Electromagnetic waves in space with three-dimensional time
V. S. BARASHENKOV(1) (*) and M. Z. YUR8IEV(2)(1) Joint Institute for Nuclear Research - 141980 Dubna, Moscow region, Russia (2) Industrial Group INTERPROM - Moscow 117418, Russia
(ricevuto il 2 Febbraio 1996; approvato il 23 Luglio 1996)
Summary. — Properties of plane waves in six-dimensional Minkowski space-time
with equal number of space and time co-ordinates are considered. In the x-subspace the wave is characterized by three orthogonal one-to-the-other vector fields two of which are transversal and the third is longitudinal with respect to the direction of the wave motion. The energy of transverse and longitudinal field components is, respectively, positive and negative; however, due to the causality principle permitting the development along all time axes only from the past to the future, the total wave energy is always positive, which prevents a spontaneous creation of waves from vacuum and exotic processes in which the final mass exceeds the initial one.
PACS 03.30 – Special relativity.
PACS 03.65.Pm – Relativistic wave equations. PACS 11.30.Cp – Lorentz and Poincaré invariance.
1. – Introduction
The Einsteinean theory of relativity was the first one to reveal a symmetry in the properties of space and time, the subsequent generalizations introducing a proper time for each particle [1] and for each space point x [2] equalized in right space and time still more. It is interesting to take a next step on this way and to consider a more consistent from a relativistic-viewpoint theory with equal number of space and time co-ordinates when
(x×)m4 (2x , t×)T
m, (x×)m4 (x , t×)Tm.
(Here and in what follows the superscript T denotes the transpose, three-dimensional vector in x- and t-subspaces will be marked, respectively, by bold symbols and by a hat, six-dimensional vectors by bold symbols with the hat; the value of Greek and Latin indices is m G6, kG3.) The most satisfactory variant of such a theory was developed by
(*) E-mail: barashenkovHlcta30.jinr.dubna.su
Cole [3-5]. His theory has an elegant mathematical form, but at the present stage it has serious difficulties due to the presence of objects with negative energy, to the spontaneous creation of particles, which results in vacuum non-stability and to the possibility of exotic decays in which the secondaries mass excels the decaying-particle mass. It is not clear whether that is an essential feature of the multi-time approach restricting its applicability only by ultra-small space-time regions as it is emphasized by Cole [5] or one may sidestep the difficulties using some additional physical conditions.
The goal of our paper is to consider the peculiarities of the multi-time theory by means of a simple example of a plane-electromagnetic-wave motion in order to more exactly understand the physical meaning of equation solutions with negative energy and to point out conditions excepting vacuum non-stability and exotic decays. It will be shown that one can gain that by means of the causality principle restricting acceptable types of time trajectories.
Section 2 considers the plane-wave solutions of the generalized multi-time Maxwell equations. In sect. 3 the six-dimensional momentum-energy vector of plane wave is derived, and in sect. 4 we discuss conditions by which the total energy of any plane wave remains always positive-definite.
2. – Plane waves in multi-dimensional time
We shall use a multi-time generalization of Maxwell equations found by Cole [6, 7]. In empty space, in the absence of charges and currents, it comes to a consideration of the equation
˘ A2A×
2 ˘(˘AA×) 40 , (1)
where (A×)m4 (2A , A×)mT is the six-dimensional vector potential, (˘A)
m4 (2˘ , ˘× )m and
˘ ×
i4 2ˇ O ˇti are del operators. The last term of eq. (1) vanishes due to the extended
Lorentz condition
˘ AA×
4 0 . (2)
As a rectilinear worldline of plane wave can be defined by the vector n× 4
( 1 , 0 , 0 , 1 , 0 , 0 )T, eqs. (1), (2) can be written in the form ˇ2A× O ˇt22 c2ˇ2A× O x24 0 , (3) ˇA4O ˇt 1 ˇA1O ˇx 4 0 . (4)
Further, just as in the known one-time theory, the first term in (4) is turned into zero by means of a gauge and we get then from eq. (3):
ˇA1O ˇt 4 const ,
i.e. the component A1bears no relation to a wave process and can be removed. It means that the potentials A and A× are transverse vectors: A× Qn×40.
