ELECTROMAGNETIC WAVES WITH NONLINEAR DISPERSION LAW P. М. Меdnis Novosibirs State Pedagogial University, Chair of the General and Theoretial Physis, Russia, 636, Novosibirs,Viljujsy, 8 e-mail: pmednis@inbox.ru Abstrat Last year ysiists in Europe have measured the veloity of the neutrinos partiles. They found the neutrinos moving faster than the speed of light in vauum. This result means that Einstein s relativity priniple and its onsequenes in modern ysis need a global additional renovation. In present paper the part of this problem is onsidered in terms of basi Maxwell s method only. By means of introdution a diffusion lie displaement urrent the new super wave equation was derived, whih permits of its solution be desribed the eletromagneti waves moving some faster than the onventional speed of light in vauum espeially in a gamma ray of a very short wave length region. The unique properties of these waves are that they undergo nonlinear dispersion law, uppermost limit of whih is restrited. Disussion of further experimental problems and a number of estimations are given for the maro ysi super wave equations also.. Introdution The lassial mirosopi Maxwell s and Lorentz s equations of eletrodynamis present the basi omponents of the Einstein s relativity priniple []. Let us write them B, B E, () E 4 B j, E 4π. () Here we have the eletri field strength vetor E and the magneti indution vetor B. The densities j and are indued the mirosopi eletri urrent density and, respetively, the volume eletri harge density whih are presented by the expressions j q v r - r, (3) q r - r. (4) Here the symbol q notes the eletri harge of -h partile. In addition to the (3) (4) the vetors r, r are the spae and the -h partile radius vetors, the vetor v r is the veloity. The - funtion is the 3-dimensional Dira, s delta funtion. The and j are obeying the harge onservation law
j. (5) The is the onventional veloity of light onstant in vauum. This is the main goal we are interested in present paper. Indeed, reently European ysiists [] have measured the veloity of the partiles alled neutrinos. They found the neutrinos moving just faster than the speed of light in vauum. This unexpeted result is enough to all it serious reason that Einstein s relativity priniple in all parts of modern ysis needs global renovation. So we must begin to revise the basi equations () and (). Our goal is to understand what hanges are to be introdued in () - (5) and in what sense the Einstein s relativity priniple were to be onserved. So, we propose here the new addition to the theory of eletrodynamis. They inlude not the only densities (3) - (4) but additional the lie diffusion virtual densities of the harge and urrent also rising the new omponents of displaement urrent. The ysial sense this supplements will be seen from the undergoing part of this paper.. The super wave equation.. The additional omponents in the displaement urrent. The eletri urrent density (3) and the eletri harge density (4) are not the only soures of the eletromagneti field. Long ago, for some reason in nonlinear optis [3] and in ourse of the field radiation onsideration [4] the enomenologial partile was onsidered may have exept the eletri harge e the own eletri dipole moment p (t) and the magneti dipole moment (t) also. Then the seleted eletri urrent density j is represented respetively by three omponents - the transfer, the polarization and the magnetization omponents. In the ysis of semiondutors [5], for example, the different inds of the diffusion urrents are onsidered. The onventional way of taing into aount an additional urrent arrier and of densities j a and a given is well nown. To be sure the onservation law is fulfilled one must write instead of () the equations E 4 B ( j j a ), E 4π( a (6)
with the additional part of the onservation law a j a. (7) It will be noted now when a new ind of harge arriers absent but the new omponents of urrent j a appears at any virtual form at best to orrespond with the Maxwell s method we must write an additional eletri field the form (6). As a result instead of (6) we have to write ( E E B E a in the displaement urrent also instead of using a ) 4 ( j j The onservation law one may write now in the alternative form a ), E 4π (8) 4 ja Ea. (9) First of all these equations permit us find the field E a or it's derivative E a and then define the additional harge density a by means of using the equation E a 4π. So, a for the diffusion lie eletri urrent of the ind j a Dgrad, () where D is the onstant oeffiient whih dimension in SI equal the square of length di- vided at the time, i.e. D m / s, we find from (8) and (9) the expression ( DE) Ea. () Here and in further expressions the symbol is the nown Laplaian operator []. The main solution of this equation gives us the new omponent of displaement urrent and the derivative of the harge as follows a expressed in terms of eletri field E and density of harge E a a DE, D. () For the mirosopi dipole diffusion lie urrent of the ind 3
j a grad (3) where is another onstant diffusion oeffiient whih dimension in SI is equal to the square of length, i.e. m, we find from (8) and (9) the expression ( E) Ea (4) As a onsequene we have the solution of this equation E a E, a. (5) As a result in summary we may write instead of (8) the equations E 4 B ( ) DE ( j ja ja ), E 4π. (6) Combine () and (3) with E 4π we may exlude the j a and j a from the left equation of (3) at all giving us the new form of equations B E 4 D E j, E 4π. (7) Also for the urrent of the partile dipole p el used in [3,4] j a pr r ( t ) (8) similar onsideration yields Ea l E, a l. (9) On its own aount this urrent all into ation the new members in () B ( Here also we may exlude j a to get E 4 l a, E 4π. () ) ( j j ) E E 4 B l j, E 4π. () Beause of the latter ase have the enomenologial harater we return to the mirosopi urrents () and (3) with the main equations (7), (3), (4) and (5). As for Fara- 4
