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On the Possibility of Faster-Than-Light Motions in Nonlinear Electrodynamics

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On the Possibility of Faster-Than-Light Motions in Nonlinear Electrodynamics Proceedings of Institute of Mathematics of NAS of Ukraine 2004, Vol. 50, Part 2, 835–842 On the Possibility of Faster-Than-Light Motions in Nonlinear Electrodynamics Gennadii KOTEL’NIKOV RRC Kurchatov Institute, 1 Kurchatov Sq., 123182 Moscow, Russia E-mail: [email protected] A version of electrodynamics is constructed in which faster-than-light motions of electro- magnetic fields and particles with real masses are possible. 1 Introduction For certainty, by faster-than-light motions we shall understand the motions with velocities v> 3 · 108 m/sec. The existence of such motions is the question discussed in modern physics. As already as in 1946–1948 Blokhintsev [1] paid attention to the possibility of formulating the field theory that allows propagation of faster-than-light (superluminal) interactions outside the light cone. Some time later he also noted the possibility of the existence of superluminal solutions in nonlinear equations of electrodynamics [2]. Kirzhnits [3] showed that a particle possessing the i tensor of mass M k =diag(m0,m1,m1,m1), i, k =0, 1, 2, 3, gab =diag(+, −, −, −)canmove faster-than-light, if m0 >m1. Terletsky [4] introduced into theoretical physics the particles with imaging rest masses moving faster-than-light. Feinberg [5] named these particles tachyons and described their main properties. Research on superluminal tachyon motions opened up additional opportunities which were studied by many authors, for example by Bilaniuk and Sudarshan [6], Recami [7], Mignani (see [7]), Kirzhnits and Sazonov [8], Corben (see [7]), Patty [9], Oleinik [10]. It has led to original scientific direction (several hundred publications). The tachyon movements may formally be described by Special Relativity (SR) expanded to the domain of motions s2 < 0. For comparison, the standard theory describes motions on the light cone s2 = 0 and in the domain s2 > 0 when v ≤ 3 · 108 m/sec. The publications are also known in which the violation of invariance of the speed of light is considered [11–17]. One can note, for example, Pauli monograph [11] where elements of Ritz and Abraham theory are presented; Logunov’s lectures [12] in which SR formulation in affine space was given; Glashow’s paper [13] discussing experimental consequences of the violation of Lorentz-invariance in astrophysics; publications [14–17] on the violation of invariance of the speed of light in SR. A version of the theory permitting faster-than-light motions of electromagnetic fields and charged particles with real masses is proposed below as continuation of such investigations. 2 Formal construction of the theory Let us introduce the space-time R4, metric properties of which may depend on the velocity of a particle being investigated and take the metric of R4 in the form: ds2 = c2 + v2 dt2 − dx2 − dy2 − dz2 0 2 2 2 2 2 2 = c0 + v (dt ) − (dx ) − (dy ) − (dz ) − invariant. (1) 836 G. Kotel’nikov Here x, y, z are the spatial coordinates, t is the time, c0, c0 are the proper values of the speed of light, v is the velocity of a body under study with respect to the reference frame K.Letus connect the co-moving frame K with this body. Let the proper speed of light be invariant 8 c0 = c 0 =3· 10 m/sec − invariant. (2) As a result, the common time similar to Newton one may be introduced on the trajectory of the frame K (when v = dx/dt) and the velocity of light c, corresponding to the velocity v: dt = dt 0 → t = t 0, (3) 2 ± v c = c0 1+ 2 . (4) c0 8 1 We shall name the value c0 =3· 10 m/sec the speed of light, the value c – the velocity of light . In accordance with the hypothesis of homogeneity and isotropy of space-time, the velocity v of a free particle does not depend on t and x. The velocity of light (4) is a constant in this case. Let dx0 = cdt and t t 2 0 ± v x = cdτ = c0 1+ 2 dτ (5) 0 0 c0 be the “time” x0 when v = const also. Keeping this in mind, we rewrite the expression (1) in the form 2 2 2 2 ds2 = dx0 − dx1 − dx2 − dx3 , (6) where x1,2,3 =(x, y, z). As is known, the metric (6) describes the flat homogeneous Minkowski 4 space-time M with gik =diag(+, −, −, −), i, k =0, 1, 2, 3. Under the condition (3), infinites- imal