBB0)z(dotted line) and uXBB0(dashed line).
24
Page 25
Figure 7: 7(1) and 7(2) show the real part of FAAΦXBB0at distances z = 10 km
and z = 100 km from the radiator located at the plane z = 0 with D = 20 m and
η = 0.005o. 7(3) and 7(4) show the real parts of FAAΦXBL0for the same distances.
25
Page 26
Figure 8: Beam plots of scalar X-waves (finite aperture).
26
Page 27
Figure 9: Peak magnitude of X-waves along the z axis.
27
Page 28
4. The Velocity of Transport of Energy of the UPWs Solu-
tions of Maxwell Equations
Motivated by the fact that the acoustic experiment of section 2 shows that the
energy of the FAA X-wave travels with speed greater than csand since we found
in this paper UPWs solutions of Maxwell equations with speeds 0 ≤ v < ∞, the
following question arises naturally: Which is the velocity of transport of the energy
of a superluminal UPW (or quasi UPW) solution of ME?
We can find in many physics textbooks (e.g.[10]) and in scientific papers[41]the
following argument. Consider an arbitrary solution of ME in vacuum, ∂F = 0. Then
if F =?E + i?B (see eq.(B.17)) it follows that the Poynting vector and the energy
density of the field are
?P =?E ×?B, u =1
2(?E2+?B2). (4.1)
It is obvious that the following inequality always holds:
vε=|?P|
u
≤ 1. (4.2)
Now, the conservation of energy-momentum reads, in integral form over a finite
volume V with boundary S = ∂V
∂
∂t
?? ? ?
Vdv1
2(?E2+?B2)
?
=
?
Sd?S.?P (4.3)
Eq.(4.3) is interpreted saying that
S = ∂V , so that?P is the flux density — the amount of field energy passing through a
unit area of the surface in unit time. For plane wave solutions of Maxwell equations,
?
Sd?S.?P is the field energy flux across the surface
vε= 1 (4.4)
and this result gives origin to the “dogma” that free electromagnetic fields transport
energy at speed vε= c = 1.
However vε≤ 1 is true even for subluminal and superluminal solutions of ME,
as the ones discussed in section 3. The same is true for the superluminal modified
Bessel beam found by Band[41]in 1987. There he claims that since vε≤ 1 there is
no conflict between superluminal solutions of ME and Relativity Theory since what
Relativity forbids is the propagation of energy with speed greater than c.
28
Page 29
Here we challenge this conclusion. The fact is that as is well known?P is not
uniquely defined. Eq(4.3) continues to hold true if we substitute?P ?→?P +?P′with
∇.?P′= 0. But of course we can easily find for subluminal, luminal or superluminal
solutions of Maxwell equations a?P′such that
|?P +?P′|
u
≥ 1. (4.5)
We come to the conclusion that the question of the transport of energy in superlu-
minal UPWs solutions of ME is an experimental question. For the acoustic superlu-
minal X-solution of the HWE (see section 2) the energy around the peak area flows
together with the wave, i.e., with speed c1 = cs/cosη (although the “canonical”
formula [eq.(2.10)] predicts that the energy flows with vε< cs). Since we can see no
possibility for the field energy of the superluminal electromagnetic wave to travel
outside the wave we are confident to state that the velocity of energy transport of
superluminal electromagnetic waves is superluminal.
Before ending we give another example to illustrate that eq.(4.2) (as is the
case of eq.(2.10)) is devoid of physical meaning. Consider a spherical conductor in
electrostatic equilibrium with uniform superficial charge density (total charge Q)
and with a dipole magnetic moment. Then, we have
?E = Qr
r2;
?B =C
r3(2cosθr + sinθθ) (4.6)
and
?P =?E ×?B =CQ
r5sinθϕ,u =1
2
?Q2
r4+C2
r6(3cos2θ + 1)
?
. (4.7)
Thus
|?P|
u
=
2CQr sinθ
r2Q2+ C2(3cos2θ + 1)?= 0 for r ?= 0.(4.8)
Since the fields are static the conservation law eq.(4.3) continues to hold true, as
there is no motion of charges and for any closed surface containing the spherical
conductor we have
?
But nothing is in motion! In view of these results we must investigate whether
the existence of superluminal UPWs solutions of ME is compatible or not with the
Principle of Relativity. We analyze this question in detail in the next section.
Sd?S.?P = 0.(4.9)
29
Page 30
To end this section we recall that in section 2.19 of his book Stratton[19]presents
a discussion of the Poynting vector and energy transfer which essentially agrees with
the view presented above. Indeed he finished that section with the words: “By this
standard there is every reason to retain the Poyinting-Heaviside viewpoint until a
clash with new experimental evidence shall call for its revision.”(∗)
5.
Principle of Relativity
Superluminal Solutions of Maxwell Equations and the
In section 3 we showed that it seems possible with present technology to launch
in free space superluminal electromagnetic waves (SEXWs). We show in the follow-
ing that the physical existence of SEXWs implies a breakdown of the Principle of
Relativity (PR). Since this is a fundamental issue, with implications for all branches
of theoretical physics, we will examine the problem with great care. In section 5.1
we give a rigorous mathematical definition of the PR and in section 5.2 we present
the proof of the above statement.
5.1
Physical Meaning
Mathematical Formulation of the Principle of Relativity and Its
In Appendix B we define Minkowski spacetime as the triple ?M,g,D?, where
M ≃ I R4,g is a Lorentzian metric and D is the Levi-Civita connection of g.
Consider now GM, the group of all diffeomorphisms of M, called the manifold
mapping group. Let T be a geometrical object defined in A ⊆ M. The diffeomor-
phism h ∈ GMinduces a deforming mapping h∗: T ?→ h∗T = T such that:
(i) If f : M ⊇ A → I R, then h∗f = f ◦ h−1: h(A) → I R
(ii) If T ∈ secT(r,s)(A) ⊂ secT(M), where T(r,s)(A) is the sub-bundle of tensors of
type (r,s) of the tensor bundle T(M), then
(h∗T)he(h∗ω1,...,h∗ωr,h∗X1,...,h∗Xs) = Te(ω1,...,ωr,X1,...,Xs)
∀Xi∈ TeA, i = 1,...,s, ∀ωj∈ T∗
(iii) If D is the Levi-Civita connection and X,Y ∈ secTM, then
(h∗Dh∗Xh∗Y )heh∗f= (DXY )ef
eA, j = 1,...,r, ∀e ∈ A.
