Homogeneous Bianisotropic Medium, Dissipation and the Non-constancy of Speed of Light in Vacuum for Different Galilean Reference Systems

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1 Progress In Electromagnetics Research Symposium Proceedings, Cambridge, USA, July 5 8, Homogeneous Bianisotropic Medium, Dissipation and the Non-constancy of Speed of Light in Vacuum for Different Galilean Reference Systems Nami Yener Technology Faculty, Umuttepe Campus, Kocaeli University, Izmit, Kocaeli 41380, Turey Abstract A procedure is developed which leads to a relation that can be used to argue the negation of the Special Relativity Theory when there exists a general homogeneous bianisotropic medium with dissipation. The unbounded general bianisotropic medium is interfaced with a perfectly conducting medium filling a half space so that the interface is an infinite plane. The perfectly conducting half space (medium (II)) is assumed to move uniformly and along the O z axis of the Galilean reference system K which is attached to medium (II), and the interface plane with medium (I), the bianisotropic medium which is at rest and to which is attached the Galilean reference system K, is assumed to be perpendicular to the O z axis. The relation found is between constitutive parameters, the direction cosines with respect to Oxyz axes of the incident plane wave impingent on the infinite plane interface, the incident wave parameters, v 1 the relative speed of K with respect to K and c the speed of light in vacuum. This relation is shown to be interpretable to falsify the Special Relativity Theory. On the other hand it is demonstrated also that when the same homogeneous bianisotropic medium without loss is considered no such relation can be obtained and the Special Relativity Theory cannot be contradicted. Three examples are presented. One for a lossless electrically uniaxially anistropic medium, one for a dissipative simple medium and another for a dissipative electrically uniaxially anistropic medium. While the first one does not lead to any contradiction of Maxwell s equations with Special relativity Theory, the other two are shown to lead to such relations. 1. INTRODUCTION A coordinate system called the DB system will be adopted to ease discussions on solutions of field vectors inside a general homogeneous medium. The DB system devised originally by J. A. Kong, consists of the vector and the DB plane [1]. The DB system has unit vectors e 1, e 2 and e 3. e 3 is taen in direction of so that = e 3. The unit vector is in the radial direction in the spherical coordinate system. In terms of the xyz coordinate system we find e 3 = e r = sin θ cos φ a x +sin θ sin φ a y +cos θ a z. The unit vector e 2 is in the e θ direction again in the spherical coordinate system. We thus have e 2 = e θ = cos θ cos φ a x + cos θ sin φ a y. The unit vectors e 1, e 2 and e 3 form a right hand orthogonal coordinate system so that e 1 = e 2 e 3 = sin φ a x cos φ a y. In the above a x, a y, a z are the unit orthogonal vectors of the Cartesian coordinate system. We assume that the uniform velocity of frame K attached to a perfectly conducting half space, with respect to K attached to a homogeneous bianisotropic medium at rest, is in the a z direction. This assumption for an unbounded medium does not impose a restriction on the validity of the results that are obtained because we can always rotate our coordinate system so that the z axis points in the direction of motion. Even though this coordinate rotation certainly affects the element values of the constitutive matrix, the form of the matrix will remain intact after the transformation. This is because the projections of the vectors on the new coordinate system axes will change numerically while the vectors are preserved in form [1]. We shall use two constitutive matrix formalisms in this paper. The first is the Lorentz covariant form given by Equation (5) [2]. The second is the one which is the result of the DB approach and from which the linear algebraic equation system of (6) can be derived. The two formalisms are equivalent in the sense they can be obtained from each other using simple matrix operations. Now we assume the interface of the perfectly conducting half space (medium (II)) and the bianisotropic medium (medium (I)) is an infinite plane perpendicular to the velocity of K with respect to K or the z axis. Then the wave vector components and frequencies of incident and reflected plane waves will be as follows due to enforcement of the phase invariance principle and the boundary condition on the interface plane after the Lorentz transformation has been applied.

