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.
The appearance of factors of i has resulted in a complex spacclimc for tachyons instead of the real spacctimc for bradyons. When transforming to tachyonic frames, the longitudinal space axis x' is real, since y is imaginary for u2 > c2, while the transverse space axes y' and z' arc both imaginary. The time axis ict' is imaginary for tachyonic frames, just as it is for bradyonic frames. Throughout Paper I it was seen that for tachyonic transformations all longitudinal variables arc real, while all transverse variables are imaginary. Examples of this include velocities, momenta, forces. electromagnetic fields, potentials and current densities. However, it must be pointed out that while the tachyonic y' and z axes are imaginary for us as bradyonic observers, tachyonic observers will regard their y' and z' axes as being real. Conversely, the y and z axes are real for us, but arc imaginary for tachyonic observers.
During the derivation of expressions describing various tachyonic effects, it will at times be neccsary to remove the imaginary part of Y, or to combine ryl with factors of i. In order to remove the possible degeneracy of signs which this entails, an "i-y convention" will be used throughout this work for u2 > c2: for an imaginary square root the factor of +i is taken outside and the square root itself becomes real. For example.
V^-«VJF^\ .(2.8) so that Y = , = ' . (2.9) •u V^f V^T
Using this convention means that the explicit form of the S.L.T.s in the present formulation is
u x x - ut , . , , cz x'= —==.. y' = iy , z' = iz . t'= , (2.10)
with the inverse S.L.T.s being given by
ux' + 2 x' -i- ut' ., ., * c nm x = - —== , y = - ty* , z = -iz', t = - . — (2.11) #~^ $
In Paper 1 it was shown that there are incrtial bradyonic reference frames in which a tachyon appeared to tra'el in the opposite direction and to carry an apparent charge which is the opposite to its rest charge. An example used to illustrate these and other points was provided by the exchange of a charged tachyon T between two bradyonic objects X and Y: sec figure 1. In frame X, using axes (x, ict), the tachyon is unswitchcd and travels from X to Y. However in frame Z', using axes (x\ ict'), the lachyon is switched and 8 appears to travel from Y to X. In order for the tachyon to conserve energy, momentum and electric charge in both frames X and X'. a "switching principle" was developed which is similar to the "Reinterprctation Principle" expounded by Bilaniuk and Sudarshan(^) and the "Stuckelberg-Feynman switching procedure" expounded by
Recami^l). An unswitched tachyon T+ always carries positive energy. A switched tachyon T. appears to travel in the opposite direction
(again sec figure 1) and appears to carry the opposite charge to T+. A more elaborate description of switched and unswitched tachyons can be found in Paper I. It is important to note that the negative
root of yu is used for switched tachyons and the positive root of yu is used for unswitched tachyons. Written explicitly, this "y-rulc" is
Yu = + r-1 (2.12a)
if the particle appears to that observer to be an unswitched tachyon, and
Y = - , l (2.12b) u V^f if the particle appears to that observer to be a switched tachyon. Note that when a particle is a bradyon relative to the observer then
there is no possibility of switching, and so for brad yon s yu is always given by eq. (2.12a). Here u is the relative speed of the final frame X' relative to the initial frame X. If v^ is the speed of the particle in the initial frame X, then the particle will appear to observer X' to be switched if
2 (i) c > u > c /vx for vx > c a'id lul < c, or
(ii) c < u < c2/v„ for v„ < c and lul > c.
The velocity transformations automatically show whether the tachyon is switched or unswitched.
Combining the y-rulc with the i-y convention gives
1 y = i.sign(yu)lyl . (2.13) u i sign(y ).i u 4 7 - 1 9
where sign(yu) = + 1 (2.14a) for unswitched tachyons, and
s«gn(Yn) = - 1 (2.14b) for switched tachyons. The y-rule determines what sign of the square root itself is used, even if the square root is real, whereas the i-y convention means that a square root which is imaginary has a factor of +i extracted to make the remainder of the square root real.
3: TRANSFORMING THE ELECTROMAGNETIC FIELDS E AND B.
It has been assumed in this theory that tachyons arc able to carry an electric charge, and so they should be able to generate an electromagnetic field which is real and observable. Therefore it is important that clcctromagnctism for tachyons be treated in great detail, as this is a possible avenue to their detection (if they exist) in terrestrial laboratories. The treatment of this subject commences with a derivation of the transformations of the components of the electromagnetic field between bradyonic and tachyonic frames.
In bradyonic frame X Maxwell's equations arc, in S. I. units,
V, B = 0 , (3.1)
•JO V x E = - ^ , (3.2)
V. E = f- , (3.3) eo '
V x B = *{* + *o |) • (3-4)
The first postulate of Hxtcndcd Relativity is that: "The laws of physics arc the same in all inertia! systems." For this to be true an incrtial tachyonic observer in tachyonic frame X' must be able to write Maxwell's equations as
V. B'= 0 , (3.5) 10
V'xE^-IJr. (3.6)
V. E' = fr , (3.7)
3 Vx B' = n'o(j' + e'0 ^ . (3.8)
Here e' and \i' are the permittivity and permeability constants in
tachyonic frame X\ while eo and u,o are the corresponding constants in bradyonic frame £. Since X' has two real spacctime axes and two imaginary spacctimc axes relative to us as bradyonic observers, then the permeability and permittivity constants used by frames X and I' cannot immediately be equated. The second postulate of E.R. states that: "The speed of light in free space has the same value c in all incrtial systems." Therefore the permittivity and permeability constants in each frame are related by
\i't' = \ = u e . (3.9)
In his formulation Rccami uses a nonzero right hand side in equation (3.1) and adds a current term to the right hand side of cq. (3.2) in order to simulate magnetic monopolc-likc behaviour for a charged tachyon(12,13) As Rccami points out, such a term only applies if one is only considering subluminal reference frames. In the present case the second frame X' is tachyonic, not bradyonic, and therefore Rccami's extra terms arc inappropriate for the present derivation of the electromagnetic field transformations.
In order to relate these two sets of Maxwell's equations to each other, the connection between the partial derivatives in I with those in X' must first be known. Using the chain rule gives, for example,
JL_MJ_ 3y_;_3_ 3zl_3_ Md_ 3x " 9x 3x' + 3y 9y' + 9z 3z' + 3t 3t* '
Applying this method to the tachyonic transformations in cq. (2.2) produces
9 1Y 3x " "^x' ' c2 3fJ ' dy ~ '3y' ' 11
1 _ .JL Ii_.fl. A.\ (3.10) 3z " ldz' ' 3t ~ Ir»^t' " ndx'j •
The inverse transformations arc
JL _ • fl JL 1\ JL _ • iL 3x' _ " xy»\dx + c2 9tJ ' 3y' " " 'dy _3_ _ . __ l__.fi 1_\ l + u (3.11) 3z* " " dz * df " ' '^St 3xJ •
The corresponding S.R. transformations of the partial derivatives omit the factor of i in cq. (3.10) and the factor of -i in cq. (3.11).
Using eq. (3.10) in the y-componcnt of (3.2) and cancelling the factor of i produces
aV " Hal7" c^ al7;= Hu-3x'' af J * and comparing this with the y'-component of cq. (3.6) gives
2 E,- = EX, E2. = Yu(Ez + uBy) , By.= Yu(By + uEz/c ) .
Similarly, using cq. (3.10) in the z-componcnt of (3.2) and again cancelling the i's leads to
H3*' " c1 at* J " a? = Hu'3x;~ dFj ' and comparing this with the z'-component of cq. (3.6) yields
E E E ,= x . y = Yu(Ey " uB.) , B,= yu(Bz - uEy/c*) .
By using eq. (3.11) and the newly derived transformations for B , and
Bz, in cq. (3.5) it can be shown that
3B,. 3Bv 3B7 u /aBj. 3E7 3E\
+ + T + "aT ay "aT " "c l"aT aV" 3zJ-
Substituting the x-componcnt of cq. (3.2) into this expression and comparing with cq. (3.1) shows that Bx, = Bx as expected. Hence for a boost along the common x,x' axes the electromagnetic field components transform between bradyonic and tachyonic frames as follows: 12
E,- = Ex . Er = Yu(Ey - uB2) , E, = YU(E, + uBy) , (3.12)
2 2 Bx. = Bx. By.= Yu(By + uEz/c ) . B^Y^-UE/C ) (3.13)
The inverse transformations are
E, = Ex. . Ey = Yu(Ey. + uB2.) . Ez = Yu(Er. - uBy.) , (3.14)
B B B 2 x = i • y = Yu(By. - uEr./C2) , Bz = Yn(Bz. + uEy./c ) (3.15)
These arc exactly the same transformations as those used in Special Relativity for transforming between two bradyonic frames. They also apply to transformations between two tachyonic frames, so that equations (3.12), (3.13), (3.14) and (3.15) arc valid for •«xu<«>. Note that the correct invarianccs for E and B hold in all frames:
E2 - c2B2 = E'2 - c2B'2 and E.B = E'.B' . (3.16)
This result is appropriate since the electromagnetic waves travel at the same speed in free space, regardless of whether the observer's reference frame is bradyonic or tachyonic.
The transformations of E and B given by equations (3.14) and (3.15) are the same as those given by Corbcn(5) when allowance is made for the conversion between Gaussian and S.I. units. The transformations of E' and B' given by equations (3.12) and (3.13) arc
different to Recami's version^2,14,15,16)t jn which the pairs of
components (Ey, Bz) and (E^, By) transform like the spacctimc coordinates x and t, while the transformations of E„ and B, behave
like the transformation of y and z such that Ex = \EX. and Bx = iB,-. The different form of the electromagnetic field transformations given by Rccami arises from his approach to tachyons using symmetrized Maxwell's equations.
In vector form the transformations (3.12) and (3.13) arc
E'„ = (E + u x B)„ , E'x = yu(E + u x B)A , (3.17)
2 2 B*„ = (B-uxE/c ),, , B^ = YU(B - u x E/c )x , (3.18) where the subscripts II and 1 refer to components parallel and perpendicular to the boost u respectively. 13
The Lorentz force on a moving electric charge q in frame X is
F = q(E + vxB) . (3.19)
In Paper I it was shown that the transformations of velocity and
force apply for - » < u < ~ and not just for the relativistic range - c < u < c. As the transformations of E and B have also been shown to be the same for both u2 < c2 and u2 > c2, then the Lorentz force in frame X' must be
F' = q'(E' + v'x B1) , (3.20) irrespective of whether X' is a bradyonic or tachyonic frame.
4: CHARGE AND CURRENT DENSITIES.
Conservation of electric charge in tachyonic frame X' is expressed by the equation of continuity:
djx. 3jy. djz. 3p' 37 + V + 37 + £ = ° • <41> where j' is the current density and p' is the charge density. It is assumed that there are no sources or sinks in this volume clement. Applying the partial derivative transformations in cq. (3.11) to cq. (4.1) and multiplying through by - 1 leads to
f d \ dj • 3j • (-&( uj ,\\ 45*.-+ up') J+ ^+ lt+ HW+ -£))=0 (4-2) Conservation of electric charge in bradyonic frame X is expressed by
3j 3i 3j 9p 3x" + ^ + ^+3T=0' <43> and comparing this with cq. (4.2) shows that
+ U 2 Jx = fyO*- P') • Jy = Uy. . \z = Uz. . P = iYu(p' + ujx./c ) . (4.4)
Therefore the transformations of current and charge densities from bradyonic frame X to tachyonic frame X' arc given by
Jx- = ' 'Y«
2 p' = - iYu(p - ujx/c ) . (4.5)
Hence for u2 > c2 the quantities j' and p' transform as a spacclikc four-vector (j\ >cp'):
c2p'2 - & - V - jj = - c2p2 + j2 + jj + k . (4.6)
The corresponding transformations of current and charge densities in S.R. arc given by
U 2 h- = YA " P) • Jy = Jy • h- = J, • P' = VU(P - "Jx/c ) . (4.7) with the inverse transformations in S.R. being
+ U + u Jc 4 8 J, = TU
The tachyonic transformations of current density and charge density have the opposite sign to the spacctimc transformations given by cq. (2.2). There arc in fact three types of tachyonic transformations: those with +i when tr ^forming from X to £', those y> th -i when transforming from £ to £' and those transformations which apply for - «> < u <«». These types will be discussed and examples of each type given in section 11, where generic supcrluminal transformations will be investigated.
The transformations of j and p derived here arc not equivalent to those of Rccami, due to the different treatment of Maxwell's equations in the two formulations. Rccami has added terms in j and p to Maxwell's equations in order to symmetrize them, which immediately leads to the idea that electrically charged tachyons appear to a bradyonic observer to behave as magnetic monopolcs. The apparent field generated by a charged tachyon in the present formulation will be discussed in detail in scc'Jon 6.