∀e ∈ M. (5.1)
(∗)Thanks are due to the referee for calling our attention to this point.
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Page 31
If {fµ= ∂/∂xµ} is a coordinate basis for TA and {θµ= dxµ} is the corresponding
dual basis for T∗A and if
T = Tµ1...µr
ν1...νsθν1⊗ ... ⊗ θνs⊗ fµ1⊗ ... ⊗ fµr, (5.2)
then
h∗T = [Tµ1...µr
ν1...νs◦ h−1]h∗θν1⊗ ... ⊗ h∗θνs⊗ h∗fµ1⊗ ... ⊗ h∗fµr.
Suppose now that A and h(A) can be covered by the local chart (U,η) of the maximal
atlas of M, and A ⊆ U,h(A) ⊆ U. Let ?xµ? be the coordinate functions associated
with (U,η). The mapping
(5.3)
x
′µ= xµ◦ h−1: h(U) → I R (5.4)
defines a coordinate transformation ?xµ? ?→ ?x
are the coordinate functions associated with the local chart (V,ϕ) where h(U) ⊆ V
and U ∩ V ?= φ. Now, since it is well known that under the above conditions
h∗∂/∂xµ≡ ∂/∂x
(h∗T)?x′µ?(he) = T?xµ?(e),
′µ? if h(U) ⊇ A ∪ h(A). Indeed ?x
′µ?
′µand h∗dxµ≡ dx
′µ, eqs.(5.3) and (5.4) imply that
(5.5)
where T?xµ?(e) means the components of T in the chart ?xµ? at the event e ∈ M,
i.e., T?xµ?(e) = Tµ1...µr
¯T = h∗T in the basis {h∗∂/∂xµ= ∂/∂x
Then eq.(5.6) reads
T
ν1...νs(xµ(e)) and where¯T
′µ1...µr
ν1...νs(x
′µ}, {h∗dxµ= dx
′µ(he)) are the components of
′µ}, at the point h(e).
′µ1...µr
ν1...νs(x
′µ(he)) = Tµ1...µr
ν1...νs(xµ(e)),(5.6)
or using eq.(5.5)
T
′µ1...µr
ν1...νs(x
′µ(e)) = (Λ−1)µ1
α1...Λβs
νsT
′α1...αr
β1...βs(x
′µ(h−1e)),(5.7)
where Λµ
In appendix B we introduce the concept of inertial reference frames I ∈ secTU,
U ⊆ M by
g(I,I) = 1 and DI = 0.
α= ∂x
′µ/∂xα, etc.
(5.8)
A general frame Z satisfies g(Z,Z) = 1, with DZ ?= 0. If α = g(Z, ) ∈ secT∗U, it
holds
(Dα)e= ae⊗ αe+ σe+ ωe+1
3θehe,e ∈ U ⊆ M,(5.9)
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where a = g(A, ), A = DZZ is the acceleration and where ωeis the rotation tensor,
σeis the shear tensor, θeis the expansion and he= g|Hewhere
TeM = [Ze] ⊕ [He]. (5.10)
Heis the rest space of an instantaneous observer at e, i.e. the pair (e,Ze). Also
he(X,Y ) = ge(pX,pY ), ∀X,Y ∈ TeM and p : TeM → He. (For the explicit form
of ω,σ,θ, see[60]). From eqs.(5.9) and (5.10) we see that an inertial reference frame
has no acceleration, no rotation, no shear and no expansion.
We introduce also in Appendix B the concept of a (nacs/I). A (nacs/I) ?xµ? is
said to be in the Lorentz gauge if xµ, µ = 0,1,2,3 are the usual Lorentz coordinates
and I = ∂/∂x0∈ secTM. We recall that it is a theorem that putting I = e0=
∂/∂x0, there exist three other fields ei∈ secTM such that g(ei,ei) = −1, i = 1,2,3,
and ei= ∂/∂xi.
Now, let ?xµ? be Lorentz coordinate functions as above. We say that ℓ ∈ GMis
a Lorentz mapping if and only if
x
′µ(e) = Λµ
νxµ(e), (5.11)
where Λµ
subset {ℓ} of GMsuch that eq.(5.12) holds true also by L↑
When ?xµ? are Lorentz coordinate functions, ?x
functions. In this case we denote
ν∈ L↑
+is a Lorentz transformation. For abuse of notation we denote the
+⊂ GM.
′µ? are also Lorentz coordinate
eµ= ∂/∂xµ, e′
µ= ∂/∂x
′µ, γµ= dxµ, γ′
µ= dx
′µ; (5.12)
when ℓ ∈ L↑
Let h ∈ GM. If for a geometrical object T we have
+⊂ GMwe say that ℓ∗T is the Lorentz deformed version of T.
h∗T = T, (5.13)
then h is said to be a symmetry of T and the set of all {h ∈ GM} such that eq.(5.13)
holds is said to be the symmetry group of T. We can immediately verify that for
ℓ ∈ L↑
ℓ∗g = g, ℓ∗D = D,
i.e., the special restricted orthochronous Lorentz group L↑
g and D.
In[62]we maintain that a physical theory τ is characterized by:
+⊂ GM
(5.14)
+is a symmetry group of
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(i) the theory of a certain “species of structure” in the sense of Boubarki[63];
(ii) its physical interpretation;
(iii) its present meaning and present applications.
We recall that in the mathematical exposition of a given physical theory τ, the
postulates or basic axioms are presented as definitions. Such definitions mean that
the physical phenomena described by τ behave in a certain way. Then, the definitions
require more motivation than the pure mathematical definitions. We call coordina-
tive definitions the physical definitions, a term introduced by Reichenbach[64]. It is
necessary also to make clear that completely convincing and genuine motivations for
the coordinative definitions cannot be given, since they refer to nature as a whole
and to the physical theory as a whole.
The theoretical approach to physics behind (i), (ii) and (iii) above is then to
admit the mathematical concepts of the “species of structure” defining τ as prim-
itives, and define coordinatively the observation entities from them. Reichenbach
assumes that “physical knowledge is characterized by the fact that concepts are not
only defined by other concepts, but are also coordinated to real objects”. However,
in our approach, each physical theory, when characterized as a species of structure,
contains some implicit geometric objects, like some of the reference frame fields de-
fined above, that cannot in general be coordinated to real objects. Indeed it would
be an absurd to suppose that all the infinity of IRF that exist in M must have a
material support.