The electric field tensor looks now as E×4 A˜×2 ˘A×4 c21
u
0 2 A . 2 2 A . 3 A.5 0 0 A.6 0 0v
. (5)(The solution of a wave equation corresponding to a wave moving along the vector n satisfies the condition ˇAmO ˇx 4 2 A.mO c.) Let us introduce the three-dimensional electric fields Ek4 (E×1 k, E×2 k, E×3 k) and E×k4 (E×k1, E×k2, E×k3). It is easy to prove that the
fields Ek and E×k for k 42, 3 are longitudinal vectors, but E1 and E×1 are transversal ones: E1Q n 4E×1Q n× 40.
From the formal point of view the appearance of the longitudinal electric field is stipulated by the impossibility to remove by means of the gauge transformation more than one component of the three-vector A× .
Using for Ei and E×i the expressions from (5), one can prove that the magnetic
field
H 4˘3A42 c21 ˇ
ˇtn 3A4n3E1 and the «time-magnetic» field
G×4 2˜×3 A×4 c21 ˇ ˇtn× 3A
×
4 n× 3E×1
are also transverse ones: H Q n4G× Qn×40. The absolute values H4NE1N , G4 NE×1N .
As we see, in the multi-dimensional world the electromagnetic wave becomes apparent in two forms: in the x-subspace it looks as a superposition of the transverse
wave (E1, H) and the longitudinal wave EL, in the t-subspace it is a sum of the
transverse wave (E×1, G×) and the longitudinal one with the vector E×L. The structure of the plane wave in the x and t subspaces is completely symmetrical—that part of the wave which in the x-subspace is transversal, in the t-subspace becomes longitudinal and vice versa (see fig. 1). Depending on their signs the longitudinal field strengths Ek
and E×k (k 42, 3) can be parallel or antiparallel to the direction of the space and time
vectors n and n× .
3. – Momentum-energy of a plane wave
The momentum-energy tensor of an electromagnetic field, derived with the help of the canonical rules,
Tmn 4 1 4 p
g
F n d Fdm1 1 4g mn F lg Flgh
is different from the respective tensor of one-time theory only by a number of compo-nents. If the electromagnetic field tensor Fmn4 ˇAmO ˇxn2 ˇAnO ˇxm is represented in
the form F × 4
u
2H × 2E×T E × G ×v
,where H× , G× , E× are 333 matrices, then Tmn can be written in a similar matrix form also: Tmnf
u
T × 1 T × 3 T × 2 T × 4v
mn 4 1 4 pu
2H×H×2 F×E× E ×T H × 1 G×E×T 2H×E×2 E×G× E ×T E × 1 G×G×v
mn 1 1 8 pg mn(H2 1 G22 E×2) , where E ×2 4!
i 41 3 E2i4!
i 41 3 E×2i .Taking into account the relations
(H×H× )ik4 HiHk2 dikH2, (G×G×)ik4 GiGk2 dikG×2,
(E×E×T)ik4 E×iE×k, (E×TE×)ik4 EiEk,
(E×TH× )ik4 2 (H×E×)ik4 (Ei3 H ) ,
(E×G×)ik4 2 (G×E×T)ik4 (E×i3 G×k) ,
which can be proved by means of the explicit expressions for the matrices E× , H× and G×, we get the following expressions for the tensors T×i:
4 pT×1ik4 2 E×iE×k2 HiHk1 1 2dik(H 2 2 G×21 E×2) , 4 pT×4ik4 EiEk1 GiGk1 1 2dik(H 2 2 G×22 E×2) , 4 pT×3ik4 4 pT×2ik4 (Ek3 H )i2 (E×i3 G×)k.