Now that the relations between the current and charge densities in X and X' arc known, the second pair of Maxwell's equations can be discussed as a check of this formulation's internal consistency. Using the partial derivative transformations (3.10) and the current and charge density transformations (4.4) in the y-componcnt of cq. (3.4) and then cancelling factors of i produces 15
dB, m u_^\_ . U^L *ILL\ dz' ' y«{dx' " c2 dt' J " ^y +c2l9t* " "• 3x* ) •
Comparing this with the y'-component of eq. (3.8) gives
B B B 2 x- = x • * = Yu(Bz - uEy/c ) . Ey. = Yu(Ey - uBz)
and also u.'o = u.c. (4.9a)
Substituting this into (3.9) shows that e'c = eo. (4.9b)
Therefore the fundamental constants u and E„ arc the same for both bradyons and tachyons. This agrees with the statement in section 1 that the same metric applies to both u2 < c2 and u2 > c2 : the fundamental properties of space time are the same for both bradyonic and tachyonic observers. Further substitutions can be used to confirm the remaining transformations of the components of E and B. Thus the second pair of Maxwell's equations yield the same transformations of E and B that were derived from the first pair, even though the tachyonic transformations of current and charge densities contain imaginary factors.
It must be pointed out that there is no choice as to the signs of the transformations, or even where the factors of i appear. For example, in deriving cq. (4.2) the minus signs in front of every term could have been retained instead of cancelling them, or the factors of i which appeared in every term could have been cancelled out. However, a high degree of internal consistency is required in the theory, so that only the choice made above allowed the second pair of Maxwell's equations to give the same transformations as the first pair, and at the same time also give the same permeability and permittivity constants for tachyons and bradyons.
5: CONSERVATION OF ELECTRIC CHARGE.
(i) Transformation of a volume clement.
A derivation of the transformation of a cubic volume clement between various lachyonic and bradyonic frames will now be given, as it is a necessary component of the proof that conservation of 16 electric charge applies to tachyons. The derivation given here is adapted from the relativistic case given by Lorrain and Corson^17).
Consider an element of volume which has speed v' in frame X' and speed v in frame X, where frame X' moves with speed u along the common x,x' axes relative to frame X. When at rest in a third
frame X0 the volume element is a cube of edge length 10, so that its 3
rest volume (i.e. proper volume) is Vo = L, . In Paper I it was shown
that a rod moving along the common x,xo axes appears to undergo Lorcntz-Fitzgcrald contraction for 0 < v2 < c2 and c2 < v2 < 2c2, while the rod appears to be dilated, or stretched, for 2c2 < v2. The apparent length 1 in frame X along the x-axis is -W 1 for v2 < c2 , .(5.1a) •=».!V^¥ for v2 > c2 . .(5.1b)
There is no length contraction or dilation in a transverse direction, so that the apparent length of the rod in frame X in the transverse direction is simply 10 for bradyons and tachyons alike. Hence the apparent volume of the clement as measured in frame X is
V= iyjl- £ = Vy/l -^ for v2 - u = V. V 1 - for v2 > c2 .(5.2b) The modulus signs are necessary for v2 > c2 as volume is always positive and real. In frame X' the cube again appears to be either contracted or dilated along the common x',x,xo axes, depending upon the relative speed. Reasoning as above, the apparent volume of the clement in frame X' is V V = „V for v'2 < c2 , .(5.3a) V* = V tfl for v'2 > c2 . ..(5.3b) 17 To find the relation between the apparent volumes V and V, the transformation between y-factors must be used. From eq. (9.5) in Paper I it can be seen that for all speeds V1 -cT = u^r— •(5-4) 1 + —T Substituting cq. (5.4) into (5.2) and then using eq. (5.3) gives the volume transformation as 7 V = v^ ? for u2 < c2 and v'2 < c2. .(5.5a) UV. 1 + u2 " c2 V = Y V. for either u2 > c2 or v'2 > c: .(5.5b) 1 + 2 c The corresponding inverse transformations arc ^ V' = for u2 < c2 and v2 < c2, .(5.6a) UV. I - V=V V^ for cither u2 > c2 or v2 > c .(5.6b) uv. 1 - The cases given by equations (5.5a) and (5.6a) are the standard S.R. version in which all three frames Z , X and X' arc bradyonic, so no modulus signs arc necessary. When one or more of Ic, I or I' arc tachyonic frames, then the requirement that volume is always positive and real causes the appearance of the modulus signs in equations (5.5b) and (5.6b). Hi) Derivation of Charge Conservation. 18 Consider the exchange of a tachyon T between two bradyonic objects X and Y, as shown earlier in figure 1. Bradyonic objects X; and Yf have charge + 4Q, while Xf and Y4 each have charge + 3Q. Therefore in bradyonic frame X the unswitched tachyon T+ carries electric charge + Q from X to Y. However, in bradyonic frame X' the tachyon is switched and appears to travel from Y to X, so that conservation of electric charge means that the switched tachyon T. appears to X' to carry electric charge - Q from Y to X. It will now be shown that the 7-rulc automatically gives conservation of electric charge in all incrtial reference frames, even for switched tachyons. Let Vo be the volume of a small element of charge as measured in an incrtial frame X0, relative to which the charge is instantaneously at rest. The total charge within the clement is equal to p0V.0, where po is the proper density of proper charge, and the current density is j0 = 0. Now assume that there is a second reference frame X, travelling with speed v = v^ along the common x,x0 axes relative to X0. Using the transformations of charge density, given by eq. (4.8) for v2 < c2 and cq. (4.4) for v2 > c2, gives the charge density in frame X as 2 2 2 P = YV(P0 + vj0/c ) = YVP0 for v < c , (5.7a) + v /c2 = i- for v2 > c2 5 7b p = 'YV(P0 j0 ) yvp0 - < - ) It is important to note that the final observer, in this case observer X, considers himself to be using a bradyonic reference frame. Therefore the transformation for a tachyonic to a bradyonic frame was used for v2 > c2. The apparent volume V of the charge clement in frame X is given by cq. (5.2), so that combining equations (5.7) and (5.2) gives the total electric charge within the volume clement as measured by X as I v^~ V 2 2 .(5.8a) pV = YvPo 0 \ 1 - ^ = poVo for v < c , n V for 2 2 V V 'g (Yv)Po o v >c , (5.8b) P = 'YvP0 o where use has been made of the \-y convention given in section 2. Hence for v2 < c2 the familiar rcsuli of S.R. has been obtained, which is that electric charge is an invariant when the rest frame and the 19 final frame are both bradyonic reference frames. For v2 > c2 the apparent sign of the electric charge depends upon the sign of the root of y, which is positive for unswitched tachyons and negative for switched tachyons. Thus the y-rulc correctly gives the apparent sign of the charge: unswitched tachyons appear to have the same charge as they have in their own rest frame, while switched tachyons appear to have the opposite charge to what they have in their own rest frame. Now suppose there :s a third inertia! reference frame X' moving along the common x',x,xo axes wilh speed v' relative io frame X0, and with speed u relative to frame X. The speeds v', u and v arc related by the velocity transformation v - u v = . (5.9) U V 1 " c2 The apparent volume of the charge clement as measured in frame X is given by cq. (5.5). The transformation of charge density for u2 < c2 was shown to be 2 2 given by cq. (4.8) and for u > c by eq. (4.4), where j' = jx>. The current density j in frame X is defined to be j' = p'v' (5.10) for both bradyons and tachyons. Now consider the possible combinations of frames X0, X and X'- Frame X is a bradyonic frame, as the final observer always considers his own reference frame to be bradyonic, so therefore there arc four possible combinations due to frames X and X each being cither bradyonic or tachyonic. In every case below the speed of frame X relative to frame X0 is v, while the speed of frame X' is u relative to X and v' relative to X . Case (i): u2 < c2 and v2 < c2. This is the standard S.R. case, in which all three frames X0, X and X arc bradyonic, and has been included purely for comparison purposes with other cases. Combining equations (5.5a) and (4.4a) gives mmmmmmmmmmmmmmmm 20 pV fV^f) = pV -<'•£> 1 + uv V c2 J 2 2 as j' = p'v'. From cq. (5.8) it is known that pV = poV0 for v < c , so that the total electric charge measured in frame Z' is pV = pV = p0Vo. Therefore the electric charge is an invariant when all frames are bradyonic, including the rest frame of the charge. Case fii): u2 < c2 and v2 > c2. In this case frame Z„ is tachyonic, while frames Z and Z arc bradyonic. Combining eq. (5.5b) with (4.8) gives pV fl = ?u(l UV <"•£> 1 + As 1 + uv > 0, even in switched frames, then 2 pV = yp'V V^ sign (T.)PV where use has been made of the Y-mle. Combining thif, with eq. (5.8) shows that the total electric charge measured in frame Z' is p'V = sign(yu)pV = sign(Yu)sign(yv)poV0 . An example of this particular case is the above mentioned example involving an exchanged tachyon as illustrated in figure 1. The rest frame of the tachyon is frame Z0, while frames Z and Z' arc both bradyonic. In this example the tachyon is unswitched in frame Z, so that sign/y \ = +1 and the electric charge measured in frame Z is pV = poVo = + Q. In frame X' the tachyon is switched and so sign/v,,} = -1- Therefore the apparent electric charge in frame Z' is measured to be P'V' = -pV = -poVo = -Q. Case (iii): u2 > c2 and v2 < c2. 21 In this case frames X0 and £ are both tachyonic, while frame Z' is of course bradyonic. Combining equations (S.Sb) and (4.4) gives VHF pv = iYU(p'+^y = iYnpV sig>gn(Tn u)pV , 1 + uv V^¥N where use has been made of the i-Y convention and the y-rulc in the form of eq. (2.13). Note that the relative speed between the two tachyonic frames Z0 and X is less than c, so that pV = p0Vo. Hence the apparent electric charge measured in frame £' is p'V = sign(Yu)pV = sign(Yu)poVo . Case (iv) : u2 > c2 and v2 > c2. In this case both frames Xo and X' are bradyonic, while the intermediate frame X is tachyonic. Combining equations (5.5b) and (4.4) gives V^F pV = = iYupV - i = '?„('P*!HT > uv Vi I + as j=pv' and 1 +~uvy > 0 for both switched and unswitchcd tachyons in this case. From cq. (2.13) it follows that pV = sign^Yu)p'V , and so the apparent electric charge measured in frame X is p'V' = sign(Yu)pV = sign(Yu)sign(Yv)poVo = p0Vo . Note that the particle appears as a bradyon in both frames X0 and X', and since bradyons do not undergo switching, then sign(Yu) = sign/Yy") in this case. If the particle is switched in frame X. il is "switched back" in the transformation from frame X to I'. The discussion of each of these four cases shows that the y-r\t\c, which was originally developed in Paper I as a device to automatically allow tachyons to obey the laws of conservation of energy and momentum, also allows tactions to obey the law of conservation of electric charge in each of the reference frames. This means that the Y-rule is not just a mathematical artifice: it has considerable physical significance. (in) Nonconservation of Total Electric Charge and Particle Number for Tachyons Between Inertial Frames. Consider a collision between two charged tachyons, as represented by a Minkowski diagram in figure 2. Observer 510 u^ing axes (x0, ict0) sees two tachyons, T, and T2, enter the collision and later sees two tachyons, T3 and T4, exit the collision. A second observer 51' using axes (x\ ict') sees three tachyons, T,, T2 and T3, enter the collision but sees only one tachyon, T4, leave the collision. Hence the apparent number of observed tachyons at any given time is not necessarily the same for different observers in a system involving collisions. In the tachyonic frame 51T. using axes (x-r, ictT), all of the particles in figure 2 appear to be bradyons, with T, and T2 colliding to produce T3 and T4. Therefore the number of particles observed before or after the collision is an invariant between observers for whom all of the particles are cither bradyons or unswitched tachyons. Some consequences of this point will be discussed shortly. Now suppose that the incoming particles Tj and T2 each have electric charge +Q, and that both T3 and T4 have charge +Q in their own rest frames. Therefore the total incoming charge in frames 51 f and ZT is + 2Q, which exactly matches the net outgoing charge. Observer 51' sees three tachyons enter the collision, but as T3 is switched it has apparent charge - Q, and so the net incoming charge in this frame is 2Q - Q = + Q. This exactly matches the charge + Q on tachyon T4, which is the only product ol the collision according to observer 51'. Thus the total electric charge is conserved in each of the frames 510, X' and 51T. even though those observers may measure 23 different total charges in the system and measure different numbers of tachyons before and after the collision. As a second example, suppose T, and T4 both have charge +Q, while both T2 and T3 have charge - Q in their own rest frames. Observers Z0 and ZT would measure a total electric charge before and after the collision of + Q - Q = 0. Observer £' would measure an initial charge of Q + Q - Q = + Q as T3 is switched, while the outgoing charge carried by T4 is +Q. Yet again it is obvious that electric charge is conserved within each frame, and that the total charge observed is different for different observers. It must be stressed that this effect is purely an artefact of the observer's relative motion, and docs not mean that some of the electric charge has been "destroyed" simply by transforming from one reference frame X0 to another reference frame £'. Transforming to any reference frame in which all of the particles appear to be either bradyons or unswitched tachyons will "recover" the