We define a spacetime theory as a theory of a species of structure such that, if
Mod τ is the class of models of τ, then each Υ ∈ Mod τ contains a substructure
called spacetime (ST). More precisely, we have
Υ = (ST,T1...Tm} , (5.15)
where ST can be a very general structure[62]. For what follows we suppose that
ST = M = (M,g,D), i.e. that ST is Minkowski spacetime. The Ti, i = 1,...,m
are (explicit) geometrical objects defined in U ⊆ M characterizing the physical fields
and particle trajectories that cannot be geometrized in Υ. Here, to be geometrizable
means to be a metric field or a connection on M or objects derived from these
concepts as, e.g., the Riemann tensor or the torsion tensor.
The reference frame fields will be called the implicit geometrical objects of τ,
since they are mathematical objects that do not necessarily correspond to properties
of a physical system described by τ.
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Now, with the Clifford bundle formalism we can formulate in Cℓ(M) all modern
physical theories (see Appendix B) including Einstein’s gravitational theory[6]. We
introduce now the Lorentz-Maxwell electrodynamics (LME) in Cℓ(M) as a theory
of a species of structure. We say that LME has as model
ΥLME= ?M,g,D,F,J,{ϕi,mi,ei}?, (5.16)
where (M,g,D) is Minkowski spacetime, {ϕi,mi,ei}, i = 1,2,...,N is the set of
all charged particles, miand eibeing the masses and charges of the particles and
ϕi: I R ⊃ I → M being the world lines of the particles characterized by the fact that
if ϕi∗∈ secTM is the velocity vector, then ˇ ϕi= g(ϕi∗, ) ∈ secΛ1(M) ⊂ secCℓ(M)
and ˇ ϕi.ˇ ϕi= 1. F ∈ secΛ2(M) ⊂ secCℓ(M) is the electromagnetic field and J ∈
secΛ1(M) ⊂ secCℓ(M) is the current density. The proper axioms of the theory are
∂F = J
miDϕi∗ˇ ϕi= eiˇ ϕi· F
From a mathematical point of view it is a trivial result that τLME has the
following property: If h ∈ GMand if eqs.(5.16) have a solution ?F,J,(ϕi,mi,ei)? in
U ⊆ M then ?h∗F,h∗J,(h∗ϕi,mi,ei)? is also a solution of eqs.(5.16) in h(U). Since
the result is true for any h ∈ GMit is true for ℓ ∈ L↑
mapping.
We must now make it clear that ?F,J,{ϕi,mi,ei}? which is a solution of eq.(5.16)
in U can be obtained only by imposing mathematical boundary conditions which we
denote by BU. The solution will be realizable in nature if and only if the mathe-
matical boundary conditions can be physically realizable. This is indeed a nontrivial
point[62]for in particular it says to us that even if ?h∗F,h∗J,{h∗ϕi,mi,ei}? can be a
solution of eqs.(5.16) with mathematical boundary conditions Bh(U), it may hap-
pen that Bh(U) cannot be physically realizable in nature. The following statement,
denoted PR1, is usually presented[62]as the Principle of (Special) Relativity in active
form:
(5.17)
+⊂ GM, i.e., for any Lorentz
PR1:Let ℓ ∈ L↑
?M,g,D,T1,...,Tm? is a possible physical phenomenon, then ℓ∗Υ = ?M,g,D,
l∗T1,...,l∗Tm? is also a possible physical phenomenon.
It is clear that hidden in PR1 is the assumption that the boundary conditions
that determine ℓ∗Υ are physically realizable. Before we continue we introduce the
statement denoted PR2, known as the Principle of (Special) Relativity in passive
form[62]
+ ⊂ GM.If for a physical theory τ and Υ ∈ Modτ, Υ =
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PR2:“All inertial reference frames are physically equivalent or indistinguishable”.
We now give a precise mathematical meaning to the above statement.
Let τ be a spacetime theory and let ST = ?M,g,D? be a substructure of Mod τ
representing spacetime. Let I ∈ secTU and I′∈ secTV , U,V ⊆ M, be two inertial
reference frames. Let (U,η) and (V,ϕ) be two Lorentz charts of the maximal atlas
of M that are naturally adapted respectively to I and I′. If ?xµ? and ?x
coordinate functions associated with (U,η) and (V,ϕ), we have I = ∂/∂x0,I′=
∂/∂x
′µ? are the
′0.
Definition: Two inertial reference frames I and I′as above are said to be phys-
ically equivalent according to τ if and only if the following conditions are satisfied:
(i) GM⊃ L↑
When Υ ∈ Modτ, Υ = ?M,g,D,T1,...Tm?, is such that g and D are defined
over all M and Ti∈ secCℓ(U) ⊂ secCℓ(M), calling o = ?g,D,T1,...Tm?, o solves
a set of differential equations in η(U) ⊂ I R4with a given set of boundary conditions
denoted bo?xµ?, which we write as
+∋ ℓ : U → ℓ(U) ⊆ V, x
′µ= xµ◦ ℓ−1⇒ I′= ℓ∗I
Dα
?xµ?(o?xµ?)e= 0 ; bo?xµ?; e ∈ U(5.18)
and we must have:
(ii) If Υ ∈ Mod τ ⇔ ℓ∗Υ ∈ Mod τ, then necessarily
ℓ∗Υ = ?M,g,D,ℓ∗T1,...ℓ∗Tm? (5.19)
is defined in ℓ(U) ⊆ V and calling ℓ∗o ≡ {g,D,ℓ∗T1,...,ℓ∗Tm} we must have
Dα
?x′µ?(ℓ∗o?x′µ?)|ℓe= 0 ; bℓ∗o?x′µ?ℓe ∈ ℓ(U) ⊆ V.(5.20)
In eqs.(5.18) and (5.20) Dα
equations in I R4. The system of differential equations (5.19) must have the same
functional form as the system of differential equations (5.17) and bℓ∗o?x′µ?must be
relative to ?x
able then bℓ∗o?x
that I ∼ I′and that ℓ∗o is the Lorentz deformed version of the phenomena described
by o.