Owing to the symmetry Tmn
4 Tnm the tensors T×ik are linked by the relations
T × 1
ik
It must be also noted that up to the sign the expressions for T×lik are symmetrical
with respect to pairs of the fields (Ek, H) and (E×k, G×): T×1ikK T×4ik, T×2ikK T×2ik under the change EkK iE×k, H KiG× and backward.
It is easy to show that by
EiK Ed1 k, (E×i)kK Eid1 k, G×K 0
all the above-written expressions turn into the known Maxwell’s theory formula. Let us define a six-dimensional momentum-energy vector
Pm
4
Tmlds n,where dsn4 nndV and dV is an element of a three-dimensional hypersurface. In the
particular case of a plane wave with the direction of its time trajectory taken as the
t1-axis, Pm4
TmktkdV 4.
/
´
T×31 kdV , T× 4 1 k dV , m 4kG3 , m 431k .So, the field momentum-energy density p × 4 (WT1 WL) n× , (6) where WT4 (A . 2 21 A . 2 3O pc24 (E211 H2) O8p and WL4 2 (A .2 51 A .2 6) O8p42 (E×121 G×2) O8p are the energies of the transverse and the longitudinal fields.
As one can see, the three-dimensional energy and momentum vectors of a plane wave are directed along its worldline x× and their transverse components pk, k c 1 , 4 ,
equal to zero always. The momentum-energy of the transverse field component (E1, H)
is positive, meanwhile not only the momentum but also the energy of the longitudinal field EL are negative.
4. – Discussion
It would seem strange that a part of the wave energy is negative however, an analogous situation takes also place in the Maxwell electrodynamics where a «scalar photon» energy is negative. Because the multi-component electromagnetic wave is a unitary object, only its total energy is physically significant and its negative part is never observed. In the six-dimensional theory the total wave energy (6) must be positive too,
i.e. WTF NWLN and the transverse component prevails always over the longitudinal one: NE1N F NE×1N . In this case the plane-wave momentum is directed along the x1-axis and the wave is developing along the t1-axis from the past into the future.
The mathematically acceptable solution with negative energy WT1 WLE 0 corre-sponding to an incoming wave with backward space and time directions must be rejected if we take into account the causality principle.
In the boundary case of equal amplitudes E2
14 E×24 H24 G×2 the complete compensation of the transverse and longitudinal components occurs (similar to the compensation of the scalar and longitudinal components of a plane wave in the one-time theory) and the momentum-energy (6) turns into zero and the wave disappears. So, as in the usual Maxwell electrodynamics the energies of all outgoing plane waves in six-dimensional world are positive and these waves develop along the positive directions of all time axes ti.
The considered example of a plane electromagnetic wave prompts that an analogous situation takes place in the general cases: due to the causality principle all wave solutions describing the motion of particles correspond to time trajectories with positive projections tiF 0 and, respectively, to energy vectors with positive
compo-nents. That forbids spontaneous creations of groups of particles from vacuum and exotic decays in which the mass of secondaries exceeds the decaying-particle mass, since every such event is accompanied always by a time-reversal motion along if only one axis ti and by a violation of causation.
It should be emphasized that the time reversibility is an approximate property of theories with finite number of particles and interconnections, meanwhile due to a non-exhaustive huge number of real interconnections in Nature time reversibility does not occur, strictly speaking, even in microscopic processes, since it would demand the time turning of all these innumerable interconnections.
R E F E R E N C E S
[1] DIRAC P. A. M., FOCK V. A. and PODOLSKY B., Zs. Sowjetunion., 2 (1932) 468. [2] TOMONAGA S., Prog. Theor. Phys., 1 (1946) 27.
[3] COLE E. A. B., J. Phys. A, 13 (1980) 109. [4] COLE E. A. B., Phys. Lett. A, 76 (1980) 371. [5] COLE E. A. B., J. Phys. A, 13 (1980) 109. [6] COLE E. A. B., Nuovo Cimento A, 60 (1980) 1. [7] COLE E. A. B., Nuovo Cimento B, 85 (1985) 105.