apparently missing charge. What must be remembered is that the conservation laws and clcctromagnctism all hold true in any given incrtial reference frame, be it a bradyonic or a tachyonic frame. This result agrees with the conclusion of RccamiOS), wno nas shown that in his formulation electric charge is always conserved in each frame, but is no longer an invariant between frames when dealing with tachyons. Examining figure 2 shows that observer X0 sees two tachyons enter the collision and later on sees two tachyons exit the collision. In this frame there arc no switched tachyons, and so the total number of tachyons is conserved. This contrasts with what observer X' sees happen in the collision: E' sees three tachyons enter the collision, but only one tachyon appears to exit. This is because one of the tachyons has undergone switching, so that instead of being a product it is now an incoming particle, and so the apparent tachyonic particle number is not conserved for observer £'. This means that apparent particle number is no longer an invariant when dealing with a collection of colliding tachyons. Since one of the standard techniques in dealing with a gas is to normalize the particle distribution by assuming that the total number of particles is constant, then it is obviously necessary to use a frame in which the apparent number of tachyons is constant. Even though particle 24 number is not conserved in X' it should be remembered that the laws of conservation of energy, momentum and electric charge all hold true in this frame. If instead the system in figure 2 is viewed by tachyonic observer XT using axes (xT,ictT), all of the particles appear to be bradyons. Observer XT sees two particles enter the collision and two particles exit, so particle number is conserved in this frame. This is similar in principle to a bradyonic observer watching a collision between two bradyons, for which particle number is obviously conserved. It is clear that the apparent number of particles before and after the collision is an invariant in reference frames in which the particles are either bradyons or unswitched tachyons. Therefore, it should be possible to carry out the usual normalization of the particle distribution in such frames. In order to be able to investigate tachyonic gases, it is therefore necessary to determine the incrtial reference frame for which all of the tachyons are unswitched, regardless of however many collisions the particles undergo during any given time interval. In analogy with investigations of the classical and rclativistic gases, it can be assumed that the gas is spherically symmetric and has a large volume. This means that the gas looks the same in any direction relative to an observer who is at rest relative to the centre of mass of the gas. Such an observer sees collisions in all directions which only involve unswitched tachyons: this frame corresponds to X0 in figure 2, and so X0 measures an invariant tachyonic particle number. An observer X' moving with speed u < c relative to the center of mass sees some of the tachyons undergo switching. As the speed of X' relative to X0 increases, more and more tachyons appear to be switched, so that X' could not determine the exact number of particles in the system due to the large number of collisions. As the tachyonic gas will have particles with speeds ranging from near c (high energy) to v » c (low energy), the switching condition c > u > c2/v shows that even when u is very small, X' will still sec some switched tachyons. Even so, these few switched tachyons will still be enough to destroy particle number invariancc for X'. 25 Thus it can be seen that a reference frame at rest with respect to the centre of mass of the tachyonic gas is the best frame to use, as there arc no switched tachyons in this frame and tachyonic particle number is constant. Using this frame would also mean that in an investigation of a tachyonic gas it will not be necessary to allow for the possibility that y may have a negative root. This last point is especially important as the tachyonic M?xwcll-Boltzmann equilibrium distribution function has been shown by Dawe ct al.(19) to contain a factor jf exp(-ay). 6: THE ELECTRIC AND MAGNETIC FIELDS OF A TACHYONIC CHARGE. In section 3 it was shown that the transformations of the electric and magnetic fields arc the same in both S.R. and E.R. However, it has also been shown that the apparent sign of a tachyonic electric charge depends upon the reference frame used by the observer. It is therefore instructive to investigate in detail the electromagnetic field produced by a uniformly moving tachyonic point charge. The following discussion is adapted from the rclativistic case given by Rcsnick(20) an(j Rosser(21). Consider a particle of charge +q' at rest in an incrtial frame £'. The field of the charge is purely electric, and so the electric and magnetic fields arc given by K' = £^' B,= 0 - <6-» o where the electric field lines diverge from q' with spherical symmetry. Here r'= Vx'2 + y'2 + z'2 (6.2) is defined to be the distance from the origin O' in frame X' to the point at which the field strength is measured. Note that the field is proportional to 1/r'2, as required by Coulomb's law. Now suppose that frame I' is moving with speed u > c with respect to a bradyonic frame X, so that the charge is tachyonic relative to 26 observer £. Using the tachyonic transformations given as eq. (2.2) in (6.2) gives r x ut 22 2 2 r"' = -"V^ - YnY»<(x -" ut)> - y - z . (6.3) Using equations (2.2) and (6.3) shows that the longitudinal component of the electric field in bradyonic frame £ is s»gn(Y )qiY (x - ut) * • — • u u p _ p _ M *— _ i L 2 2 2 C C 3 22 i " * ~ 4JCE/ " „4«e„ of| - „Y ,U(„X - ...ut)M - ..y2 - „z2 13/V 2 where q' = signf? \q. Making use of the Y-rulc, given by cq. (2.13), leads to q|Yu|(* " "t) E 64a x = Ane—-T-T, ut~ 2 2, ,13/2 2 • < > 0\- ?«(* - ) - y - z 1 Note that the sign change due to switching in both the gamma factor and the electric charge cancels out. In frame £ the transverse electric field components arc K|qy E y = T.(E, + uB,) = —r -^ (6.4b) v 2 2 2 ' 4neo{-YU(X - ut) - y - z } 64c \ = T.(E, - uBy.) = ——r 2 3/2 < > v 7 2 2 2 4ne0{-Yu(x - ut) - y - z } The magnetic field measured in frame X is Bx = Bx. = 0 , (6.5a) uE A uE. ( uE A uE B*=yJ**+^L) = j? • <65c> If the moving charge is considered to be at the origin of X at the instant t = 0, then the electric field is 27 E = 66 ' ' 2 2 2 2 213/2 3 2 • < > 4iteo|-Y«x -y -z } At other times the field will look the same but tra- slated along the x-axis by the distance ut. By defining 6 to be the angle made by r with the x-axis, then x = r.cosG , y2 + z2 = r2sin2G . (6.7) 22 2 m.. . J 2,2 ,2 ,2 2 ,2 /, uu sin 9A' This leads to yux + y + z = yur jl - j— , .(6.8) and substituting this into eq. (6.6) produces qrJY^I sign(Yu)qr^- l) E = T7T = -1/7 • (6.9) where cq. (2.13) has been used to simplify the expression. The equivalent expression in S.R. for u2 < c2 is qr E = s ^^rjz , (6.10) i 2 Nj/Z J,3 u sin G 4rc eor | 1 so that the form of the expressions describing the field in S.R. and E.R. arc very similar. In both cases the field in frame X is still an inverse square one, as its strength is proportional to 1/r2 in any direction. Examining eq. (6.9) shows that when c2 > u2sin28 all the components of E and B are real, while for c2 < u2sin20 all the components of E and B are imaginary. Since I is an incrtial frame used by a bradyonic observer, who can only detect real quantities with his instruments, then the charged tachyon's electromagnetic field is only detectable for > IsinGI. The electromagnetic field is imaginary and therefore undetectable in frame £ for IsinGI > These cases arc represented in figure 3, which is a Mach cone similar to those generated by aircraft travelling at supersonic speeds. The circles in 28 figure 3 represent electromagnetic wavefronts emitted by the charged tachyon as it passes through a numbered sequence of points. Inside the cone the field is real and detectable, whereas for all points outside the cone the wavefronts have not yet arrived and so the tachyon's field is imaginary and undetectable. When Isinei = - the observer in frame £ is at the edge of the u cone and E, B„ and B, are instantaneously infinite. At any later time IsinGI < and the field is real and finite. In analogy with the "sonic boom" produced by aircraft flying faster than the speed of sound, the observer's initial contact with the cone of electromagnetic wavefronts is described as an "optic boom"(9). At the initial contact the observer receives a wavefront from one direction, but at later times the observer receives wavefronts from two separate directions. This "two source effect" is shown in figure 3 as the intersection of any two wavefronts. For example, an observer at A detects the wavefronts emitted by T when it passed through positions 1 and 2, an observer at B detects the wavefronts radiated from posiUons 2 and 3, etc. Therefore any numerical determination of the electromagnetic field must account for the fact that the test charge experiences a superposition of fields generated by the tachyon from two separate positions. As figure 3 clearly shows, there arc two combinations of r and 6, i.e. (r,,6,) and (r2,02), which arc permissible in eq. (6.9). Indeed, the definitions given by eq. (6.7) apply to both pairs (r,, 6,) and (r2, 82). Hence the total electric field for any point inside the Mach cone is actually given by the superposition sign(Y„)q[c2 E = (R1 + R2) .(6.11) 4ne„ where R, = 2 2 R 2 2 .(6.12) u sin 8,\3/2 * 2~ u sin e„\3/2 r 1 rill(.."WL- J 2 For points outside the cone or on the edge of the cone, only one pair of values (r, G) applies and so cq. (6.9) is appropriate in those 29 instances. The "two source effect" has also been discussed by Rccami et al.(8). It is important to note that bradyons do not exhibit either the "two source effect" or an "optic boom" when travelling through a vacuum. These effects are distinctive features of charged tachyons. In Paper I it was shown that u = ± V2c is a "Newtonian limit" for tachyons. When u = the electric field generated by the charged tachyon is "s'gn^Y^q" E = + .(6.13) 3/2 3 3/2 4JIE. 2 2 r,(l-2sin 6,) r2( 1- 2sin 02) If the observer is on the tachyon's line of motion then sinB, and sin82 are zero, and so the expression for the electric field simplifies to si g"(y„)q (r, r,\ E = ..(6.14) 4tl £ U r2 This expression is similar in form to the field in the particle's rest frame, with added complications caused by the possibility of switching and the superposition due to the two source effect. Continuing with the main example, v = u, v = v? = 0 so that the magnetic field can be generalized to / i n 1 v x E g (Yu)q^- ) B = xR + v xR .(6.15) A* y . 2) • where uo 2 . Here again the two source effect causes a e c superposition of fields generated by the chargcJ tachyon. In (he limit as u -» °° the electrically charged tachyon exhibits behaviour similar to that of a magnetic monopolc. Assuming that 9, and 9? arc not close to either 0° or 180°, the electric field in (6.11) is proportional to 1/u for u -> °°. For v = u -» °° the magnetic field in (6.15) becomes approximately 30 /A A _ ' u x r u x r, 3 T 3 .(6.16) 471 l_ + L. 3 3 r|Sin 0, r2sin 82 V / where u is a unit vector in the direction of u. Hence B is not dependent on the magnitude of u in this limit, and so the expression for the magnetic field of the tachyon looks similar to the expression for the field of a magnetic monopolc, with the extra term again arising from the two source effect. Note that B in (6.16) is imaginary so that the "monopolc field" is undetectable. The semivcrtex angle of the Mach cone is given by sinB = c/u, so that as u -» °° the field is only real and detectable inside a narrow wedge. Now consider the same limit u -> <*> but with 6, and 92 being either 0° or 180°, so that the observer is now on the line of motion of the charged tachyon. In the case v = u the magnetic field is negligible due to the cross products, so that the only remaining term is purely electric and the tachyon looks like an electric charge with an effective field strength which diminishes as 1/u for u -> «>. In his work Rccami claims that a tachyon which has an electric charge in its rest frame behaves like a supcrluminal magnetic monopolc('2), and \i has been shown here that for u -> °° there arc some similarities. A quote from Rccami's Review II is appropriate^); "This docs not mean, of course, that a Supcrluminal charge is expected to behave just as an ordinary monopolc, due to the difference in speeds (one sub-, the other Supcr-luminal!)". The differences arc evident if u is finite, as charged tachyons have their own distinct form of behaviour when compared to charged bradyons or bradyonic magnetic monopolcs. v x E Using B = j^~ in the Lorentz force law, eq. (3.19), gives „ ,„ qu x (v x E} ,, ,_s y L F = qE +— f2 , (6.17) so that the second term on the right ham! side is of order uv/c2. For charged bradyons uv < c2 always, but for enarged tachyons uv can be less than, equal to, or greater than c2 depending upon the reference frame. This is similar to the "dual speed" condition discussed in Paper I. 31 The fact that the tachyon appears to a bradyonic observer to outstrip the electromagnetic field it is radiating gives rise to further distinctively tachyonic features and complications. One such complication is of course the fact that in some bradyonic frames the tachyon is switched, and so appears to carry the opposite sign charge and travel in the opposite direction. In order to determine what happens in such cases, the following four diagrams, i.e. figures 4 to 7, represent the electromagnetic field radiated by a charged tachyon as it approaches and recedes from a test charge. Suppose the tachyon has charge +Q, constant speed v = +2c and is approaching a test charge +q which is at rest relative to an