?xµ?and Dα
?x′µ?mean α = 1,2,...,m sets of differential
′µ? the same as bo?xµ?is relative to ?xµ? and if bo?xµ?is physically realiz-
′µ?must also be physically realizable. We say under these conditions
Since in the above definition ℓ∗Υ = ?M,g,D,ℓ∗T1,...,ℓ∗Tm?, it follows that
when I ∼ I′, then ℓ∗g = g,ℓ∗D = D (as we already know) and this means that the
35
Page 36
spacetime structure does not give a preferred status to I or I′according to τ.
5.2
and PR2
Proof that the Existence of SEXWs Implies a Breakdown of PR1
We are now able to prove the statement presented at the beginning of this sec-
tion, that the existence of SEXWs implies a breakdown of the Principle of Relativity
in both its active (PR1) and passive (PR2) versions.
Let ℓ ∈ L↑
ℓ∗F = RˇFR−1, whereˇFe= (1/2)Fµν(xδ(ℓ−1e))γµγνand where R ∈ secSpin+(1,3) ⊂
secCℓ(M) is a Lorentz mapping, such that γ
?xµ? and ?x
dx
+⊂ GM and let F, F ∈ secΛ2(M) ⊂ secCℓ(M), F = ℓ∗F. Let F =
′µ= RγµR−1= Λµ
αγα,Λµ
α∈ L↑
+and let
′µ? be Lorentz coordinate functions as before such that γµ= dxµ,γ
′µ= xµ◦ ℓ−1. We write
Fe=1
2Fµν(xδ(e))γµγν;
′µ=
′µand x
(5.22a)
Fe=1
2F′
µν(x
′δ(e))γ
′µγ
′ν; (5.22b)
Fe=1
2Fµν(xδ(e))γµγν; (5.23a)
Fe=1
2F′
µν(x
′δ(e))γ
′µγ
′ν. (5.23b)
From (5.22a) and (5.22b) we get that
F′
αβ(x
′δ(e)) = (Λ−1)µ
α(Λ−1)ν
βFµν(xδ(e)). (5.24)
From (5.22a) and (5.23b) we also get
Fαβ(xδ(e)) = Λµ
αΛν
βFµν(xδ(ℓ−1e))(5.25)
Now, suppose that F is a superluminal solution of Maxwell equation, in par-
ticular a SEXW as discussed in section 3. Suppose that F has been produced
in the inertial frame I with ?xµ? as (nacs/I), with the physical device described
in section 3.F is generated in the plane z = 0 and is traveling with speed
c1 = 1/cosη in the negative z-direction.
spacetime, according to the observers in I. Now, there exists ℓ ∈ L↑
It will then travel to the future in
+such that
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ℓ∗F = F = RFR−1will be a solution of Maxwell equations and such that if the
velocity 1-form of F is vF = (c2
F is vF= (c
As its is well known F carries negative energy according to the observers in the I
frame.
We then arrive at the conclusion that to assume the validity of PR1is to assume
the physical possibility of sending to the past waves carrying negative energy. This
seems to the authors an impossible task, and the reason is that there do no exist
physically realizable boundary conditions that would allow the observers in I to
launch¯F in spacetime and such that it traveled to its own past.
We now show that there is also a breakdown of PR2, i.e., that it is not true that
all inertial frames are physically equivalent. Suppose we have two inertial frames I
and I′as above, i.e., I = ∂/∂x0, I′= ∂/∂x
Suppose that F is a SEXW which can be launched in I with velocity 1-form as
above and suppose F is a SEXW built in I′at the plane z′= 0 and with velocity
1-form relative to ?x
?
c2
1− 1)−1/2(1,0,0,−c1), then the velocity 1-form of
1− 1)−1/2(−1,0,0,−c′
′2
1), with c′
1> 1, i.e., vFis pointing to the past.
′0.
′µ? given by vF= v
′µγ′
µand
vF=
1
?
1− 1
,0,0,−
c1
?
c2
1− 1
?
(5.26)
If F and F are related as above we see (See Fig.10) that F, which has positive
energy and is traveling to the future according to I′, can be sent to the past of the
observers at rest in the I frame. Obviously this is impossible and we conclude that
F is not a physically realizable phenomenon in nature. It cannot be realized in I′
but F can be realized in I. It follows that PR2does not hold.
If the elements of the set of inertial reference frames are not equivalent then there
must exist a fundamental reference frame. Let I ∈ secTM be that fundamental
frame. If I′is moving with speed V relative to I, i.e.,
I′=
1
√1 − V2
∂
∂t−
V
√1 − V2∂/∂z ,(5.27)
then, if observers in I′are equipped with a generator of SEXWs and if they prepare
their apparatus in order to send SEXWs with different velocity 1-forms in all pos-
sible directions in spacetime, they will find a particular velocity 1-form in a given
spacetime direction in which the device stops working. A simple calculation yields
then, for the observes in I′, the value of V !
In[65]Recami argued that the Principle of Relativity continues to hold true even
though superluminal phenomena exist in nature. In this theory of tachyons there
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Page 38
exists, of course, a situation completely analogous to the one described above (called
the Tolman-Regge paradox), and according to Recami’s view PR2is valid because
I′must interpret F a being an anti-SEXW carrying positive energy and going into
the future according to him. In his theory of tachyons Recami was able to show that
the dynamics of tachyons implies that no detector at rest in I can detect a tachyon
(the same would be valid for a SEXW like F) sent by I′with velocity 1-form given
by eq.(4.26). Thus he claimed that PR2is true. At first sight the argument seems
good, but it is at least incomplete. Indeed, a detector in I does not need to be at
rest in I. We can imagine a detector in periodic motion in I which could absorb
the F wave generated by I′if this was indeed possible. It is enough for the detector
to have relative to I the speed V of the I′frame in the appropriate direction at the
moment of absorption. This simple argument shows that there is no salvation for
PR2(and for PR1) if superluminal phenomena exist in nature.
The attentive reader at this point probably has the following question in his/her
mind: How could the authors start with Minkowski spacetime, with equations car-
rying the Lorentz symmetry and yet arrive at the conclusion that PR1and PR2do
not hold?