2 490 PIERS Proceedings, Cambridge, USA, July 5 8, 2010 Here subscript i indicates incident, while r indicates reflected. iz = α( z ω r/c) ix = x iy = y ω i = α(ω rc z) rz = α( z + ω r/c) rx = x ry = y ω r = α(ω + rc z) (1a) (1b) (1c) (1d) (2a) (2b) (2c) (2d) Primed quantities denote those measured from K while unprimed ones denote those measured from K. ω represents angular frequency while represents wave numbers. α, r are defined in [3] and c is the speed of light in vacuum. In the DB system, the incident wave vector will have the following components. i1 = 0, i2 = 0, i3 = i (3) In Cartesian coordinates these will transform into: ix = i sin θ cos φ = x = rx iy = i sin θ sin φ = y = ry iz = i cos θ = α( z ω r/c) (4a) (4b) (4c) On the surface of the perfect electric conductor (at z = 0) tangential electric field component must vanish. This implies that the equalities (1b), (2b) and (1c), (2c) must have the same right hand sides. On the other hand the incident electric field vector observed from K will read: E i = [E ix a x + E iy a y + E iz a z ] exp{j( xx + yy + zz ω t )}, with i = x a x + y a y + z a z while the reflected electric field observed from K will read: E r = [E rx a x + E ry a y + E rz a z ] exp{j( xx + yy zz ω t )}. Since our interface plane is smooth, the result of specular reflection will yield the following wave vector for the reflected wave if i is as given above: r = x a x + y a y z a z. Notice components of i and r tangential to the interface surface are equal. Note that all these properties are implicit in (1) and (2). Then the relation r E r = jω B r must hold, and we conclude that B r = 1 jω [ a x ( ye rz + ze ry) a y ( xe rz + ze rx) + a z ( ye rx + xe ry)]. Furthermore according to [1] we have [ ] [ cd P L H = M Q ] [ E c B ], (5) where the coefficients matrix on the right hand side is the constitutive matrix of the bianisotropic medium observed from K. Entries of this matrix are given in [1]. In other words D r = P E r /c + L B r and H r = M E r + c Q B r are true. On the other hand inserting this H r into the Maxwell s equation r H r = jω D r, we shall obtain another expression for D r. Equating these two vectors will yield three linear algebraic homogeneous equations for E rx, E ry, E rz with coefficients as functions of x, y and z. For a non-zero solution for the set E rx, E ry, E rz, the coefficient matrix must have a vanishing determinant. This vanishing determinant will yield the dispersion equation for the reflected wave observed from K. All field components will thus have been expressed in terms of one scalar function assuming the coefficients matrix has ran 2. Otherwise the field components will have been expressed in terms of two scalar functions. These scalar functions are elements of the set of E rx, E ry, E rz field components and they will be constrained by the initial conditions.

3 Progress In Electromagnetics Research Symposium Proceedings, Cambridge, USA, July 5 8, The exact same procedure will yield the solution for the incident wave. The boundary condition on the infinite perfectly conducting interface plane will require E iy = E ry, E ix = E rx. Since all the field components will again be determined by one or two scalar functions, choosing these functions as E ry and/or E rx will yield the solution for the other field components. Also the determinant of the pertinent linear algebraic equation system will yield the dispersion relation for the incident wave observed from K. This completes determination of the field vectors that also satisfy the boundary conditions on interface of the two media. 