observer £. A second observer £' is moving with speed + 3c/4 relative to observer £. Figures 4 through 7 depict schematically the electromagnetic wavefronts emitted by the tachyon as it passes through a numbered sequence of points. The numbered wavefront emitted from each point consists of photons and so is unaffected by whether the tachyon is switched or unswitched. The tachyon is of course continuously radiating the electromagnetic field, which propagates at speed c in all inertia! frames. Figures 4 and 5 apply to frame X, while figures 6 and 7 apply to frame £'. Figures 4 and 5 show a positive energy tachyon T+ radiating an electromagnetic field as it approaches (fig. 4) and recedes (fig. 5) from the test charge, which is at rest in frame Z. From figure 4 it is apparent that wavefronts emitted during approach arc received by the test charge in reverse order to their order of emission, while from figure 5 it can be seen that wavefronts emitted during recession arc received by the test charge in the same order as their order of emission. Note that the tachyon has already gone past before the test charge starts to experience the tachyon's electromagnetic field. Figures 6 and 7 show a switched tachyon T as it approaches (fig. 6) and recedes (fig. 7) from the test charge. In this frame the observer X' is moving with speed +3c/4 relative to the test charge and the negative root of yu is used as the tachyon is switched, in frame £' the lachyon T 'as apparent charge - Q and speed 32 _ v* ' u _ 5c v»' " uv ~ " 2 • Note that in figures 6 and 7 the horizontal scale has been expanded by the amount necessary to counteract Lorcntz contraction. The deflection of the test charge has been ignored in the diagrams for simplicity. From figure 6 it can be seen that wavefronts emitted by T during its approach arc received by the test charge in the same order as their order of emission, while figure 7 shows that wavefronts emitted during recession arc received by the test charge in reverse order to their order of emission. This is the opposite of the ordering for wavefronts emitted by T . Once the test charge has come into contact with the electromagnetic field of the unswitched tachyon T+ the net force can be calculated, but it should be noted that the superposition of a delayed field due to the two source effect requires a vectorial addition of two forces. The same is true when the test charge comes into contact with the electromagnetic field generated by T. In this case it is important to remember that switching has changee the effective sign of E in cq. (6.9) due to the factor of sign^y,,)- When allowance is made for the relative motion between the two reference frames, the test charge itself cannot tell the difference between a tachyon T+ going in one direction and its switched counterpart T which appears to be going in the opposite direction. Figure 8 is a Minkowski diagram representing a system similar to the one depicted in figures 4 to 7. Bradyonic observer X uses axes (x, ict) while the second bradyonic observer X' uses axes (x\ ict'). The test charge is at rest in frame X and so it moves along the ict axis. Each photonic worldlinc is numbered according to the point on the x-axis from which it was emitted. In frame X the test charge receives the photons emitted during the approach of T+ in the order 6, 4, 3, which is in reverse order to their order of emission. According to frame X' this same section of the tachyon's worldlinc shows the switched lachyon T receding from the test charge, which receives the photons in the order 6, 4, 3. In frame X the test charge receives 33 photons from the receding tachyon T+ in the order 9, 10, 12, 13, so that they are received in the same order as that in which they were emitted. In frame Z' this section of the switched tachyon's worldlinc represents the approach of T. Since the photons are received in the order 9, 10, 12, 13, observer X' considers the photons to be received by the test charge in the same order in which they were emitted. Note that figure 8. which is inspired by figure 15 of Recami's Review Il(23)) clearly shows the two source effect experienced by the bradyonic observers. The above results contrast with what the test charge would experience if the charged object is not a tachyon T, but is instead a bradyon B with charge + Q. In this case the electromagnetic wavefronts always arrive at the test charge before B. The two source effect docs not apply for bradyons, because for each point inside any given wavefront there is only one apparent source of the electromagnetic field. Note that the test charge can detect the field radiated by any receding charged object, regardless of whether it is bradyonic or tachyonic, but can only detect the approach of charged bradyons and not charged tachyons. Moreover, wav^fronts emitted by charged bradyons arc always received by the test charge in the same order as their order of emission, so that receiving wavefronts in reverse order only happens if the waves are generated by tachyons. Receiving wavefronts in reverse order is interpreted as a negative frequency^,24^ and so any equation describing the observed frequency as a function of initial frequency and relative speed must account for this. The transformation of frequencies will be discussed further in the next section, which deals with the Dopplcr effect for tachyons. 7: TACHYONIC OPTICS. (a) Doppler Effect: Firsj Derivation. From figures 4 to 8 it can clearly be seen that tachyons exhibit a Dopplcr effect, in which the spacing between wavefronts is compressed in the tachyon's forward direction of motion and dilated in the opposite direction. The following derivation of the tachyonic 34 Dopplcr effect has been adapted from the relativistic case given by Helliwcll(25). Consider a source which is at rest in frame X', as shown in figure 9. The source emits a photon at angle 6' with respect to the x'-axis. In a bradyonic frame X the photon has angle 6 w.r.t. the x-axis. Frame X' has speed u relative to frame X and moves along the common x,x' axes. For photons in frame X' the energy to' is related to the frequency v' by the relation lkl=^=-^, (7.1) c c hvcose and so k = . (7.2) * c Both the energy and momentum of the photon should transform in the same manner as for the energy and momentum of any other particle. Hence the energy transformation is 2 2 0)' = Yu(o> - ukx) for u < c , (7.3) 2 2 or (o* = iyu(o) - ukx) for u > c . (7.4) Substituting appropriate expressions for ©', co and kx into cq. (7.3) for u2 < c2 and rearranging terms gives the rclativistic Dopplcr formula: W--S " = u.cose <75> C For the tachyonic case substituting expressions for (o\ w and kx into the energy transformation gives the tachyonic Dopplcr formula: IV v = V^: rF— = WITT— . (7.6) u.cosO u.cosO c c The corresponding inverse transformations arc: 35 (S.R.) 'V^¥ •(7.7) v = U.COS0' * 1 + 1 (E.R.) v = ~7777~ = ~~~7. ..(7.8) U.COS0' u.cosO' 1 + 1 + Once again the i-y convention has been used for u2 > c2. The expression for the tachyonic Dopplcr effect given here differs from that given by Recami^") by a sign in cq. (7.8), and also differs due to the possibility of switching changing the effective sign of the square root. Note that the tachyonic Dopplcr effect derived here is real and therefore measurable, which shows that the formulation has internal consistency. There are three main cases of interest: (i) 6=0, when the source approaches the observer, (ii) 9 = n, when the source recedes from the observer, and (iii) 9 = ± n/2, when the source moves perpendicular to the observer. Each of these cases can be compared with the diagrammatic representation of figures 4 to 7. In each case v' is the proper frequency of the source and v is the observed frequency. Case (i); 9 = 0 (approach). In this case the relative motion results in the source and the observer directly approaching each other. The S.R. case, obtained from eq. (7.5), is c_ = V (u2 < c2) ..(7.9) 0-UcX'-Uc)J 0- h The corresponding E.R. case, obtained from eq. (7.6), is (»- \n 2 /y_+]Y c 2 2 v = - v >M = - v (u > c ) . .(7.10) *"£••) 36 Examining equations (7.9) and (7.10) shows that Ivl > Iv'l for -«» < u < «>, so that relative approach of source and observer always results in the observed light being blueshiftcd. For an unswitched tachyonic source the positive root of eq. (7.10) is used, which means that the observed frequency is negative. This agrees with figures 4 and 5, which show that light emitted from the unswitched tachyonic source T+ as it approaches the observer is received in reverse order to its order of emission. For a switched tachyonic source the negative root of cq. (7.10) is used, so that the observed frequency is positive. This agrees with figures 6 and 7, which show that light emitted from the switched tachyonic source T as it approaches the observer is received in the same order as its order of emission. Each of these four diagrams clearly shows the wavefronts emitted during approach by either T+ or T as being compressed together, which indicates that they arc blueshifted. In the limit as u -» °° then v -> - sign(yuW, while in the limit as u -> c+ then v -> - sign/y^ x <», and for u -» c then v —> + <». It is important to note that negative frequencies do not necessarily mean the photons have negative energy. The photons themselves always have positive energy. The concept of a negative frequency is useful only for explaining and/or interpreting the apparent reversal of the order of the wavefronts: it has no real significance beyond that. The energy of the observed light is in effect given by the relation E = hlvl for u2 > c2. Case (ii): 6 = n (recession). In this case the relative motion results in the source and observer moving directly apart. Eq. (7.5) gives the S.R. version as (f. uV. ^\\2 / u_\i/2 2 2 V = V = v* . (u < c ) . (7.11) V The corresponding E.R. version is found from cq. (7.6) to be 37 ((* .Vu .\ \n /u_ \n 2 2 v = v (u > c ) . (7.12) v..*V Examining equations (7.11) and (7.12) shows that Ivl < Iv'l for -« < u < °o, and so the observed light is rcdshiftcd regardless of whether the source is bradyonic or tachyonic. For an unswitched tachyonic source the observed frequency is positive. This agrees with figure S, in which the observed wavcfronts emitted during recession arc received in the same order as their order of emission. For a switched tachyonic source the observed frequency of the receding source is negative, which agrees with figure 7. Figures 5 and 7 show that the wavcfronts emitted by T+ or T during recession display increased separation, which indicates that they are redshifted. In the limit as u -» «» then v -» sign^Yu^v', while for the limits u —> c* then v -» 0. Case (nil: Q = ± nil (transverse motion). In this case the relative motion results in the source and the observer moving at right angles to each other. The S.R. case is v-W77ii. .(7.13) while the E.R. case is v = Wi£'! • (7-14) For both u2 < c2 and c2 < u2 < 2c2 it can be seen that Ivl < Iv'l, and so the light appears to be rcdshiftcd. However, for u2 > 2c2 the light is instead blucshifted as Ivl > Iv'l. This "colour change" is another effect of having \y\ = 1 at u2 = 2c2, along with the contraction/dilation effect noted in Paper I. Once again the possibility of switching can change the apparent sign of v, so that v is positive for unswitched sources and negative for a switched tachyonic source. In the limit as u -» <*> uv' then v becomes proportional to sign/^}— . In the limits u -> c1 then v-> 0. Figure 10 is a plot of each of these three cases of the Dopplcr effect for both the rclativistic (u/c < 1) and tachyonic (u/c > 1) speed 3 8 i ranges. Relative approach (8 = 0) always results in a blueshift, relative recession (0 = n) always results in a redshift, while transverse motion is a redshift for u/c < \2 and a blucshift for u/c > The sign in brackets denotes whether the tachyonic source appears to be switched (-) or unswitched (+). This figure compares favourably with Recami's Dopplcr effect graph^27). (b) Dopplcr Effect: Second Derivation. <, There arc several different derivations of the relativistic Dopplcr effect, so there should be just as many derivations of the tachyonic Dopplcr effect. Giving all of these derivations would not be constructive, however, so only one more will be given which shows how tachyonic quantities combine to give the correct result. This particular method is adapted from the rclativistic case given by Muirhcad(28) In frame X' the radiating particle of proper mass m0 is at rest, so that its four-momentum Pv is (0,0, 0, imoc). A photon emitted by this object has four-momentum Kx, = (k\ iw'/c) where co' is the photon's energy. In Special Relativity the scalar product £ Px-Kx. is a Lorcntz invariant such that 4 T P..K.. = - m co' in frame I' (7.15) 4 Ed) and I PX.KX. = p.k - ^ (7.16) in frame X. Here frame X is a bradyonic frame, relative to which frame X' has speed u < c along the common x,x' axes. E is the emitting particle's energy as measured in frame X. Now suppose that frame X' has speed u > c relative to frame X, so that the radiating particle is a tachyon. It has been shown in Paper I that the sign of the square of the spacctimc and energy-momentum four-vectors changes when transforming between frames when u2 > c2. However, it was shown earlier in this paper that electromagnetic transformations, i.e. transformations involving photons, arc the same for both u2 < c2 and u2 > c2. This is a direct consequence of the two 39 postulates given in section 1. Therefore there is no clear precedent in determining the net sign of a product of a momentum four-vector and an electromagnetic four-vector, and so in this case the sign choice on the product of the four-vectors in eq. (7.16) for u2 > c2 at first appears to be arbitrary. However, only one of the two available choices gives Doppler frequencies consistent with figures 4 to 7 and gives the correct sign for the apparent frequency. For the tachyonic case the product of the four-vectors Pv and Kx. is also given by equations (7.15) and (7.16), so that these equations apply for both u2 < c2 and u2 > c2. For photons the usual relations apply: <•* hv ,_ „„, G) = hv and lkl=-=— . ...r..