The reason is that the Lorentzian structure of ?M,g,D? can be seen to exist
directly from the Newtonian spacetime structure as proved in[66]. In that paper
Rodrigues and collaborators show that even if L↑
tonian dynamics it is a symmetry group of the only possible coherent formulation of
Lorentz-Maxwell electrodynamic theory compatible with experimental results that
is possible to formulate in the Newtonian spacetime(∗).
We finish calling to the reader’s attention that there are some experiments
reported in the literature which suggest also a breakdown of PR2 for the roto-
translational motion of solid bodies. A discussion and references can be found in[67].
+is not a symmetry group of New-
6. Conclusions
In this paper we presented a unified theory showing that the homogeneous wave
equation, the Klein-Gordon equation, Maxwell equations and the Dirac and Weyl
equations have solutions with the form of undistorted progressive waves (UPWs) of
arbitrary speeds 0 ≤ v < ∞.
We present also the results of an experiment which confirms that finite aperture
approximations to a Bessel pulse and to an X-wave in water move as predicted by
(∗) We recall that Maxwell equations have, as is well known, many symmetry groups besides
L↑
+.
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Figure 10:¯F cannot be launched by I′.
our theory, i.e., the Bessel pulse moves with speed less than cs and the X-wave
moves with speed greater than cs, csbeing the sound velocity in water.
We exhibit also some subluminal and superluminal solutions of Maxwell equa-
tions. We showed that subluminal solutions can in principle be used to model
purely electromagnetic particles. A detailed discussion is given about the superlu-
minal electromagnetic X-wave solution of Maxwell equations and we showed that it
can in principle be launched with available technology. Here a point must be clear,
the X-waves, both acoustic and electromagnetic, are signals in the sense defined by
Nimtz[74]. It is a widespread misunderstanding that signals must have a front. A
front can be defined only mathematically because it implies an infinite frequency
spectrum. Every real signal does not have a well defined front.
The existence of superluminal electromagnetic waves implies in the breakdown
of the Principle of Relativity.(∗)We observe that besides its fundamental theoretical
implications, the practical implications of the existence of UPWs solutions of the
main field equations of theoretical physics (and their finite aperture realizations) are
(∗)It is important to recall that there exists the possibility of propagation of superluminal sig-
nals inside the hadronic matter. In this case the ingenious construction of Santilli’s isominkowskian
spaces (see[68−73]) is useful.
39
Page 40
very important. This practical importance ranges from applications in ultrasound
medical imaging to the project of electromagnetic bullets and new communication
devices[33]. Also we would like to conjecture that the existence of subluminal and
superluminal solutions of the Weyl equation may be important to solve some of the
mysteries associated with neutrinos. Indeed, if neutrinos can be produced in sublu-
minal or superluminal modes — see[75,76]for some experimental evidence concerning
superluminal neutrinos — they can eventually escape detection on earth after leaving
the sun. Moreover, for neutrinos in a subluminal or superluminal mode it would be
possible to define a kind of “effective mass”. Recently some cosmological evidences
that neutrinos have a non-vanishing mass have been discussed by e.g. Primack et
al[77]. One such “effective mass” could be responsible for those cosmological evi-
dences, and in such a way that we can still have a left-handed neutrino since it
would satisfy the Weyl equation. We discuss more this issue in another publication.
Acknowledgments
The authors are grateful to CNPq, FAPESP and FINEP for partial financial sup-
port. We would like also to thank Professor V. Barashenkov, Professor G. Nimtz,
Professor E. Recami, Dr. E. C. de Oliveira, Dr. Q. A. G. de Souza, Dr. J. Vaz Jr. and
Dr. W. Vieira for many valuable discussions, and J. E. Maiorino for collaboration
and a critical reading of the manuscript. WAR recognizes specially the invaluable
help of his wife Maria de F´ atima and his sons, whom with love supported his varia-
tions of mood during the very hot summer of 96 while he was preparing this paper.
We are also grateful to the referees for many useful criticisms and suggestions and
for calling our attention to the excellent discussion concerning the Poynting vector
in the books by Stratton[19]and Whittaker[78].
Appendix A. Solutions of the (Scalar) Homogeneous Wave
Equation and Their Finite Aperture Realizations
In this appendix we first recall briefly some well known results concerning the
fundamental (Green’s functions) and the general solutions of the (scalar) homoge-
neous wave equation (HWE) and the theory of their finite aperture approximation
(FAA). FAA is based on the Rayleigh-Sommerfeld formulation of diffraction (RSFD)
by a plane screen. We show that under certain conditions the RSFD is useful for
designing physical devices to launch waves that travel with the characteristic veloc-
ity in a homogeneous medium (i.e., the speed c that appears in the wave equation).
More important, RSFD is also useful for projecting physical devices to launch some
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of the subluminal and superluminal solutions of the HWE (i.e., waves that propa-
gate in an homogeneous medium with speeds respectively less and greater than c)
that we present in this appendix. We use units such that c = 1 and ¯ h = 1, where
c is the so called velocity of light in vacuum and ¯ h is Planck’s constant divided by 2π.
A1. Green’s Functions and the General Solution of the (Scalar) HWE
Let Φ in what follows be a complex function in Minkowski spacetime M:
Φ : M ∋ x ?→ Φ(x) ∈ I C .(A.1)
The inhomogeneous wave equation for Φ is
2Φ =
?∂2
∂t2− ∇2
?
Φ = 4πρ , (A.2)
where ρ is a complex function in Minkowski spacetime.
Green’s function for the wave equation (A.2) as a solution of
We define a two-point
2G(x − x′) = 4πδ(x − x′) .(A.3)
As it is well known, the fundamental solutions of (A.3) are:
Retarded Green’s function: GR(x − x′) = 2H(x − x′)δ[(x − x′)2];
Advanced Green’s function: GA(x − x′) = 2H[−(x − x′)]δ[(x − x′)2];
(A.4a)
(A.4b)
where (x − x′)2≡ (x0− x
x0= t,x
We can rewrite eqs.(A.4) as (R = |? x − ? x′|):
′0)2− (? x − ? x′)2, H(x) = H(x0) is the step function and
′0= t′.
