UM-P-91/35 • "A New Formulation of the Theory of Tachyons. Part II: Tachyon Electrodynamics." ROSS L. DAWE* and KENNETH C. HINES School of Physics, University of Melbourne, Parkville, Victoria 3052, Australia. * Now at Maritime Systems Division, WSRL, P.O. Box 1700, Salisbury, South Australia 5108, Australia. PACS. Numbers: 03.30.+p Special Relativity 03.50.De Maxwell Theory 14.80.Pb Hypothetical Particles (Tachyons) ABSTRACT: A new formulation of the theory of tachyons using the same two postulates as in Special Relativity is applied to electrodynamics. Use is made of a "switching principle" to show how tachyons automatically obey the law of conservation of electric charge in any inertial reference frame, even though the observed electric charge is not an invariant for tachyons. Tachyonic transformations of electromagnetic fields E, B, D, H, P and M arc rigorously derived from Maxwell's equations and arc shown to be the same as for bradyonic transformations. Tachyonic transformations of current and charge densities and scalar and vector potentials arc also derived and discussed. The electromagnetic field produced by a charged tachyon takes the form of a "Mach cone", inside which the electromagnetic field is real and detectable, while outside the cone the field generated by the tachyon is imaginary and undetectable. Tachyons also exhibit a Doppler effect: a blueshift for approach and a redshift for recession, just as for bradyons. The transverse Dopplcr effect is a redshift for c2 < u2 < 2c2 but is a blucshift for 2c2 < u2. Tachyons also cause an "optic boom" effect when the observer contacts the edge of the Mach cone, while inside the cone the observer experiences the "two source effect" whereby the tachyon appears to be in two separate and mutually receding places. This means that electromagnetic fields and potentials generated by tachyons arc given by a superposition of an "earlier" and a "later" source. Further examples include calculations of the magnetic dipolc moment of a tachyonic current loop and of the speed of light in a tachyonic dielectric. Constitutive equations for a tachyonic dielectric arc also given. The Lagrangian and Hamiltonian for charged tachyons arc discussed, as well as generic tachyonic trai sformations. 3 1: INTRODUCTION. This paper is the sequel to an earlier paper by Dawc and Hincs^1), hereafter referred to as Paper I, which detailed a new formulation of the theory of tachyons and applied it to tachyon mechanics. The aim of the present paper is to expand this formulation to cover the subject of electrodynamics for tachyons. Recami writes in his Review II that(2); "Within the classical theory of tachyons, it would be important to evaluate how charged tachyons would clcctromagnctically interact with ordinary matter: for instance, with an electron. The calculations can be made on the basis of the generalized Maxwell equations, cither in Corbcn's form or in Mignani and Rccami's . .". The investigation in this paper is intended to determine just how charged tachyons behave and interact with bradyonic matter. It is assumed that tachyons carry ordinary electric charges which must be conserved, and so they shou'd be able to generate real electromagnetic fields and potentials as described by Maxwell's equations. The formulation presented in this paper will follow the ideas set out in Paper I, which were themselves based upon work by Bilaniuk and Sudarshan(3), Corbcn^-5-6' and RccamK7). The idea is to build up the formulation using only minimal extensions to the theory of Special Relativity, and this is achieved simply by using the same two postulates for tachyons and bradyons. Postulate 1: The laws of physics arc the same in all incrtial systems. Postulate 2: The speed of light in free space has the same value c in all incrtial systems. The term "incrtial system" now refers to any system travelling at a constant velocity with respect to an incrtial observer, irrespective of whether the system is travelling slower than or faster than the speed of light. The laws of physics that arc treated as being the same in all incrtial frames arc the same as those in Special Relativity, namely the conservation laws of energy, momentum and electric charge and Maxwell's equations in free space. The conservation laws have all been discussed in Paper I, although a far more rigorous 4 •j derivation of the conservation of electric charge for tachyons will be given in section S. Some terms will be in common usage throughout this paper, so they will be defined here. "Special Relativity" refers to all currently accepted physics for particles which travel slower than the speed of light relative to the observer. These particles will henceforth be called "bradyons". A "tachyon" is defined to be a particle which is travelling at a speed greater than the speed