(7.17) \f C Since the source is at rest in X' thcr. its momentum in frame X is 2 2 px = yum.u , py = pz = 0 for u > c . Y m.uhv.cosG Hence p.k = pJklcosG = — , (7.18) where 9 is the angle made by the light ray with the x-axis. Equating the right hand sides of (7.15) and (7.16) gives Yum.uhv.cos0 E(0 -mo(o'= -» - -^ . (7.19) As the source is a tachyon in frame L, its proper mass is given by 2 m, = imo and its energy is E=Yum,c . Substituting these expressions and the ones for a and w' into cq. (7.19) leads to . Ai.cos8 \ 2 2 - v = IYUV 1 , for u > c . This of course can be rearranged using the i-Y convention to give cq. (7.6). This demonstrates how the imaginary factors in various terms combine to cancel out and give a real answer, and also shows that the present formulation is internally consistent. ££j Aberration. 40 Another effect caused by the relative motion of the source and the observer is that of the aberration of light, whereby the apparent direction of propagation of light differs between two different inertia) frames. The following derivation of tachyonic aberration of light is adapted from the relativistic case given by Muirhcad^29). Consider a source at rest in frame £' which emits a photon at an angle 0' to the x'-axis, as shown in figure 9. The photon has momentum k' and energy a' = hv' = Ik'lc. Frame L' moves along the common x,x' axes with speed u relative to a bradyonic frame X. In this frame the photon has momentum k and energy w. The angle of emission in each frame is determined by the relations k k tan9 = r* and tanG" = r^ . (7.20) The transformations of k and (o from £ to X' will have the same form as the corresponding transformations of p and E, so that for u2 >c2: ky = - iky. = - ilk'IsinG' (7.21) U and k, = -iYu(kx. + ^j = - iy^lk'IcosG' + ^ . (7.22) k (sin8'r\/ 1 - ^j Therefore tanG = £* = • (7.23) x cosG' + - c Equation (7.23) describes the aberration of light due to a tachyonic light source. It is identical to the relativistic aberration formula except for the range of u. The inverse transformation is (sinermery i tanG' = , (7.24) cosG u_ c which is also valid for - °° < u < °° . s 2 2 Since yu ' imaginary for u > c , then cq. (7.23) gives an imaginary value for tanG. This imaginary factor is due to the transverse axes being imaginary for tachyonic frames and real for 41 bradyonic frames. Hence the factors of + i and - i are interpreted as a 90° rotation between an imaginary axis and a real axis in complex spaced me. For example, in eq. (7.23) the actual angle 8 is given by \ (sinGT^ I 9 = Tan ..(7.25) cosG' + — V c / for u2 > c2. The factor of + i which should appear in front of the square root has been implicitly removed to account for the 90° rotation from an imaginary axis to a real axis. Furthermore, the sign of the square root can also change due to switching: it is positive for unswitched sources and negative for switched sources. Another complication arises because of the two source effect, which was discussed at length in section 6. This means that there arc two aberration formulae for each point inside the Mach cone, one for each apparent tachyonic source: / \ (sinG' tan9, •»VF ,.(7.26a) - cos8'i1 + V c / ( \ (si noy^l - 1 and tan92 = .(7.26b) , cos8'2 + ~ V c ) (d) Plane waves. The wavefunction describing a plane wave is of the form(38,31) fr. /x.cos8 + y.sin8 \ "\ ¥ =A.cos[2n( 5p v t 1 j , (7.27) where A is the amplitude, X is the wavelength, v is its frequency and 8 is the angle in the x-y plane between the wave's direction of propagation and the x-axis. Now consider a train of monochromatic light waves of unit ampliludc, emitted by a source at the origin O' of frame X': sec figure 42 9. The wavefronts are sufficiently distant from O' to be considered as plane waves. The propagation is of the form /_ fx'cosQ' + v'sinB' , ,\\ HH—*—' ))• ..(7.28) The wave speed is of course XV = c. In a second inertia! frame £ the wavefronts are still planes, as both S.R. and E.R. transformations turn a plane in J.' into a plane in £. Therefore the propagation of the wavefronts with unit amplitude in frame X is described by x.cos8 + v.stn9 COSI2JI( vt ..(7.29) ))• where the wave speed is Xv = c. Now suppose that frame Z' moves with speed u > c along the common x.x' axes relative to a bradyonic frame £. The appropriate transformations for x', y' and t' for u2 > c2 arc given by cq. (2.2). Substituting these transformations into cq. (7.28), rearranging terms and using the identity cos9 = cos(-G) gives the propagation of the waves as \ -ivxfcosG' + —I ' V U jy-sine' , f, u.cos9'\ cos 9ff + i .(7.30) As both equations (7.29) and (7.30) describe the same plane wave as seen in frame X, the coefficients of x, y and t from each equation can be equated. Comparison of the x-componcnt gives C08e + cosO «Yu| ' c) .(7.31) V The y-componcnt gives sinQ j.sin9' .(7.32) while the t-componcnt gives . /, u.cos6^ v = -iYuv(l + ——J. .(7.33) 43 On rearrangement cq. (7.33) gives the inverse transformation of the tachyonic Dopplcr effect, eq. (7.8). This agrees with the result derived earlier using the energy and momentum of the photons, and demonstrates the internal consistency of this formulation of the tachyon theory. Dividing equation (7.32) by (7.31) gives eq. (7.23), which is the expression describing the aberration of light due to a tachyonic source. This expression was derived in the previous subsection using a different method, again showing the consistency of the theory. Despite these successes, there is in fact a small problem here. Using v = cA. and v' = cA/ in cq. (7.33) gives 'V. , u.cose'N = TM,1 + c J - (7-34) and using this to eliminate X and X' from cq. (7.31) leads to u cos8 + — cos9 = , u.cosV • <7-35> 1 + This is the expression describing the transformation of cosines between the two frames, and is the same for u2 < c2 and u2 > c2. Inserting test values for u and 0' into cq. (7.35) shows that IcosGI < 1 for u2 < c2 and IcosBl > 1 for u2 > c2. Hence the formula produces a mathematical absurdity for u2 > c2, although in spite of this, cq. (7.35) appears to be the "correct" transformation. For example, it gives the correct conversion for the Dopplcr formula when exchanging frames. Substituting cq. (7.35) into the expression for the Dopplcr effect, cq. (7.6), and rearranging the resulting expression gives eq. (7.8), which is the inverse transformation of the Dopplcr effect. This is analogous to the treatment of "absurd angles" in the treatment of Frcsncl relations encountered in S.R/32) Even though the transformations may give IsinG'l greater than 1 or cos8' as being imaginary for specific cases, the Frcsncl relations still give meaningful results with regard to reflection and refraction of electric and magnetic fields at a plane boundary. Similarly, in the present E.R. case it is the final result which counts, not the intermediate steps. 44 Equation (7.34) can also be used in cq. (7.32) to obtain sinO"\/ 1 - ^1 sine" sinO = .(7.36) (. u.cosG^ u.cosG* 1 + which is the sine transformation, applicable for both u2 < c2 and u2 > c2. Note that sinO is imaginary for u2 > c2, due to the tachyonic transverse axes being imaginary. The factor of +i is the conversion between an imaginary axis and a real axis in complex spacctimc, and can therefore be implicitly removed. The sine transformation then becomes sin0"V 7* 1 sinG = ..(7.37) u.cosG' 1 + Inserting test values for 6' and lu/cl > 1 shows that many combinations do give IsinGI < 1 as required, while combinations such as 6' -» ± n/2 and u » c, or cosG' -» - c/u, give IsinGI > 1. The inverse of the sine and cosine transformations are respectively cosG - cosQ' = ..(7.38) u.cosG 1 - 5inG-\/ 1 - "- and sinG' = .(7.39) u.cosG where both transformations apply for - °» < u < «>. Note that the same problems of having IcosG'l > 1 and IsinG'l > 1 occur with the inverse transformations but, just as with the Frcsncl relations in S.R., it is the net result which is important and not the intermediate steps. 8: ELECTROMAGNETIC FOUR-POTENTIAL. 45 Earlier sections in this paper have dealt with the electric and magnetic fields generated by a charged tachyon. In this section the electromagnetic scalar and vector potentials produced by such a tachyon will be investigated. The scalar and vector potentials in bradyonic frame X are P)A E =-v In tachyonic frame X' the potentials $' and A' arc related to the fields E' and B' via the same equations: E'= -V'f-^r , (8.3) B' = V'xA' . (8.4) Using the transformations of the partial derivatives given by cq. (3.11) and the electromagnetic field transformations given by cqns. (3.12) and (3.13), it can be shown by substitution into equations (8.3) and (8.4) that for u2 > c2 the potentials transform as follows: 2 A,. = iy./A, - u<)./c) . Ay. = iAy , Az. = iAz , •' = iY|1(* - uAx) (8.5) The inverse transformations for u2 > c2 arc A, = " JYu(\' + u*'/c2) • \ = - iAy • \ = " *\- • 4 = - iy^' + uAx.) . (8.6) Hence for u2 > c2 the potentials <(>' and A' transform as a spacclikc four-vector (A', i<>Vc): 2 2 2 2 j (j>' A2 2 2 i * 2 ,2 2 /n _ - ^ - A,- - A,. - Az. = + ~^~ + A, + Ay + A* . (8.7) The transformations of components, i.e. Ax and In bradyonic frame £ the potentials $ and A satisfy the equations VA + hfi= ° ' (8"8) 1 32A v2A-^a£ = -^ • (89) 1 320 o V2*-c^ = X' (8-,0> while in tachyonic frame E' the potentials $' and A' satisfy identical equations with primed quantities replacing unprimed quantities. Equation (8.8) is known as the Lorcntz condition. Each of these equations applies to potentials in free space: the modified forms for the potentials in a material medium will appear in section 19. In section 6 the electromagnetic field produced by a charged tachyon was investigated in some detail, with several interesting features being discussed. Having just given the tachyonic transformation of the scalar and vector potentials, it is therefore of interest to apply them to a point charge moving with uniform velocity. The two source effect also applies to the potentials, and will be discussed shortly. The following derivation has been adapted from the rclalivistic case given by Rosscr(34). 47 Consider a tachyon carrying charge +q* which is moving with uniform speed, as shown in figure 11(a). The tachyon is moving with speed u > c in bradyonic frame X (fig. 11(a)), but is at rest in tachyonic frame X* (fig. 11(b)). In frame X' the charge has position (x\ y\ z') at a distance r' relative to the point of observation O'. The origins O' and O of frames X' and X are synchronized so that they coincide at the instant when t' = t = 0. In frame X the point of observation is at O. Figure 11(a) clearly shows that if the tachyon produces potentials at position A, then the tachyon has moved to position B by the time the information collecting sphere, travelling with speed c, has reached the origin O from position A. This sphere has speed c due to the second postulate, even though the source producing the potentials is a tachyon. Hence the position of the tachyon when it actually produced the scalar and vector potentials measured at 0 is the "retarded position" (x, y, z) at a time t - r/c, as the information requires an elapsed time equal to r/c to travel from A to O. Frame X' has a uniform velocity u parallel to the common x,x' axes relative to bradyonic frame X, so that the charge is at rest in frame X' but has velocity u in frame X. Due to the possibility of switching, the apparent charge in frame X is q = sign/-? \q'. The potentials in frame X' at the origin 0' arc given by •' = T^—; and A'= 0 , (8.11) where r'2 = x'2 + y'2 + z'2 . (8.12) The values of A and <|> at the origin of X at t = 0 for u2 > c2 can be found using the transformations given by equations (8.6) and (8.11): 'Ho"Yu-sign(yu)q A* = " 4rcr4rcr"' • (813a) Ay = A7 =0 , (8.13b) 'Yu-sign(yu)q * = - w • (8-13c) 48 The corresponding equations for u2 < c2 can be obtained by removing the factor - i.sign(v). •u' As these arc electromagnetic potentials, in frame Z' the information travels with speed c from the particle's stationary position to the point of observation at the origin O'. Therefore it takes an elapsed time equal to r'/c for the information to %o from position (x\ y\ z') to 0', and hence the distance r' is measured at a time t' = - r'/c. This means that the inverse S.L.T.s given by eq. (2.5) lead to x=-nr„[x -—J. y = -iy . z = -.z . t=--^-r+— J. Substituting these expressions into r2 - c2t2 = x2 + y2 + z2 - c2t2 leads to r2 - c2t2 = r'2 - x'2 - y'2 - z'2 =0 . Of the two possible solutions r = ± ct, the one chosen is r = - ct as it corresponds to the position of the charge at the time t = - r/c in frame £ and time t' = - r'/c in frame X'. This is the same result as in S.R., once again showing how the effect of the imaginary factors disappears when necessary. Note that the vector r measured at the time t = - r/c is taken from the retarded position of the charge to the point of observation. 2 Using t = - r/c in the transformation t' = iyu(t - ux/c ) gives r' = - cf = iyu(r + ux/c) . (8.14) As u.r = ur.cosG = - ux, where G is the angle between u and r (sec fig. 11(a)), then .(8.15) The corresponding S.R. case can be obtained by deleting the factor of i. Using cq. (8.15) in equation (8,13) with t = 0 gives the potentials measured in frame I for u2 > c2: Aiq.sign(Y \\ A - - M— ^-^ (8.16a) u . r r - V c 49 Ay =AZ =0 , (8.16b) 1 fsignfj,,^ .