2. DISPERSION RELATIONS FOR INCIDENT AND REFLECTED WAVES AND THE RELATION TO NEGATE SPECIAL RELATIVITY THEORY The dispersion relation of medium (I) as observed from K can be obtained from the determinant of following coefficient matrix for D 1 and D 2, the first two components of the displacement flux density vector in the DB system. {[ ] [ ] [ ] [ ] } [ ] χ 11 χ 12 u ν22 ν 12 γ 11 γ 12 + u κ11 κ 12 D1 N = 0 (6) χ 21 + u χ 22 ν 21 ν 11 γ 21 u γ 22 κ 21 κ 22 D 2 where N = ν 11 ν 22 ν 12 ν 21. The meanings of particular symbols used in (6) can be found in [1]. Except for u which is equal to ω, all others are functions involving the angles φ, θ which appear in the direction cosines of the wave vector with respect to the xyz coordinate system, and the constitutive parameters in the xyz coordinate system, which may include frequency dependence as well. This determinant will yield a quartic in ω i i as follows when the dependence on the constitutive parameters are suppressed in the coefficients, while frequency dependence is exposed. ( ) 4 ( ) 3 ( ) 2 ( ) A (φ, θ, ω) +B (φ, θ, ω) +C (φ, θ, ω) +D (φ, θ, ω) +E (φ, θ, ω) = 0 (7) i i 2 ix + 2 iy Here φ = arctan( iy / ix ), θ = arctan( iz ). φ will be common for both incident and reflected waves whereas θ will be a function of ω for reflected wave and will be the same function of ω for the incident wave when (1a) and (2a) are noted. So we can write ( ω ) 4+ ( ω ) 3+ ( ω ) 2+ ( ω ) Ã(φ, ω, ω) B(φ, ω, ω) C(φ, ω, ω) D(φ, ω, ω) +Ẽ(φ, ω, ω)=0 (8) Upper signs refer to incident wave when ω, possess subscript i and lower signs refer to reflected wave when ω, possess subscript r. The tildes are used to indicate the different coefficient functions that appear when θ is replaced by ω as the independent variable Non-dissipative Case The question rises how the ω dependences of the coefficients à through Ẽ will affect (8). Because of (1d) and (2d) ω i = α(ω rc z), ω r = α(ω + rc z) hold, and replacing ω by ω in either one gives the quantity for the other one except for a minus sign. However noting that a minus sign for frequency in the entries of constitutive matrices is tantamount to a complex conjugation of the corresponding entries for the same frequency with the plus sign [1, p308], we conclude that replacing ω by ω in the coefficients à through Ẽ will switch us from ω i to ω r but at the same time also give the complex conjugates of the true coefficients à through Ẽ for the relevant wave incident or reflected. Here one needs also to note that θ is real in the lossless case so that it does not tae part in the complex conjugation operation. Therefore ω r ( ω ) r ( ω ) = ω i (ω ) i (9) (ω ) holds where ( ) indicates complex conjugation. On the other hand by the definitions of ω i, ω r, i, r we can easily see that ω i ( ω ) i ( ω ) = ω r(ω ) r (ω ). (10) Equation (10) when considered with (9) yields Re{ ω i(ω ) i (ω ) } = 0 for simultaneous satisfaction of (8) for incident and reflected waves. Here Re{} indicates the real part of the complex number i i

4 492 PIERS Proceedings, Cambridge, USA, July 5 8, 2010 within braces. Now if we notice that i has to be real for a lossless medium, this last condition requires Re{ω i } = 0. The medium is non-dissipative and exponential decay in time can not be expected. Because frequency cannot be complex when there are no damping terms in the differential equations [4]. Hence this implies Im{ω i } = 0 also. Here Im{} indicates the imaginary part of the complex number within braces. Then since ω i = 0 holds, due to (10) ω r = 0 must hold also. But ω r = ω i = 0 together with (1d) and (2d) implies ω = = 0 for a common solution of (8) for incident and reflected waves. I.e., there exists no relation to base an argument on to negate the Special Relativity Theory because then due to (1) and (2) the fields become static and we no longer have incident and reflected waves that simultaneously satisfy (8). As an example we tae as medium (I), a lossless uniaxially anisotropic medium. We have for the extra ordinary wave ω2 = 1 2 µ [ 1 ε 1 cos 2 θ + 1 ε 3 sin 2 θ] as the dispersion relation. Here the permittivity matrix has ε 1 as the first two diagonal entries and ε 3 as the third [5]. Medium has scalar constant magnetic permeability µ. Since all quantities on the right are positive and real, in this dispersion relation, Re{ ω i(ω ) i(ω ) } = 0 can simply not hold and by the arguments above there exists no relation to argue against the Special Relativity Theory. We neglect the ordinary wave case for conciseness, but it will also yield ω = = 0, i.e., no relationship to argue against the Special Relativity Theory Dissipative Case In this case i will be