day s equation in () to ensure the further wave solutions in absene of any ind of magneti harge we must generalize this equation for symmetry reason to the form B E DB, B () or in the equivalent form B E D B, B. (3) The group of equations (7), (3) and (5) presents the new equations of eletrodynamis taing into aount the two types of diffusion lie urrents. It is seen that eletromagneti field in vauum is haraterized not only by the onstant but in addition aount must be taen of the pare onstant and D. It will be our tas below to disuss and estimate them... Field super wave equation. Exlude from the equations (7) and (3) in turn respetively eletri and magneti fields we may write the E E D E 4 j D j (4) B B D 4 B j For the free of harge spae we get the equations D D. (5) E E, E (6) B B, B. (7) For the further development suh equations are named as super wave equations (SWE). In the free spae SWE permits the plane wave solutions E E ir t, B ir t e B e (8) 5
with the onstant omplex amplitudes E, B, whih are perpendiular to the wave vetor, i. e. the transverse wave onditions E and B must be satisfied. The symbol notes a irle frequeny in ourse of time proess. More over this frequeny depends on the additional onstants and D. The dispersion law is D. (9) Consider now the sore of the possible solutions and their graial presentation with the help of MathCad omputer program. In values region of wave vetors /D the frequeny is real, so the wave solution (8) with arbitrary sign of (9) exists. If the ondition /D taes plae the frequeny beame imaginary. A ysial damped solution (8) in this ase must be onstruted losely with hoose of the sign of (9). As we an see from denominator of (9) the singular point exists if the wave vetor obeys the ondition /. For the not damped regular wave solution (8) the ondition /D / realized. In this ase the dependeny of seen at the Fig.. D/ on D / for the /D. 5 as example is.77 M i.5.5 xind i Figure : The ounter plot of the D/ versus the D/ for the /D. 5 The ase speed V of this wave is given as usual by expression 6
V D. (3) The dependeny of V / on D/ for the same /D. 5 is seen at the Fig...5.5 M i.5.5 xind i Figure : The ounter plot of the V / versus the D/ for the /D. 5 For the not damped wave solution (8) under ondition / /D the point of singularity presents inside the region. In this ase the dependeny of /D.5 as example is seen at the Fig. 3. D/ on D/ for the 5 4 M i.5 xind i Figure 3: The ounter plot of the D/ versus the D/ for the /D. 5 The dependeny of V / on D/ for the same /D. 5 is seen at the Fig. 4. 7
5 4 M i.5 xind i Figure 4: The ounter plot of the V / versus the D/ for the /D. 5 It seems the inherent damped waves and the interruption in the spetrum for the lassial vauum spae are to be the nonysial solutions. To avoid suh solutions we must demand the ondition D/ to be taen plae. For this ase the dispersion law is in form /,. (3) The dependeny of / on / is seen at the Fig. 5. 5 4 M i.5 xind i The ase speed Figure 5: The ounter plot of the V in this ase is / versus the / V /. (3) The dependeny of V / on / is seen at the Fig. 6. 8
5 4 M3 i.5 xind i.999 Figure 6: The ounter plot of the V / versus the / In addition as another option to avoid any damped waves in a broad manner we must suggest also that the onstant D = in () at all. Then instead of (3) and (3) we must write respetively /, (33) and V /. (34) solely under ondition. The grais of that funtion are similar in a sense to of Fig. 5 and Fig. 6 respetively. What ase is realized really must be solved by a future experiment. My own preferene to be related to the last ase beause of some its natural simpliity. Indeed, the maximal value of wave vetor exists beause of the minimal spatial parameter or a alled minimal spae length exists and broadly are disussed until now. The further part of this paper is onsidering under ondition that D..3. Field potentials super wave equations. From the () by means of standard method we derive the onnetions of fields and potentials, B A A E. (35) 9
Here respetively A and are the vetor and salar potentials. Then from (6) follow the equations for potentials A 4 A j A, (36) A 4. (37) As well nown the potentials A and are not unique values haraterizing the fields. The gradient or alibration transformation in (35) to another pair of potentials A ~ and ~ with any arbitrary funtion should be written in the form A A ~, ~. (38) If needed this property permits us to simplify (36) and (37). So in partiular new Lorentz s alibration is presented now in the form However the Coulomb s alibration beame unhanged A. (39) A. (4) 3. Further more 3.. How we an verify the theory. As a result of our theory for the ase D also we are thus led to the onept that the speed of light in vauum is no more the onventional onstant value. The dependene of the speed of light versus the relative wave number / is given by (34). The real value of is unnown now. In the full lot of early experiments it has been showed that the speed of light in vauum is the onstant value beause I thin of it was the onstant by the definition mostly. Indeed, this result may mean no more that the value of is very large or as seen from (33) the value of is a very small value also. To estimate the let us tae the relative speed differene of the same order as in [], for example, i. e.