of light relative to the observer. "Extended Relativity" is the theoretical framework which describes the motion and interactions of tachyons. A "bradyonic observer" travels at a speed less than c, while a "tachyonic observer" travels at a speed greater than c. It is important to note that the second postulate means that electromagnetic effects travel at the same speed in free space, regardless of whether the effects were generated by a charged tachyon or a charged bradyon. Just as in Paper I, it is assumed throughout this work that there arc no large gravitating masses present which would deflect the paths of photons. This paper will not consider any extension of General Relativity for tachyons, nor will it consider tachyonic quantum mechanics. The Euclidean metric (+1, +1, +1, +1) will be used throughout this paper, with the space time coordinates being given by x, = x, x2 = y, x3 = z and x4 = ict. The conventions that Latin indices i, j, ... run from 1 to 3 and Greek indices X, \i, v, ... run from 1 to 4 will also be used. The position four-vector is written as Xx = (x,t x2, x3, x4) = (x, ict). Note that the metric (+1, +1, +1, +1) applies to both bradyonic and tachyonic inertial reference frames, as the metric is independent of the observer's relative motion and is the same regardless of whether the observer is dealing with bradyons or tachyons. Furthermore, there is no distinction made in the following work between covariant and contravariant quantities. Electrodynamics will be shown in this paper to be almost the same for tachyons and bradyons, with only minor modifications being required in the tachyonic case. Great care is taken in the derivation of the transformations of the field vectors E and B (section 3) D and H (section 14) and P and M (section 15). Section 4 contains derivations of the tachyonic transformations of charge and current densities. A rigorous derivation of the conservation of electric charge between all types of incrtial reference frames appears in section 5, while section 6 discusses the electric and magnetic fields of an electrically charged tachyon in great detail. It is also shown using an example and diagrams how the "two source effect" and the "optic boom" arise when dealing with tachyons: both these effects have already been theoretically demonstrated by Rccami(8.9,10) Section 7 consists of a discussion of tachyonic optics, including the Dopplcr effect and aberration. In section 8 the electromagnetic scalar and vector potentials arc discussed for tachyonic sources. Sections 9 and 10 deal with Lagrange's equations and Hamilton's equations for charged tachyons respectively. There is a discussion of generalized four-vector transformations in section 11, followed by a discussion of electromagnetic four-tensors in section 12. Section 13 contains a discussion of why tachyons can be considered to be localized particles, and section 14 points out that switched tachyons arc not antitachyons. Two examples of how tachyonic media behave arc also given in this paper: the electric dipolc moment of a tachyonic current loop is the subject of section 17, while the velocity of light in a tachyonic dielectric is investigated in section 20. Electromagnetic invarianccs for tachyonic speeds and a compact form for Maxwcl">'s equations arc discussed in section i8, while section 19 gives the constitutive equations which apply for media travelling at tachyonic velocities. 2: BACKGROUND FROM PAPER I. The two postulates given above lead to the l.orcntz transformations for u2 < c2: 2 x' = yu(x-ut) , y'= y , z* = z, f = yu(t - ux/c ) . (2.1) It was shown in Paper I that these same postulates also lead to the supcrluminal Lorcntz transformations (S.L.T.s) for u2 > c2 : 2 x' - iyu(x - ut) , y' = iy , /.' = iz , t' = iyu(t - ux/c ) . (2.2) In both cases the y-(ic\ot is defined as Y = J-— • (2-3) 0 ^f Here u is the relative speed along the common x.x' axes of an inertial frame Z' with respect to an incrtial reference frame Z- The inverse Lorentz transformations arc 2 x = YU(X' + uf) , y = y* , z = z* . t = Yu(f + uxVc ) , (2.4) and the inverse S.L.T.s are 2 x s - IYU(X' + uf) , y = - iy' , z = - iz' , t = - iyu(f + ux'/c ) (2.5) These transformations lead to the rclativistic and tachyonic transformations of numerous quantities such as velocity, energy, momentum, force and acceleration, as shown in Paper I. The velocity transformations are the same for u2 < c2 and u2 > c2: v . = —s , v . = * . v . = - . (2.6) 1 2 l " C 1»{ • If) H - C2 J The inverse velocity transformations also apply for - <» < u < °°: v + u v . v v, = —» , vv = * , v7 = * . (2.7) 1 + c2 ^u^1 + irj y\- + -^r) Note that transforming between two tachyonic frames is equivalent to transforming between two bradyonic frames, as the relative speed between two tachyonic frames is less than c.