(8.16c) 4jte0 u . r r - ^ c J To find the corresponding equations for the potentials if instead u2 < c2, simply delete the factor of - signfy^ in cq. (8.16). The above derivation is of course incomplete as the two source effect is once again applicable, just as it was for the discussion of the fields produced by a charged tachyon. Figure 12 shows how the two source effect applies to the retarded scalar and vector potentials. The information collecting sphere, travelling with speed c, intersects the path of the tachyon at two points, instead of a single point if the charged particle was instead a brad yon. As figure 12 clearly shows, there arc two retarded positions for the tachyon, given by (r(,6,) and (r2,62). The derivation above applies to th_ point whose coordinates arc (x2, y2, z2), so that t2 = - r2/c. The particle is stationary in tachyonic frame E\ so that t' = - r'/c as before. The earlier retarded position has coordinates (x,, y,, z ) so that t, = - TJC. The derivation above also applies to this point, up until cq. (8.14). This equation becomes r' = -cf= iy^r, + ^ , (8.17) for the earlier retarded position, with r* = -cf = iyu(r2 + "-^j , (8.18) for the later retarded position. Earlier equations involving quantities measured in frame Z can also be written as two separate equations with subscripts "1" or "2" depending upon the retarded position being discussed. For the earlier position u.r, = ur^osG, = + ux, (sec figure 12), so that r' / u-r,>\ ..(8.19) 50 Hence the vector and scalar potentials measured in frame X due to the earlier retarded position arc /uq^sign£Y\ \ .(8.20a) u.r, '. + —, Ay =AZ =0 , ..(8.20b) /sign (^ ,\ .(8.20c) 4» = - 4ne„ r, + . V > c J The net vector and scalar potentials measured at 0 arc a superposition of the potentials produced at each of the retarded positions 1 and 2, so that UqSign Y 1 J \ %X(tOUI) .(8.21a) fe( (-))| u.r, u.r2 r, + r, - 1 c 2 c j ^(lolal) "~ A,/,„!,i\E(M>UI\) — U , .(8.21b) si'gn(y„)i q / 1 J >t ..(8.21c) (total) 4xce„ u.r. u.r. r + i c ) These expressions can be generalized to give the tachyonic Licnard-Wicchcrt potentials: s| n > '», S'gn(Yu,) "2 g (^2) '(total) .(8.22a) 4JT u,.r, U2'r2 r. + 2 c ) 1 c ^sign/yu]^ sign^y^A .(8.22b) * «o.ai) - - 4ne u r r i U2'r2 c y Each set of values (rru,) and (r2, u2) refer to the retarded positions of the tachyonic charge. The different values u, and u2 refer to the 2 tachyon's velocity at each of the retarded positions. Note that if u(> 51 2 2 2 c then u2 > c , as a tachyon in one bradyonic frame is a tachyon in any other bradyonic frame. Now consider a slightly different situation, in which the scalar and vector potentials arc produced by the same tachyonic charge as produced the electric and magnetic fields in section 6. The particle has charge +q' and is at rest in tachyonic frame X'. The potentials in frame X' are again given by eq. (8.11), with r' being defined by cq. (8.12). Both frame X' and the charged tachyon move with speed u > c along the common x,x' axes relative to a bradyonic frame X. Using the tachyonic transformations given by eq. (2.2) and the i-y convention produces r' = \^ ylx2 + y2 + z2 , (8.23) where it is assumed, purely for convenience, that the moving charge is at the origin at the instant t = 0. The origin of the coordinate systems used by frames X and X' coincide at the instant t = t' = 0. Using cq. (6.7) to relate cartesian coordinates to polar coordinates leads to 2 2 2 , . u sin2 eV'2 .__.. . Mr u sin ey r = >YurM - —15— • (8.24) •T.r[l - —-T-J Equations (8.11), (8.23) and (8.24) can be combined with the tachyonic transformations of the vector and scalar potentials given by (8.6) to obtain sign(Y„)Hou Ay= A, = 0 , (8.25b) 2 2 (t u sin 9V sign(yu)q * = - V(1 - \ -cT-; , J . (8.25c) ., u2sin29V'2 4rce„r 1 These potentials arc evaluated for the charge passing through the origin at the instant t = 0. The charge moves a distance ut = ur/c 52 during the time it takes for the potentials to propagate with speed c from the point of emission to the observer. As the tachyonic charge is moving along the x-axis in this example, then the expression describing the vector potential can be generalized to s»gn(Y„)u0qu A = - , . , • (8.26) , r u2sm2ey/2 4K\l - —*-) If instead the charge was bradyonic and not tachyonic, then the factor of - sign(Y„) should be deleted in equations (8.25) and (8.26). Once again the two source effect causes a modification of the equations describing the potentials, due to the superposition of the potentials generated from each of the apparent sources. Assuming that the charge travels with a constant velocity u, the net vector potential is given by A - - sign(,„)(M^ • £) , (8.27, u2sin28. V 1 - —^—L (8.28a) 2 2 . I u sin e2 and s2= ^ 1 - —^j—~ (8.28b) The net scalar potential is given by • - - M*")(£:I^+ 7k) • <8-29) An examination of equations (8.27), (8.28) and (8.29) shows that both potentials A and 9: LAGRANGE'S EQUATIONS. Lagrange's equations arc defined in the same way for both S.R. and E.R. as 4®-*"°' where the q{ arc generalized coordinates q,, q2, .... q„ (i = 1. 2, ..., n). The Lagrangian for a bradyon carrying charge q is 2 2 C„ = - m c^ 1 - v^ 7 - q* + q(v.A) , (9.2) v0 V where v is the velocity of the particle. The particle is assumed to be moving in a region containing an electromagnetic field described by the scalar and vector potentials 4> and A respectively. The Lagrangian for a tachyon carrying charge q is CT = - m.c*-y I - ;£ - q* + q(v.A) . (9.3) Note that both CR and £T arc real, with m. = im0 for tachyons. Taking the partial derivative of CT with respect to x and remembering that $ and A depend upon the position of the charge, but not its velocity, leads to 3£T 3* ( 3A. 3A 3A \ &T = - o* + \y*lt + yy it+ v» it J • (94) The r.h.s. of this expression is the same for the bradyonic Lagrangian, so that 3CT/3x and 3CB/3x arc the same except for the allowed range of v and the possibility of switching. Partially differentiating the tachyonic Lagrangian with respect to v„ gives 3CT m.v, — ~= + qAx , (9.5) * 4*'i- £ and so 54 ( ni.v, \ ..(9.6) W%+ q dt Since Ax is a function of the spacctime coordinates x, y, z and t then 3A. 3AX d\ 9AX dA = "5fdx + ^~dy + -r"dz + ^~dt , x 3x 3y dz dt and so dividing by dt gives dA, _ 3A, 3AX BAX 3A, dt = "aT"* + "aT^ + ~dP* + at • •(9.7) Combining equations (9.5), (9.6) and (9.7) shows that the l.h.s. of Lagrange's equation for tachyons leads to dt . | 3x ( m.vx ^ q(~ 1fc" "aT") "qVy(v x A)z + qVz(V x A)y (98) The relations between the electromagnetic fields and potentials arc given by equations (8.1) and (8.2), so that eq. (9.8) now becomes /dfty\ 3CT (J / m.vx \ 8 v B dt 9x dl q£* -qvy ^ + q z , V3xy v-i dp = 77 - qEx - q(v x B)x . ..(9.9) The x-componcnt of the force is defined to be dpx d( m.vx -\ dt ~ dt v^ while the x-componcnt of the Lorcntz force is F x(i.) = qEx + q(v x B)x . Therefore the x-componcnt of Lagrange's equation for tachyons gives d/d*V\ ^T F dl . ax = «-•>> = <>• ..(9.10) Ux 55 Thus the tachyonic Lagrangian leads directly to the Lorentz force equation, which demonstrates the high degree of internal consistency in this formulation. Similar derivations apply for the y- and z- componcnts of the Lagrangian. 10: HAMILTON'S EQUATIONS. Now that the tachyonic Lagrangian has been given and confirmed to be correct, the next step is to develop the tachyonic Hamiltonian. It is defined to be HT = I PjQi - C-r . (10.1) i where again q( is the generalized coordinate and i = 1, 2, .... n. The generalized momentum p( is defined by Pi = —~ • (10.2) These definitions have exactly the same form for bradyonic Lagrangians and Hamiltonians. Indeed, the only differences between the bradyonic and tachyonic equations arc the range of allowed speeds and the definition of the particle's proper mass. Hamilton's equations for tachyons can be written as 3HT PI = - aqf . <10-3) 3HT . * = ' 3p7 • (,0-4> If a single tachyon has charge q and moves through a region in which scalar and vector potentials arc present, then the Hamiltonian is 2 2 2 4 HT = q<}» + A/c (p - qA) - m c . (10.5) If the scalar and vector potentials arc set to zero, it can be seen that the Hamiltonian then becomes equivalent to the tachyon's energy: 56 "T=^/ p c - m„c = Ej. . (10.6) The second of Hamilton's equations, eq. (10.4), gives c • ™z (Px- qAx) ,1ft„ dPx f '(p - qA) - m0c with similar expressions for the y- and z-components of v. Substituting for v using eq. (10.7) allows the first of Hamilton's equations to be written as dp. a* / 3AX 3AV 3A, , dpx 3* / 3AX 3A„ 3AZ\ P* H "dT = ~%x + y*~37 + vy ax + v* 3x~J The expansion of dAx/dt given by cq. (9.7) can now be used so that dpx dA„ aj, 3AX /3A^ dA,\ (d\ 3AX\ "dT - ^"dT = -q3x " q~3T + qv*U " ay"J + qVzl^" ^"J • which can be written in more compact notation using (8.1) and (8.2) as ^(Px - qAx) = qEx + q(v x B)x . (10.9) From equations (9.5) and (10.2) it can be seen that d ( m'vx "\ d = ^(Px " qA,) = qEx + q(v x B)x . (10.10) dt / v^~ Therefore the Hamiltonian HT gives the correct equation of motion in the x-dircction for a charged tachyon. Similar calculations also apply to the y- and z-componcnts, so that for tachyons the general case is d/ m.v \ d dt V^l = ^P - qA) = qE + q(v x B) . (10.11) Thus it can be seen that the Hamiltonian formalism gives the same results as those that would be obtained using the ordinary mechanical formalism developed in Paper I. This means that an extension of rclativistic quantum mechanics into the tachyonic realm 57 should have excellent prospects of success if the Hamiltonian formalism is used. 11: GENERALIZED FOUR-VECTOR TRANSFORMATIONS IN E.R. Looking back over the transformations of various four-vectors already derived in this formulation, it is readily apparent that the great similarity in the form of the transformations indicates that they can be expressed as a more generalized matrix equation. The tachyonic four-vector transformations are very similar to the corresponding rclativistic transformations, in that they differ only in factors of + i or - i and have a different allowable range of speeds for u and the y-factor. Therefore it is expected that the generalized matrix equation which applies for u2 > c2 will be only slightly changed from the u2 < c2 case, and so the matrix equation for E.R. can be found by once again carrying out a minimal extension of the corresponding equation in S.R. It must of course be remembered during what follows that the Euclidean metric (+1, +1, +1, +1) is being used with spacctimc coordinates given by (x, y, z, ict), and so no distinction is made here between covariant and contravariant quantities. In Special Relativity the transformation of any four-vector B^ can be expressed as(35,36) B\ = £kvBV (H.l) v=l where LXv is a 4x4 matrix such that iuy /c Yu 0 0 u 0 1 0 0 kv = 0 0 1 0 .(11.2) -iuyu/c 0 0 - Yu The corresponding inverse transformation is Bv = 1 L'vXB; , (11.3) where the matrix L'vX is the inverse of LXv and is given by 58 Yu 0 0 - iuYu/c 0 10 0 .(H.4) "-* = 0 0 1 0 iuyu/c 0 0 Y0 Hence LXyL'vX = I4, where I4 is the 4x4 identity matrix. Here of course both frames Z and £' are bradyonic frames. Examples of four-vectors which obey equations (11.1) and (11.3) include the following: (i) spacctimc position: Xx = (x, ict) , (ii) energy-momentum: Px = (p,iE/c) , (iii) charge and current density: Jx = (j, icp) , (iv) electromagnetic potential: Ax = (A, i<|>/c) , (v) partial derivatives: Dx = lr^ , r^H . For tachyonic transformations it is necessary to include the factor of cither + i or - i which appears in each term, so that the four- vectors transform according to either the upper or lower signs in equation (11.5): B-x = ±ilLB¥. (11.5) v=l The inverse transformation is Bv= + iXL'^. (11.6) It must of course be remembered that frame X' is now a tachyonic frame, while frame X is still a bradyonic frame. Note that the matrices LXy and L'yX are still defined by equations (11.2) and (11.4), even though the generalized transformations given by equations (11.5) and (11.6) apply for u2 > c2. It turns out that each of the signs in equations (11.5) and (11.6) describe some of the possible transformations. The upper sign applies to the four-vectors Xr Px and Ax, while the lower sign applies to the four-vcclors Jx and Dr Henceforth these four-vectors will be grouped by the type of tachyonic transformation they obey. "Type I" four- vectors such as Xx, Px and Ax obey the transformations (11.5) and 59 (11.6) using the upper signs. "Type II" four-vectors such as Jx and Dx obey the transformations (11.5) and (11.6) with the lower sign. The square of all of the four-vectors listed above is 2 2 1(B\) = ± I(BX) , (11.7) where the upper (+) sign applies for u2 < c2 and the lower (-) sign applies for u2 > c2. There is, however, a class of transformations which provides the exception to the above discussion of generic transformations in S.R. and E.R. These arc transformations of quantities which normally only make up three-vectors, such as velocity and force. These quantities have the common properties that their transformations arc exactly the same for u2 < c2 and u2 > c2, and that transformations of the components perpendicular to the boost contain y-factors. As these quantities have the same transformation for - °° < u < °°, then the square of their four-vector Bx is 2(B\)2 = I(B/ . (11.8) One example is the velocity four-vector Ux = (YUU, icyu), for which 2 UXUX = - c (11.9) for -°° i F u> ( Yu - \ IYUF, 1. Note that the vector component of U^ and F^ is imaginary for u2 > c2 for all three coordinates, and is real in the fourth component. This contrasts with Types I and II four-vectors, for which the vector component is real and the fourth component is imaginary. 12: ELECTROMAGNETIC FOUR-TENSORS. Having developed the tachyonic transformations of various four- vectors and electromagnetic quantities, it is now possible to discuss 60 some of the electromagnetic four-tensors and their transformations. The first such tensor is the electromagnetic field tensor Fap, given by B * B - iE /c - o - , X 0 Bx - iEy/c F<* = 0 iE c .