V.5 5. (4) Then for the very high energeti gamma rays whih length of wave is equal m we get for the value 5 m. This is the value of the order of the fundamental length, whih estimated value supposedly lies now in this region and less indeed. From (33) it follows that 7.5 m. Hene the dependene (34) an be identified for the very short wave length gamma rays region only. For maroysis ase of fields we must onsider the new equations (6) () or (7) (3) as the averaged one by mirosopi initial states with any distribution funtion [4]. Up to visual opti region we may hope to he the additional urrent effets by means of polarizations effets only beause of a small new addition in boundary onditions for eletri field strength vetor E depending on. 3.. The ways and means to renovate the Einstein s relativisti priniple. The oordinates and time transformation of equations (36) and (37) are the more so interesting for us now beause of the transformation from one inertial system to another inertial system of referenes. In aordane with the priniple of relativity the Maxwell s equation () and () are unhanged in form under the linear Lorenz s transformation []. Let be the group of spatial oordinates x, y, z and the time t are given in an inertial system of referenes K in whih the equations () and () are given. Let be another group of spatial the oordinates x, y, z and the time t are given in an inertial system of referenes K whih is moving with the onstant speed V relatively to a system of referenes K in the positive diretion of x axis so the all diretions of axis x, y, z and of axis x, y, z remains parallel to eah other. For this ase the Lorenz s transformations are x x Vt V, y y, z z, V t x t. (4) V
The reverse transformations are found from (4) if the all hanges of quantities x, y, z, t x, y, z, t plus the hange V V are realized respetively. For beginning in our ase all we an write now for equations (36) and (37) the symboli transformations in the form x x( x, t), y y, z z, t t( x, t), (43) x x( x, t), y y, z z, t t( x, t). (44) To transform the potentials in the equations (36) and (37) we must ombine the vetor and salar potentials and hange variables as follows Ax Ax, Ay Ay, Az Az, (45) Ax. (46) In this equations the parameters ij ( i, j, ) are not depending on the oordinates and the time. The primed potential A and in these equations are depending on x, y, z, t. The starting potentials on the left side of (36) and (37) are depending on x, y, z, t. Obviously the reverse transformation of (45) and (46) may be found easily. Substitute the (45) and (46) in the expression standing in square braet of (36) and (37) equations. Then we an find that this expression is invariant, i.e. A A inv, (47) if the rules of transformation of the differentials are defined as follows x x y y,, z, (48) z. (49) x The reverse transformations of these differentials are written respetively in form x x, y y, z z, (5)
t x. (5) In addition, speial aount must be taen of Laplaians on the left sides of (49) and (5), they were expressed by means of (48) and (5) respetively. Also we suppose that both reverse operators and exist. After substitution (45) and (46) in (36) and (37) we an see they are transformed to invariant form if the following onditions for ij taes plae,,. (5) However the expliit sight of the transformation funtions x x( x, t) and t t( x, t) may not be found so easily as we want. I stop further development of searhing the solutions for a variety of reasons. First of all the funtions x x( x, t) and t t( x, t) are expeted to be the nonlinear funtions it may be of a linear ombinations of x and t resulting some tehnial diffiulties for defining the onstants ij and the integration onstants also. Besides that to renovate Einstein s relativisti priniple by means of the simple exhange of Lorenz s transformations (4) by any new transformations only does not possible in priniple. The main reason is that equations (36) and (37) are to be written in a new invariant form to define the alled ation and then to find a new equation of motions for harged partile using the priniple of minimal ation [6]. Unfortunately this proedure needs intra generalization to be applied to the ase of derivatives of higher than seond order in the equations (36) and (37). It will be very interesting to solve both these problems stated due to the full sope of interests mainly for the wide range of new marosopi appliations. Naturally the starting point of a solution may differ from the suggested one. 4. Referenes. J. D. Jason. Classial Elerodynamis, 3 rd ed. Wiley, New Yor, 999.. Measurement of the neutrino veloity with the OPERA detetor in the CNGS beam. Arxiv.org/abs/9.4897 3
3. V.N. Genin, P.M. Mednis. About the own eletri dipole and quasi-orbital magneti moments of a ondutivity eletron. Izv. Vuzov, GGU, Radio-fizia, (967), 585. 4. V.L. Ginzburg. The theoretial ysis and astroysis. Mosow, Naua, 98. 5. C.F. Klingshirn. Semiondutor optis. 3 rd ed. Springer, Berlin 7. 6. L.D. Landau, E.M. Lifshitz. The lassial theory of fields. Oxford, Butterworth Heinemann, 995. 4