GR(x0− x
′0;? x − ? x′) =1
Rδ(x0− x
′0− R) ; (A.4c)
GA(x0− x
′0;? x − ? x′) =1
Rδ(x0− x
′0− R) . (A.4d)
We define the Schwinger function by
GS= GR− GA= 2ε(x)δ(x2); ε(x) = H(x) − H(−x) .(A.5)
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It has the properties
2GS= 0; GS(x) = −GS(−x); GS(x) = 0 if x2< 0 ;
∂GS
∂xi
For the reader who is familiar with the material presented in Appendix B, we
observe that these equations can be rewritten in a very elegant way in CℓC(M). (If
you haven’t read Appendix B, go to eq.(A.8′).) We have
(A.6a)
GS(0,? x) = 0;
????xi=0= 0;
∂Gs
∂x0
????x0=0= δ(? x) .(A.6b)
?
σ⋆dGS(x − y) = −
?
σdGS(x − y)γ5= 1, if y ∈ σ,(A.7)
where σ is any spacelike surface. Then if f ∈ sec I C ⊗?0(M) ⊂ secCℓC(M) is any
function defined on a spacelike surface σ, we can write
?
Eqs.(A.7) and (A.8) appear written in textbooks on field theory as
σ[⋆dGS(x − y)]f(x) = −
?
dGs(x − y)f(x)γ5= f(y) .(A.8)
?
σ∂µGS(x − y)dσµ(x) = 1 ;
?
σf(x)∂µGS(x − y)dσµ(x) = f(y) . (A.8’)
We now express the general solution of eq.(A.2), including the initial conditions, in
a bounded constant time spacelike hypersurface σ characterized by γ1∧ γ2∧ γ3in
terms of GR. We write the solution in the standard vector notation. Let the constant
time hypersurface σ be the volume V ⊂ I R3and ∂V = S its boundary. We have,
?t+
1
4π
1
4π
0
Φ(t,? x) =
0
dt′
? ? ?
Vdv′GR(t − t′,? x − ? x′)ρ(t′,? x′)
?
? ?
+
? ? ?
?t+
Vdv′
GR|t′=0∂Φ
∂t′(t′,? x′)|t′=0− Φ(t′,? x′)|t′=0
∂
∂t′GR|t′=0
?
+dt′
Sd?S′.(GRgrad′Φ − Φgrad′GR),(A.9)
where grad′means that the gradient operator acts on ? x′, and where t+means that
the integral over t′must end on t′= t+ε in order to avoid ending the integral exactly
at the peak of the δ-function. The first term in eq.(A.9) represents the effects of
the sources, the second term represents the effects of the initial conditions (Cauchy
problem) and the third term represents the effects of the boundary conditions on
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the space boundaries ∂V = S.This term is essential for the theory of diffraction and
in particular for the RSFD.
Cauchy problem: Suppose that Φ(0,? x) and∂
in space, and assume that there are no sources present, i.e., ρ = 0. Then the solution
of the HWE becomes
?
∂tΦ(t,? x)|t=0are known at every point
Φ(t,? x) =
1
4π
? ? ?
dv′
GR|t′=0∂
∂tΦ(t′,? x′)|t′=0−∂
∂tGR|t′=0Φ(0,? x′)
?
. (A.10)
The integration extends over all space and we explicitly assume that the third term
in eq.(A.9) vanishes at infinity.
We can give an intrinsic formulation of eq.(A.10). Let x ∈ σ, where σ is a
spacelike surface without boundary. Then the solution of the HWE can be written
Φ(x) =
1
4π
?
σ{GS(x − x′)[⋆dΦ(x′)] − [⋆dGS(x − x′)]Φ(x′)}
(A.11)
=
1
4π
?
σdσµ(x)[GS(x − x′)∂µΦ(x′) − ∂µGS(x − x′)Φ(x′)]
where GS is the Schwinger function [see eqs.(A.7, A.8)]. Φ(x) given by eq.(A.11)
corresponds to “causal propagation” in the usual Einstein sense, i.e., Φ(x) is in-
fluenced only by points of σ which lie in the backward (forward) light cone of x′,
depending on whether x is “later” (“earlier”) than σ.
A2. Huygen’s Principle; the Kirchhoff and Rayleigh-Sommerfeld Formu-
lations of Diffraction by a Plane Screen[79]
Huygen’s principle is essential for understanding Kirchhoff’s formulation and the
Rayleigh-Sommerfeld formulation (RSF) of diffraction by a plane screen. Consider
again the general solution [eq.(A.9)] of the HWE which is non-null in the surface
S = ∂V and suppose also that Φ(0,? x) and∂
∂tΦ(t,? x)|t=0are null for all ? x ∈ V . Then
eq.(A.9) gives
Φ(t,? x) =
1
4π
? ?
Sd?S′.
1
Rgrad′Φ(t′,? x′) +
?R
R3Φ(t′,? x′) −
?R
R2
∂
∂t′Φ(t′,? x′)
t′=t−R
.
(A.12)
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Page 44
From eq.(A.12) we see that if S is along a wavefront and the rest of it is at infinity
or where Φ is zero, we can say that the field value Φ at (t,? x) is caused by the field
Φ in the wave front at time (t − R) earlier. This is Huygen’s principle.
Kirchhoff’s theory: Now, consider a screen with a hole like in Fig.11.
Figure 11: Diffraction from a finite aperture.
Suppose that we have an exact solution of the HWE that can be written as
Φ(t,? x) = F(? x)eiωt, (A.13)
where we define also
ω = k (A.14)
and k is not necessarily the propagation vector (see bellow). We want to find the
field at ? x ∈ V , with ∂V = S1+S2(Fig.11), with ρ = 0 ∀? x ∈ V . Kirchhoff proposed
to use eq.(A.12) to give an approximate solution for the problem. Under the so
called Sommerfeld radiation condition,
lim
r→∞r
?∂F
∂n− ikF
?
= 0, (A.15)
where r = |? r| = ? x − ? x′, ? x′being a point of S2, the integral in eq.(A.12) is null over
44
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S2. Then, we get
F(? x) =
1
4π
? ?
S1dS′
?∂F
∂nGK− F∂GK
∂n
?
; (A.16)
GK
=
e−ikR
R
, R = |? x − ? x′|,? x′∈ S1. (A.17)
Now, the “source” is opaque, except for the aperture which is denoted by Σ in
Fig.11. It is reasonable to suppose that the major contribution to the integral arises
from points of S1in the aperture Σ ⊂ S1. Kirchhoff then proposed the conditions:
(i) Across Σ, the fields F and ∂F/∂n are exactly the same as they would be in
the absence of sources.