(12.1) By Bf - J L iEx/c iE /c iEz/c 0 The following discussion is adapted from the rclativistic case given by Lawdcn^7). Thc four-vector potential Ax in incrtial frame X is defined to be Ax = (A, i 2 • AX = -^0JX' (12.2) ^, a2 a322 a322 I1 a322 2 (12.3) where Q = ^ + ^ + ^ - ^^ and ix = (j. icp) is the four-current density. The corresponding equations in tachyonic frame X' arc ..(12.4) m_iL A ii J-iL ..(12.5) wherc G - 3x>2 + 9y,2 + dz<2 - c2 3t,2 and i\ = (j', icp'). By using Ax and equations (8.1) and (8.2), it can be shown that the field tensor components may be written as dAa 3A Fa& " ax axp- .(12.6) Since the individual components of F. obey the same set of transformations for u2 < c2 and u2 > c2, then the field tensor in frame X' can be written as 0 B - iE* /c B'z '> x B'> 0 B' - iE'y/c ?« = .(12.7) B'„ B'. - iE'7/c L iE' /c iE* /c IE'JC Ay/. It can also be shown by using A'^ and equations (8.3) and (8.4) that the electromagnetic field tensor F' in tachyonic frame X' is 61 3A' 3A' F*„ = ..(12.8) op " 3X"_ ax1. By substituting the tachyonic transformations for the partial derivatives eq. (3.11), and electromagnetic potentials eq. (8.5), into the expression for F-, it can be shown that iE 0 B U U X. *.( *- ^) -*.(B, + £) C uB - Y •(B«-V) 0 B. —1 E - 0 *% = c V y »Y - B —(E z + uB ) c V y iE »T. * u (Ey - uB.) ~f(Ez + uBy) 0 .(12.9) This same result could have been obtained simply by transforming the individual tensor components according to equations (3.12) and (3.13), which are valid for - « < u <«. This shows that for u2 > c2 the electromagnetic field tensor transforms according to .(12.10) r aaB? - 2, 2, La|l^v^ vp H=lv=l As this is exactly the same transformation as for u2 < c2, then cq. (12.10) is valid for - °° < u < °o. The tachyonic four-vectors involved in this transformation arc A^ and D^, which arc respectively Type I and Type 11 four-vectors. This means that the +i in the transformation involving Ax (upper sign in (11.5)) combines with the -i due to the transformation involving Dx (lower sign in (11.5)) to give a net effect of +1. This is yet another example of how the imaginary factors, which appear throughout this theory, usually cancel out at the appropriate stage so as to produce reasonable and consistent results. The components of the electromagnetic stress-energy tensor T . arc given by(38,39,40) 5../ o2x / B B.\ 62 i ,5i T4 = T. = —(E x B). = — , .(12.11b) •* 4| CU ' C (12.11c) where S is Poynling's vector and Uem is the energy density of the electromagnetic field. As the components of Ta_ undergo the same transformations for u2 > c2 as they do for u2 < c2, then T must be related to T_ by T.^I^XL^Lp, (12.12) for - oo < u < oo. Further electromagnetic four-tensors involving electric displacement and magnetic field strength will be discussed in section 18. 13: WHY TACHYONS ARE EFFECTIVELY LOCALIZED PARTICLES. Throughout sections 6 to 8 it was pointed out that the "two source effect" meant that tachyons appeared to a bradyonic observer to be in two places at once. In Recami's formulation of the theory of tachyons^41), this effect is taken to be evidence that tachyons arc noniocalizcd particles. Moreover, the fact that tachyons can have infinite speed relative to certain observers means that those observers cannot fix the tachyon's postion in space, and so this is also taken to be evidence that lachyons are noniocalizcd. This point is one of the major differences between the Rccami formulation and the new formulation presented here. It will be argued here that tachyons arc effectively localized particles, even though they appear to bradyonic observers to be in two places at once. Consider the example in section 13 of Paper I, in which it was shown that a tachyonic cube moving perpendicularly to the observer at a large distance will appear to be rotated so that its side face can be seen. The face ABCD which normally was facing towards the observer was contracted for v2 < 2c2 and dilated for 2v2 > c2. The side face ADFE appeared to be rotated into the view of the observer and appeared to be dilated as the speed increased. At no stage did the 63 image of the cube lose its integrity and appear to be fragmented into cones or hyperbolae or any other peculiar shape. The same is true if instead the object under consideration is a sphere in its rest frame. In any frame in which the sphere is moving with a speed v such that vz > c2 then it appears to be an ellipsoid which undergoes rotation and contraction/dilation effects. The equation describing the shape of the object which is spherical in its rest frame Z' is 0 < x'2 + y'2 + z'2 < r'2 . (13.1) In frame X the apparent shape of this object when moving with speed v such that v2 < c2 is (x - vt)2 0 < * Y~ + y2 + z2 < r2 . (13.2) V 1 - ~Tc2 In his formulation Rccami claims that such a particle would, if instead it were a tachyon, occupy the whole space bounded by a double, unlimited cone and a two-shected hypcrboloid connected at a point^42). This conclusion was reached because Rccami argues that if the object is seen as a tachyon, its coordinates should transform according to the (Rccami) S.L.T.s so that, for v2 > c2, (x - vt)2 2 2 2 0 > - ^—2 ~ + y + z > - r , (13.3) c2 which yields the hypcrboloid to which he refers. The sign change on the transverse components y2 and z2 is due to the way the y- and z- coordinalcs transform in the (Rccami) S.L.T.s: y' = ± iy, z' = ± iz. It is here that another subtle difference between the two formulations becomes apparent. Even though tachyons use a complex spacctimc rather than the real spacctimc normally used by bradyons, the Icnglh, area and volume of a tachyon must be real and positive due to the properties of numbers in the complex plane. In Paper I it was shown that for v2 > c2 the apparent length I of a rod moving parallel to the boost is given by 1 = l0 where 1„ is the rest length. The apparent Icnglh of the rod measured perpendicularly 64 i to the boost is 1 = |il0j and not 1 = - il0. Therefore the components of equation (13.1) arc transformed according to 2 2 x22 (x " vt) 2 2 2 x' -> (iYu(x - vt)) = ^ ~ . y' -* y . z* -> z . c^" * and so the equation describing the apparent shape of the object in frame £ in the present formulation for v2 > c2 is not (13.3), but is instead (x - vt)2 0 < ^-5 *- + y2 + z2 S r2 . (13.4) h - i c2 This equation describes an ellipsoid and allows that the image is still connected, in agreement with the discussion of the visual appearance of the tachyonic cube. It docs not take into account the apparent elongation of the side as the speed increases (this corresponds to the face ADFE in the cube example): this must be left to a more detailed analysis than the one presented here. The problem has been caused by a subtle misinterpretation of one of the intrinsic properties of tachyons and tachyonic observers: they consider their transverse axes to be real, but bradyonic observers consider these same transverse axes to be imaginary. Likewise, transverse axes which are real for bradyonic observers arc imaginary for tachyonic observers. However, lengths, areas and volumes arc always positive and real for both bradyons and tachyons, and so one always has to consider carefully exactly what is being transformed or measured in each frame. The discussion of the electromagnetic field of a tachyonic charge in section 6 used the S.L.T.s without the modulus signs for transforming the coordinates because it was not necessary to calculate any lengths, areas or volumes. Similarly, it was not necessary to use modulus signs when discussing the scalar and vector potentials in section 8. It is important to distinguish cases where it is necessary to calculate the magnitude of a quantity, such as length, area and volume, as these cases arc due to the requirement that lengths are real and positive, even for numbers in the complex plane. Note that 65 this is still consistent within the present formulation, as shown by the detailed proof given in section 5 of the conservation of electric charge in all incrtial frames. In that derivation the modulus signs were used on the volume transformations for v2 > c2 and were shown to yield the correct sign for the observed charge, even when the volume factor was combined with other terms which did not have modulus signs. The above discussion shows that the image of the tachyonic object remains spatially connected and localized, so that tachyonic objects can effectively be treated as localized objects just like bradyons. The discussion in sections 6 to 8 docs, however, show tha* apart from the rotation and contraction/dilation effects (which normally apply in both S.R. and E.R.), there are extra effects with regard to the tachyonic object. When outside the "Mach cone" the information about the tachyon has not yet reached the observer. At the instant of optical contact the observer experiences the "optic boom". After this initial contact the observer sees the "two source effect", in which two images of the tachyonic object arc seen receding from each other. For example, if the tachyon is a sphere when at rest then each of the images will be of an ellipsoid which undergoes apparent rotation and contraction/dilation effects. If the tachyonic object is a cube when at rest, the discussion in Paper I on its visual appearance applies to both of the observed images of cubes, with two differences between the the two images. If the cube is passing the observer to the right, then the right hand image shows the front face ABCD and the side face ADFE to the observer. The left hand image receding in the opposite direction shows the front face ABCD and the face of the cube opposite to ADFE to the observer. That is, each image shows the face which is to the rear of its apparent direction of motion. The second difference allows the images to be labelled as "advancing" and "receding" images. Suppose the tachyonic object is moving to the right relative to the observer. The "advancing" image is on the right and moves in the same direction as the object. The image moving to the left is the "receding" image and moves in the direction opposite to the actual direction of motion of the object. Hence the tachyonic Dopplcr effect discussed earlier can be used to determine the actual direction of motion of the tachyonic object. 66 It is important to note that the two source effect is simply due to the inherent time delay involved in the propagation of signals. The second image occurs because the information travels at the fixed speed of c (in free space) from the source to the observer. This second image is real to the extent that it produces real and measurable effects and appears because the tachyon moves faster than the intermediary carrying the effect. This is analogous to the effect produced with sound waves by a supersonic aircraft. Thus it can be seen that tachyons arc effectively located particles for the purpose of calculations, and have implicitly been treated that way earlier in this work. Apart from the two source effect and their speed, tachyons behave just like normal point particle bradyons for the purpose of generating fields and potentials. This has the distinct advantage of removing the inhcrciu difficulties associated with dealing with nonlocalizcd particles, for example the problem of finite time extension for tachyons(43). 14: SWITCHED TACHYONS ARE NOT ANTITACHYONS. In his formulation Recami uses a third postulate of Extended Relativity to assist in the resolution of causal problems. This third postulate, which is quoted verbatim from page 15 of Review Il(52)r js: "The principle of retarded causality": Positive-energy objects travelling backwards in time do not exist; and any negative-energy particle P travelling backwards in time can and must be described as its antiparticlc P, endowed with positive energy and motion forward in. time (but goiag the opposite way in space)." This highlights one of the subtle differences between the present formulation and that proposed by Rccami. The third postulate above indicates that when a tachyon P undergoes switching, then it effectively becomes its own antiparticlc P. However, in the formulation proposed in the present work, when a particle undergoes switching it docs not appear to become its own antiparticlc. A switched particle has a subtle but nevertheless important difference from a true antiparliclc. 