space-time transformations, retaining invariance of the form (6), are accompanied by the transformation of the velocity of light [17]: i i k 2 dx = L kdx ,c= c(1 − β · u)/ 1 − β ,i,k=0, 1, 2, 3. (7) i Here L k is the matrix of Lorentz group with β = V /c = const [11], u = v/c. Corresponding homogeneous integral transformations in the case of the one-parametric Lorentz group L1 are x0 − βx1 x1 − βx0 1 − βu1 x0 = ,x1 = ,x2 = x2,x3 = x3,c = c , (8) 1 − β2 1 − β2 1 − β2 1 1 0 with β =(V/c,0, 0), u = dx /dx = vx/c. For inertial motions in the space-time (1) we have 2 x − Vt 1 − Vvx/c t = t, x = ,y = y, z = z, c = c , 1 − V 2/c2 1 − V 2/c2 where we take into account vx = x/t. The transformations (8) are induced by the operator 1 X = x1∂0 − x0∂1 − u c∂c which is the sum of Lorentz group L1 generator J01 = x1∂0 − x0∂1 and the generator D = c∂c of scale transformations group 1 of the velocity of light c = γc.We can say that these generators act in 5-space M 4 × V 1 where V 1 is a subspace of the velocities of light, and that the transformations (8) belong to the group of direct product L1 ×1. The generators J01 and D and transformations (8) are respectively the symmetry operators and 1In the form of c = c(1 − β2)1/2 expression (4) was obtained by Abraham in the model of Aether [11]. On the Possibility of Faster-Than-Light Motions 837 symmetry transformations for the equation of the zero cone s2 = 0 in the 5-space V 5 = M 4 ×V 1 where |c| < ∞ includes the subset c0 < |c| < ∞: 2 0 2 1 2 2 2 3 2 2 2 s = x − x − x − x =0,J01 s =0,Ds =0, [J01,D]=0. The relationships between the partial derivatives of the variables (t, x, y, z)and(x0,x1,x2,x3) are as follows: ∂ ∂x0 ∂ ∂xα ∂ ∂ = + = c , ∂t ∂t ∂x0 ∂t ∂xα ∂x0 α ∂ ∂x0 ∂ ∂xα ∂ ∂c ∂ ∂ = 0 + α = dτ 0 + 1 ,α=1, 2, 3. (9) ∂x ∂x ∂x α ∂x ∂x ∂x ∂x ∂x The expressions for ∂/∂y and ∂/∂z are analogous to the expression ∂/∂x. Further we restrict ourselves by the case of positive values of the velocities of light and by studying a variant of the theory in which the velocity of light in the range of interactions may only depend on the time “t”, i.e. c = c(t) ↔ c = c(x0). The relationship between x0 and t may be deduced from the solution of equation (5). Then ∇c x0 =0,c= c x0 ,u2 = u2 x0 ↔∇c(t)=0,c= c(t),v2 = v2(t). (10) Let us note some features of motions in this case. 1. As in SR, the parameter β = V/c in the present work is in the range 0 ≤ β<1. 2. As in SR, the value dx0 is the exact differential in view of the condition ∇c =0. 3. As distinct from SR, the “time” x0 = ct in the present work is a function of the time t only for the case of a free particle. In the range of interaction the velocity of light may depend on time t, and the value x0 becomes the functional (5) of the function c(t). 4. The parameter β = V/cof the transformations (8) may be constant not only at the constant velocity of light, but also with c = c(t). Indeed we may accept that 0 ≤ β = V (t)/c0(1 + 2 2 1/2 v (t)/c0 ) =const< 1, which is not in contradiction with V = V (t),c= c(t). This property i permits one to use the matrix L k from (7) for constructing Lorentz invariants in the range of interaction where c = c(t). 5. The condition (10) is invariant on the trajectory of a particle because t = t in this case. Replacing space-time variables in the expression ∇c =0(c = γc) we find that the condition ∇c = 0 follows from ∇c =0if∇γ =0→ (β − u1)∂β/∂xα +(β3 − β)∂u1/∂xα =0,α =1, 2, 3. The system contains the solution β = const, u1 = u1(x0) in agreement with (10) and item 4. Keeping this in mind, let us construct in the space (6) a theory like SR, reflect it on the space-time (1) by means of the formulas (9), (10) and consider the main properties of the theory. Following [18], we may construct the integral of action in the form: e i 1 ik 4 S = Sm + Smf + Sf = −mc0 ds − Aidx − FikF d x c0 16πc0 2 e 0 1 2 2 3 0 = −mc0 1 − u + (A · u − φ) dx − (E − H )d xdx c0 8πc0 1 i 4 1 ik 4 = −mc0 ds − Aij d x − FikF d x. (11) c0 16πc0 0 2 1/2 0 Here in accordance with [18] Sm = −mc0 ds = −mc0 (c0/c)dx = −mc0 (1 − u ) dx is ik 4 the action for a free particle; Sf = −(1/16 πc0) FikF d x is the action for free electromagnetic i i 4 field, Smf = −(e/c0) Aidx = −(1/c0) Aij d x is the action corresponding to the interaction between the charge e of a particle and electromagnetic field; Ai =(φ, A) is the 4-potential; 838 G.
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