complex, and to satisfy Re{ ω i i } = 0, ω i will have to be complex as well, but then we cannot write down (9). We rewrite (7) in the following form A(φ, θ, ω)ω 4 + B(φ, θ, ω)ω 3 + C(φ, θ, ω)ω D(φ, θ, ω)ω 3 + E(φ, θ, ω) 4 = 0 (11) Now, ω = 0, = 0 is a solution of this equation. (11) is valid for both the ω i, i and ω r, r pairs with the coefficients depending on the respective frequency (ω i or ω r ). Therefore ω i = i = 0 or ω r = r = 0 choices are possible for a solution of the equation. We pic the second solution since ω i = i = 0 would mean a static field is incident on the interface. Now setting ω r = r = 0 in (11) when it is written for ω i, i pair, observing that z = ω /(rc), ( ) ω ix 2 + iy 2 2 = (12) αrc will hold now because ω r = r = 0, one has also A(φ, θ, 2αω )c 4 + B(φ, θ, 2αω )c 3 + C(φ, θ, 2αω )c 2 + D(φ, θ, 2αω )c + E(φ, θ, 2αω ) = 0 (13) This gives a quartic relating c, ω, φ, θ and frequency independent part of constitutive parameters. This relation can always be used to justify that c depends on ω and by writing ω as ω = α(ω i v 1 iz ) to justify that c depends on v 1 as well. To see the latter part of this statement, import from (12), (1a) and (1d) into ω = α(ω i v 1 iz ) and one will obtain an identity independent of ω. This indicates that ω is arbitrary and can be chosen at will with a corresponding iz and ω i established from (12), (1a) and (1d). Because ω i = 2αω, fixing ω will result in a change of 2αω with v 1. When considered with (13), this will mean a change in the value of cin order to eep (13) satisfied while coefficients à through Ẽ vary with v 1 through 2αω. Notice that the solution that is the subject of this sub-section is a solution that satisfies (7) simultaneously when it is written for incident and reflected waves. Two examples we shall cite for this subsection are 1) a dissipative simple medium 2) dissipative uniaxially anisotropic medium. 1) We tae as medium (I) of above development, a dissipative simple medium. We have ω2 ω µ 1 ε 1 ω+jµ 1 σ 1 for the dispersion relation [6]. Here imposing ω r = r = 0 will cause ω i = 2αω and i = 2αω /c to hold so that c 2 2αω = µ 1 ε 1 2αω +jµ 1 σ 1 will be found from the dispersion relation for the incident wave. When rearranged this reads 2αω c (1 µ 2 1 ε 1 c 2 ) = jµ 1 σ 1 which is the basic equation obtained in [3] and used to negate the Special Relativity Theory. 2) We now tae as medium (I), a dissipative uniaxially anisotropic medium. We have for the extra ordinary wave ω2 = 1 2 µ [ 1 ε 1 cos 2 θ + 1 ε 3 sin 2 θ] as the dispersion relation [1]. Here the permittivity 2 =

5 Progress In Electromagnetics Research Symposium Proceedings, Cambridge, USA, July 5 8, matrix has ε 1 as the first two diagonal entries and ε 3 as the third. Medium has scalar constant magnetic permeability µ. Also ε 1 = ε 11 + jσ 11 /ω and ε 3 = ε 33 + jσ 33 /ω hold. When these and relations (12) are substituted in the dispersion relation for the incident wave we obtain the following quadratic for ω i. c 2 µ [ 2 ε 11 ε 33 + σ 11 σ 33 jω i (σ 11 ε 33 + σ 33 ε 11 ) ] = 2 ( ε33 cos 2 θ + ε 11 sin 2 θ ) ( jω i σ33 cos 2 θ + σ 11 sin 2 θ ) (14) (14) can always be explicitly solved for ω i = 2αω. Because ω i = 2αω, when ω is fixed and v 1 varies, c has to change to eep up with the variation of ω i with v 1. This relation therefore can be used to argue against the Special Relativity Theory and to prove it cannot account for the loss in medium (I). The ordinary wave case is omitted because it reduces to Example 1 of this subsection. REFERENCES 1. Kong, J. A., Electromagnetic Wave Theory, EMW Publishing, Cambridge, USA, Kong, J. A., Theorems of bianisotropic media, Proc. of IEEE, Vol. 60, No. 9, , Yener, N., On the non-constancy of speed of light in vacuum for different Galilean reference systems, Journal of Electromagnetic Waves and Applications, Vol. 21, No. 15, , Courant, R. and D. Hilbert, Methods of Mathematical Physics, Vol. 2, , Interscience Publishers, New Yor, USA, Ramo, S., J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, John Wiley, New Yor, Balanis, C. A., Advanced Engineering Electromagnetics, John Wiley, USA, 1989.