(ii) Over the portion of S1that lies in the geometrical shadow of the screen the
field F and ∂F/∂n are null.
Conditions (i) plus (ii) are called Kirchhoff boundary conditions, and we end
with
FK(? x) =
? ?
ΣdS′
?∂F
∂nGK− F∂
∂nGK
?
, (A.18)
where FK(? x) is the Kirchhoff approximation to the problem. As is well known, FK
gives results that agree very well with experiments, if the dimensions of the aperture
are large compared with the wave length. Nevertheless, Kirchhoff’s solution is in-
consistent, since under the hypothesis given by eq.(A.13), F(? x) becomes a solution
of the Helmholtz equation
∇2F + ω2F = 0 ,
and as is well known it is illicit for this equation to impose simultaneously arbitrary
boundary conditions for both F and ∂F/∂n.
A further shortcoming of FK is that it fails to reproduce the assumed bound-
ary conditions when ? x ∈ Σ ⊂ S1. To avoid such inconsistencies Sommerfeld pro-
posed to eliminate the necessity of imposing boundary conditions on both F and
∂F/∂n simultaneously. This gives the so called Rayleigh-Sommerfeld formulation of
diffraction by a plane screen (RSFD). RSFD is obtained as follows. Consider again
a solution of eq.(A.18) under Sommerfeld radiation condition [eq.(A.15)]
?∂F
(A.19)
F(? x) =1
4
? ?
S1
∂nGRS− F∂GRS
∂n
?
dS′, (A.20)
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Page 46
where now GRS is a Green function for eq.(A.19) different from GK. GRS must
provide an exact solution of eq.(A.19) but we want in addition that GRSor ∂GRS/∂n
vanish over the entire surface S1, since as we already said we cannot impose the
values of F and ∂F/∂n simultaneously.
A solution for this problem is to take GRSas a three-point function, i.e., as a
solution of
(∇2+ ω2)G−
RS(? x,? x′,? x′′) = 4πδ(? x − ? x′) − 4πδ(? x − ? x′′). (A.21)
We get
G−
RS(? x,? x′,? x′′) =eikR
R
−eikR′
R′, (A.22)
R = |? x − ? x′|; R′= |? x − ? x′′|,
????S1
(A.23)
where ? x ∈ S1and ? x′= −? x′′are mirror image points relative to S1. This solution
gives G−
RS
?= 0.
????S1= 0 and ∂G−
Another solution for our problem such that G+
realized for G+
RS/∂n
RS
????S1?= 0 and ∂G+
RS/∂n
????S1
= 0 is
RSsatisfying
(∇2+ ω2)G+
RS(? x,? x′,? x′′) = 4πδ(? x − ? x′) + 4πδ(? x − ? x′′). (A.24)
Then
G+
RS(? x,? x′,? x′′) =eikR
R
+eikR′
R′
, (A.25)
with R and R′as in eq.(A.23).
We now use G+
? n = −?k,?k being the versor of the z direction,?R = ? x − ? x′,?R.? n = z′− z cos(? n,?R) =
F(? x) = −1
2π
RSin eq.(A.25) and take S1as being the z = 0 plane. In this case
(z′− z)/R and we get
? ?
S1dS′F(x′,y′,0)
ikzeikR
R2−eikR
R3z
. (A.26)
A3. Finite Aperture Approximation for Waves Satisfying Φ(t,? x) = F(? x)e−iωt
The finite aperture approximation to eq.(A.26) consists in integrating only over
Σ ⊂ S1, i.e., we suppose F(? x) = 0 ∀? x ∈ (S1\Σ). Taking into account that
k = 2π/λ, ω = k, (A.27)
46
Page 47
we get
FFAA=1
λ
? ?
ΣdS′F(x′,y′,0)eikR
R2z +
1
2π
? ?
ΣdS′F(x′,y′,0)eikR
R3z.(A.28)
In section A4 we show some subluminal and superluminal solutions of the HWE
and then discuss for which solutions the FAA is valid. We show that there are indeed
subluminal and superluminal solutions of the HWE for which (A.28) can be used.
Even more important, we describe in section 2 the results of recent experiments,
conducted by us, that confirm the predictions of the theory for acoustic waves in
water.
A4. Subluminal and Superluminal Solutions of the HWE
Consider the HWE (c = 1)
∂2
∂t2Φ − ∇2Φ = 0 .(A.2′)
We now present some subluminal and superluminal solutions of eq.(A.2′).[80]
Subluminal and Superluminal Spherical Bessel Beams. To introduce these beams we
define the variables
ξ<= [x2+ y2+ γ2
1
?
ξ>= [−x2− y2+ γ2
γ>=
?
We can now easily verify that the functions Φℓm
subluminal and superluminal solutions of the HWE (see example 3 below for how
to obtain these solutions). We have
<(z − v<t)2]1/2;
<; v<=dω<
(A.29a)
γ<=
1 − v2
<
; ω2
<− k2
<= Ω2
dk<
; (A.29b)
>(z − v>t)2]1/2;
>= −Ω2
(A.29c)
1
v2
>− 1
; ω2
>− k2
>; v>= dω>/dk>. (A.29d)
<and Φℓm
>below are respectively
Φℓm
p(t,? x) = Cℓjℓ(Ωpξp)Pℓ
m(cosθ)eimθei(ωpt−kpz)
(A.30)
where the index p =<, >, Cℓare constants, jℓare the spherical Bessel functions,
Pℓ
mare the Legendre functions and (r,θ,ϕ) are the usual spherical coordinates.
47
Page 48
Φℓm
<
function jℓ(Ω<ξ<) [jℓ(Ω>ξ>)] moves with group velocity v<[v>], where 0 ≤ v<< 1
[1 < v>< ∞]. Both Φℓm
term has been introduced by Courant and Hilbert[1]; however they didn’t suspect of
UPWs moving with speeds greater than c = 1. For use in the main text we write
the explicit form of Φ00
[Φℓm
>] has phase velocity (w</k<) < 1 [(w>/k>) > 1] and the modulation
< and Φℓm
> are undistorted progressive waves (UPWs). This
<and Φ00
>, which we denote simply by Φ<and Φ>:
Φp(t,? x) = Csin(Ωpξp)
ξp
ei(ωpt−kpz); p =< or > .(A.31)
When v<= 0, we have Φ<→ Φ0,
Φ0(t,? x) = CsinΩ<r
r
eiΩ<t, r = (x2+ y2+ z2)1/2. (A.32)
When v>= ∞, ω>= 0 and Φ0
>→ Φ∞,
Φ∞(t,? x) = C∞sinhρ
ρ
eiΩ>z, ρ = (x2+ y2)1/2. (A.33)
We observe that if our interpretation of phase and group velocities is correct,
then there must be a Lorentz frame where Φ<is at rest. It is trivial to verify that
in the coordinate chart ?x
(v</
1 − v2
ative to I = ∂/∂t, Φpgoes in Φ0(t′,? x′) given by eq.(A.32) with t ?→ t′, ? x ?→ ? x′.
′µ? which is a (nacs/I′), where I′= (1 − v2
<)∂/∂z is a Lorentz frame moving with speed v<in the z direction rel-
<)−1/2∂/∂t +
?
Subluminal and Superluminal Bessel Beams. The solutions of the HWE in cylindrical
coordinates are well known[19]. Here we recall how these solutions are obtained in
order to present new subluminal and superluminal solutions of the HWE. In what
follows the cylindrical coordinate functions are denoted by (ρ,θ,z), ρ = (x2+y2)1/2,
x = ρcosθ, y = ρsinθ. We write for Φ:
Φ(t,ρ,θ,z) = f1(ρ)f2(θ)f3(t,z) . (A.34)
Inserting (A.34) in (A.2′) gives
ρ2d2
dρ2f1+ ρd
dρf1+ (Bρ2− ν2)f1= 0;
?d2
?d2
(A.35a)
dθ2+ ν2
dt2−∂2
?
f2= 0;
?
(A.35b)
∂z2+ Bf3= 0.(A.35c)
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Page 49
In these equations B and ν are separation constants. Since we want Φ to be periodic
in θ we choose ν = n an integer. For B we consider two cases:
(i) Subluminal Bessel solution, B = Ω2
In this case (A.35a) is a Bessel equation and we have
<> 0
Φ<
Jn(t,ρ,θ,z) = CnJn(ρΩ<)ei(k<z−w<t+nθ), n = 0,1,2,..., (A.36)
where Cnis a constant, Jnis the n-th order Bessel function and
ω2
<− k2
<= Ω2
<. (A.37)
In[43]the Φ<
Bessel beams are examples of undistorted progressive waves (UPWs). They are
“subluminal” waves. Indeed, the group velocity for each wave is
Jnare called the nth-order non-diffracting Bessel beams(∗).
v<= dω</dk<, 0 < v<< 1 ,(A.38)
but the phase velocity of the wave is (ω</k<) > 1. That this interpretation is correct
follows from the results of the acoustic experiment described in section 2.
It is convenient for what follows to define the variable η, called the axicon
angle[26],
k<= k<cosη , Ω<= k<sinη , 0 < η < π/2 .(A.39)
Then
k<= ω<> 0(A.40)
and eq.(A.36) can be rewritten as Φ<
An≡ Φ<
Jn, with
Φ<
An= CnJn(k<ρsinη)ei(k<z cosη−ω<t+nθ). (A.41)
In this form the solution is called in[43]the n-th order non-diffracting portion of
the Axicon Beam. The phase velocity vph= 1/cosη is independent of k<, but, of
course, it is dependent on k<. We shall show below that waves constructed from the
Φ<
Jnbeams can be subluminal or superluminal !
(ii) Superluminal (Modified) Bessel Solution, B = −Ω2
In this case (A.35a) is the modified Bessel equation and we denote the solutions by
>< 0
Φ>
Kn(t,ρ,θ,z) = CnKn(Ω>ρ)ei(k>z−ω>t+nθ), n = 0,1,...,(A.42)
(∗)The only difference is that k< is denoted by β =
ω/c > 0. (We use units where c = 1).
?ω2
<− Ω2
<and ω< is denoted by k′=
49
Page 50
where Knare the modified Bessel functions, Cnare constants and
ω2
>− k2
>= −Ω2
>. (A.43)
We see that Φ>
v>= dω>/dk>such that 1 < v>< ∞ and phase velocity 0 < (ω>/k>) < 1. As in
the case of the spherical Bessel beam [eq.(A.31)] we see again that our interpretation
of phase and group velocities is correct. Indeed, for the superluminal (modified)
Bessel beam there is no Lorentz frame where the wave is stationary.
The Φ>
motion. Band proposed to launch the Φ>
radius r1on which there is an appropriate superficial charge density. Since K0(Ω>r1)
is non singular, his solution works. In section 3 we discuss some of Band’s statements.
We are now prepared to present some other very interesting solutions of the
HWE, in particular the so called X-waves[43], which are superluminal, as proved by
the acoustic experiments described in section 2.
Knare also examples of UPWs, each of which has group velocity
K0beam was discussed by Band[41]in 1988 as an example of superluminal
K0beam in the exterior of a cylinder of
Theorem [Lu and Greenleaf ][43]: The three functions below are families of exact
solutions of the HWE [eq.(A.2′)] in cylindrical coordinates:
Φη(s) =
?∞
?π
0
T(k<)
?1
2π
?1
2π
?π
?π
−πA(φ)f(s)dφ
?
dk<; (A.44)
ΦK(s) =
−πD(η)
−πA(φ)f(s)dφ
?
dη ; (A.45)
ΦL(ρ,θ,z − t) = Φ1(ρ,θ)Φ2(z − t) ; (A.46)
where
s = α0(k<,η)ρcos(θ − φ) + b(k<,η)[z ± c1(k<,η)t]
?
In these formulas T(k<) is any complex function (well behaved) of k<and could
include the temporal frequency transfer function of a radiator system, A(φ) is any
complex function (well behaved) of φ and represents a weighting function of the inte-
gration with respect to φ,f(s) is any complex function (well behaved) of s (solution
of eq.(A.29)), D(η) is any complex function (well behaved) of η and represents a
weighting function of the integration with respect to η, called the axicon angle (see
eq.(A.39)), α0(k<,η) is any complex function of k<and η,b(k<,η) is any complex
function of k<and η.
(A.47)
and
c1(k<,η) = 1 + [α0(k<,η)/b(k<,η)]2. (A.48)
50
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