67 Consider an ordinary, unswitched tachyon which is an electron of charge - q in its own rest frame. In all other tachyonic frames it 2 appears to be a bradyon with speed vT such that vT < c and it exhibits all of the properties appropriate to an electron in motion. In 2 bradyonic frames with relative speed less than c /vT the tachyon is unswitched, has apparent charge - q and obeys all of the laws of mechanics and electromagnctism. In bradyonic frames with relative 2 speed greater than c /vT the tachyon is switched and appears to have electric charge + q. In such frames the tachyonic electron still obeys the laws of mechanics and electromagnctism, but docs not have all of the properties of a positron. Figure 13 is a Minkowski diagram representing a tachyon travelling through space. The tachyon emits a photon at a certain time and loses positive energy, which causes the tachyon's worldline to bend away from the light cone. Note that photons always have positive energy, regardless of what type of particles emit them. Bradyonic observer X0, using axes (x0, ict0), sees the unswitched tachyonic electron ejn+ emit a photon y and become cout+ in the reaction Cjn+ -* eoUl+ + y. A second bradyonic observer X', using axes (x\ ict'), sees two incoming tachyons collide and mutually annihilate to produce a single photon y. One of the tachyons is switched and the other is unswitched, so that observer X' describes the reaction as $„+ + Cjn. -» y. Here superscripts refer to the apparent sign of the charge on the particles and the subscripts denote whether the particle is incoming or outgoing, and whether the particle is unswitched (+) or switched (-). If the switched tachyon seen by X' were a true antiparticlc, in this case a positron, there would be two photons produced in the mutual annihilation with the tachyonic electron. Instead there is only one photon produced, and so there is a subtle distinction in the effects of a switched tachyon and a tachyonic antiparticle. If a tachyonic electron collided and mutually annihilated with a tachyonic positron there would be two photons produced, just as if the particles were bradyons instead. Note that in a tachyonic frame a collision between two tachyonic particles looks like a collision between two bradyons according to that observer. Hence there would be two photons produced by the electron-positron pair in the tachyonic frame. The number of photons docs not change 68 when crossing the light barrier, so a bradyonic observer must also see two photons produced by a particlc-antiparticle collision. It should also be pointed out that in the example illustrated by figure 13, observer X0 sees the net charge - q before and after the interaction, whereas observer X' sees net charge zero before and after the interaction. Total electric charge is conserved in each frame, but is no longer necessarily the same between different frames. This point was explained in detail in section 5. IS: TRANSFORMATIONS OF D AND H. The first postulate of Extended Relativity, given in section 1, has been taken to mean that the form of Maxwell's equations in free space arc the same for both bradyonic and tachyonic inertia! observers. It will now be extended to cover the form of Maxwell's equations which use the electric displacement vector D and the magnetic field intensity vector H. Two of Maxwell's equations can therefore be written as(44) VxH = ^ +j . (15.1) V.D = p , (15.2) in bradyonic frame X, while in tachyonic frame X' the corresponding equations arc v"xH' = ~ +j' , (15.3) V'.D' = p' . (15.4) Note that these are just restatements of equations (3.3), (3.4), (3.7) and (3.8) with the replacements E -» D/e0, E' -> D7e0, B/u0 -> H and B7u0 -> H . (15.5) It must of course be remembered that e0 and \i0 arc unchanged when transforming between tachyonic and bradyonic frames. 69 The tachyonic transformations of 3/dz, 3/dx, 3/3t and jy can be substituted into the y-component of eq. (15.1) and, after rearranging terms and cancelling a common factor of i, the result is 37 ' Y-3xT(H--uDy) =Jy + Y«al\Dy- ~^) • Comparison with the y'-component of eq. (15.3) shows that 2 Hx. - I?, . Hz. = yu(Hr - uDy) , Dy. = Y„(ny - uHz/c ) . This process can be repeated by substituting the tachyonic transformations of d/dx, d/dy, d/dt and jz into the z-component of eq. (15.1). The resultant expression is 3H. a / uH„ 3 , x 3HX a/ uHv\ Yu ^r(Hy + uDz) - ^7 = ]'z + YU^DZ + -^j . and comparing it with the z'-component of cq. (15.3) shows that 2 Hx. = Hx , Hy. = Yu(Hy + uDz) , Dz. = Y„(DZ + uH/c ) . By substituting appropriate transformations into eq. (15.2) and comparing the result with cq. (15.4) it can be shown that Dx = Dx.. Hence the tachyonic transformations of D' and H' arc 2 2 Dx. = Dx , Dy. = Yu(Dy - "H,/c ) , Dz. = YU(DZ + "Hy/c ) (15.6) Hx. = Hx . Hy. = Yu(Hy + uDz) , Hz. = Yu(Hz - uDy) , (15.7) while the inverse tachyonic transformations are 2 2 Dx = Dx. , Dy = Yu(Dy. + uHz./c ) , Dz = YU(DZ. - uHy./c ) (15.8) Hx = Hx. , Hy = yu(Hy. - uDz.) , Hz = Yu(Hz. + uDy.) . (15.9) These transformations have exactly ihe same form in both S.R. and E.R. and so they arc valid for - °° < u < °°. This is the same result as for the transformations of E and B, and could have been obtained directly from equations (3.12) to (3.15) by making the substitutions listed in (15.5). In vector form the transformations (15.6) and (15.7) arc 2 2 D',, = (D + u x II/c )„ , D'j. = YU(D + u x H/c )i , (15.10) 70 H*„ = (H-ux D), . H'x = yu(H -uxD), , (15.11) where the subscripts II and X refer to components parallel and perpendicular to the boost u respectively. Of course (uxH)„ = (uxD), =0 . (15.12) but these terms have been explicitly included for the appearance of symmetry. Note that the axes perpendicular to the boost are imaginary, while the axis parallel to the boost is real. This conforms with earlier comments on the nature of various axes in E.R. 16: TRANSFORMATIONS OF P AND M. For stationary matter the polarization vector P is defined as the dipolc moment induced per unit volume of a dielectric due to the influence of an applied electric ficld(45). The relationship between P, D and E is expressed as P = D-e0E , (16.1) and is applicable to both stationary and moving frames. For stationary matter the magnetization vector M is defined as the magnetic dipolc moment induced per unit volume by an applied magnetic field. The relationship between M, B and H is expressed as M = J- -H , (16.2) which also applies in both stationary and moving frames. Now suppose that the medium is at rest in tachyonic frame £', so that its polarization is P = D'-E0E' , (16.3) ind its magnetization is M'= ^- -H' . (16.4) In the bradyonic frame X (laboratory frame) the apparent polarization is given by cq. (16.1) above. By substituting the 71 transformations of D and E from equations (3.14) and (15.8) into each component of P, it can be shown that P. = D.'-e.E,. , Py = Yu(Dy.-e0Ey.) -f^-H,.). UY /B . \ P* = Y„(Dr. - e„E,.) +-^-Hy.J. Comparing each of these three relations with the components of equations (16.3) and (16.4) shows that 2 2 Px = P„. , Py = yu(Py. - uMz./c ) . Pz = Y.(P«. + uMy./c ) (16.5) The inverse transformation of each component is 2 2 Px. = P„ , Py. = yu(Py + uMz/c ) , Pz. = YU(PZ - uMy/c ) . (16.6) Just as for the transformations of E, B, D and H, the transformation of the electric polarization is the same for both S.R. and E.R. This is to be expected considering how P is directly related to D and E. Now consider the magnetization of the material. In tachyonic frame Z' the medium is at rest so that its magnetization M' is given by cq. (16.4). The magnetization in bradyonic frame X is given by cq. (16.2). Substituting the transformations of B and H, given by equations (3.15) and (15.9) respectively, into each component of cq. (16.2) and comparing each of the resulting relations with the corresponding components of equations (16.3) and (16.4) shows that Mx = M„. . My = yu(My. + uPz.) , Mz = YU(MZ. - uPy.) . (16.7) The inverse transformations arc obviously uP Mx.= Mx , My = Yu(My " *) • Mz. = yu(Mz + uPy) . (16.8) Hence the transformation of the magnetization has the same form for both u2 < c2 and u2 > c2, as expected. The transformations of P' and M' can be written in vector form as 72 2 2 P'„ = (P - u x M/c ),, , ?\ = yu(P - u x M/c )x . (16.9) M'„ = (M + u x P)„ . M*x = YU(M + u x P)x . (16.10) The transverse axes arc of course imaginary, whereas the longitudinal axis is real. This is in agreement with the earlier discussion on the nature of tachyonic axes. An observer in tachyonic frame £' considers all of the quantities E'H, E'x, B'„, B'x, D'l|t D'x, H'H, H'x, P'n, P'x, M'n and M'x to be real, whereas a bradyonic observer in frame X considers all of the primed quantities with subscript "II" to be real and all primed quantities with subscript "1" to be imaginary. Conversely, the bradyonic observer in frame £ considers all of the quantities E„, Ex, BM, Bx, DH, Dx, H„, Hx, PH, Px, M,, and Mx to be real, but tachyonic observer X' considers all of the unprimed quantities with subscript "II" to be real and all unprimed quantities with subscript "1" tc be imaginary. 17: ELECTRIC D1POLE MOMENT OF A TACHYONIC CURRENT LOOP. In this section the electric dipole moment of a simple current loop moving with speed u > c will be investigated. As with much of the material in this paper, the following discussion is adapted f.om the corresponding relativistic case given by Rosscr(4^). Suppose there is a current-carrying coil at rest in frame X\ as shown in figure 14(a). For simplicity the coil is treated as a rectangle with area EB x CD = a' x b'. The coil is in the x'-y' plane and carries a conduction current i'c. The wire itself has a uniform rectangular cross-scctional area A', so that the current density inside the wire is J,=J/= *f. • (17.D The magnetic moment of the coil in frame X' is m'^n'i'.a'b' , (17.2) where n' is a unit vector normal to the plane of the coil and m' is real. For the coil illustrated in figure 14(a) the components of m' arc 73 nV = ny = 0, mz. = - i'ca'b'nz. , (17.3) where nz. is a unit vector in the z'-direction. Figure 14(b) shows the same coil as seen by bradyonic observer X In this frame the coil has speed u along the x-axis, so that due to length contraction (or dilation) the apparent length, a, of sides EB and DC is a = a"y 1 - j^T • for u2 < c2 , (17.4a) a = z'lyi - Jjf , for u2>c2 . (17.4b) Even for tachyonic objects all lengths arc positive and real: sec Paper I. For the transverse sides DE and BC the apparent length is b = b' for u2 < c2 , (17.5a) b = l-ib'l = b' for u2 > c2 . (17.5b) In frame X the wire's apparent cross-sectional area. A, is also affected by length contraction (or dilation): i) for sides EB and CD the area is A = A' for - oo < u < °°, (17.6) ii) for sides BC and DE the area is A = A"\j 1 - ^7 . for u2 < c2 , (17.7a) A = A' -\j\ - ^7 , for u2>c2 . (17.7b) Now assume that there is an equal number of positive and negative charges in each of the four sides of the coil, so that the effective charge density in frame X' is p' = 0. In frame X the electric charge density is P = Y»(p" + 7f) = "7^ • for u2 < c2 , (17.8a) / UJ.A iuY.J,. 2 2 P = iYu[p' + TTJ = —£*- • ^ u > c . (17.8b) 74 The total electric charge Q along the arm EB is apA, so that for u2 < c2: Q = apA = a"\/l " ^(^j*' = ^ • <17-9a> while for u2 > c2: I / u^| /iu .j,.^ sign( .)a'ui' Q = apA = a'|-y 1 - ^| (-^-JAT ' = ^ Y c . (17.9b) The y-rulc and i-y convention have been used in obtaining cq. (17.9b). As the current is in the opposite direction along the arm CD, the total electric charge along that arm in frame £ is a'ui' -Q = -—j-* for u2 a sign(Yu) 'ui'c -Q = ±-f foru2>c2. (17.10b) Therefore in frame X there is an effective charge separation between the arms EB and CD, and so there must be an electric dipolc moment in the y-dircction, given by py = Qb = apAb (17.11) for - oo < u < oo, with Q being given by cq. (17.9). The apparent magnetic moment as observed in frame X can be affected by switching, due to the possible reversal of the unit vector. In figure 14(a) the unit vector n* points out of the page. When the coil moves, as shown in figure 14(b), the arm BC leads DE in the increasing x-dircction for unswitched frames. However, in frames in which the coil appears to have undergone switching, the arm BC still leads DE but in the decreasing x-dircction. (See Paper I for a full discussion of this effect for rods). Hence the unit vector normal to the plane of the coil in frame X points out of the page for an unswitched coil and into the page for a switched coil: this is expressed as n = sign(Yu)n' . (17.12) 75 Using cq. (17.12) in (17.3) shows that the magnetic moment is , , , m, = - sign(y.)i ca b n, . (17.13) Combining the above relations for mz., b and apA gives the electric dipolc moment of the coil as a'ui'cb' urn,,. — 2 2 Py = ~2— = - ~^~ for u < c . (17.14a) sign^Wui^b1 for u2>c2 (,714b) Py = ^2 =-~^ - This result can be generalized, so that a current loop which is at rest in frame Z' has an electric dipolc moment in bradyonic frame Z equal to u x m' P =—cl (17.15) for - oo < u < «>. Here m' is the magnetic moment measured in frame Z' and u is the uniform velocity of frame Z' relative to frame Z- This example shows that the tachyonic transformations and switching give reasonable and internally consistent results. 18: ELECTROMAGNETIC 1NVARTANCES FOR TACHYONIC OBSERVERS. The electromagnetic field tensor F(lv in frame Z was defined in section 12 as 0 B, By " iEx/c Bz 0 B, - iEy/c F = ..(18.1) J1V 0 By -B, - iEz/c L iEx/c iEy/c iEz/c 0 It was shown in section 12 that the corresponding field tensor F'^ in tachyonic frame Z' could be found using cither the general transformation matrices such that 4 4 f^aP ~ L, L, ^an^v^ vp • ..(12.10) H=lv=l 76 or by simply transforming each component of F^ into the corresponding component of Fap using the transformations of the fields E and B. This was a direct result of the fact that the transformations of E and B arc the same for u2 < c2 and u2 > c2. In order to demonstrate that numerous clectrodynamic invarianccs hold for u2 > c2, the following four-tensors arc defined^47): 0 - iEz/c iEy/c iEL/c 0 • iE./c B F = y ..(18.2) iEy/c iEx/c 0 B, - B. - B„ -B, 0 Hy -icD, H 0 H, - icD, G.„. = z .(18.3) '(IV -icD "y 0 z icD. icD„ icD, 0 0 -icDz icDy H, 0 icD, G*..„ = "y (iv ..(18.4) - icDy icDx 0 - H, - H„ - H, 0 The tensors G^v and G*^ can be obtained from F^ and F* by using 2 2 the replacements B% -» Hx, By -» Hy, Bz -» H7, Ex -» c Dx, Ey -» c Dy and Ez -> 2 c Dz. The corresponding tensors in frame X' arc F^, F\v, G'^ and G'*(1V and arc defined similarly to equations (18.1) to (18.4), except that primed quantities replace unprimed quantities. It must of course be remembered that the Euclidean metric (+1, +1, +1, +1) is being used with coordinates given by (x, y, z, ict), and so there is no distinction made between covariant and contravariant quantities here. There arc six combinations of B, E, D and II which arc invariant for u2 > c7, all of which arc listed below. These arc the same six combinations which arc invariant for u2 < c2. 2 2 2 I I F V = 2B2.y= 2 B' -T=H F> V . .(18.5) (i=lv=l V L J \ C J H=lv=l 4 4 4iB.E 4JB.E' « « „, „,. y y p p* F .(18.6) (iv* nv • l-i i- ' UV* 11V ~: = - : = n=iv=XI i uv^ (l=1v=l X Z Z ZG^GV = -4icH.D = -4icH'.D' = £ IG;,G';, . (18.8) H= 1 v= 1 |i= I v= 1 G F G ivZFV ,v = -2i(cD.B + ^ = - 2i(cD\B' + ^j = £vt, V Vv • (18.9) Z Z F^G^ = 2(B.H - D.E) = 2(B'.H' - D'.E) =21 V* (1810) (1=1 V=I |i=lv=l Each of these six invariants holds for - °° < u < °°. In the Rccami formulation^2-'2) these quantities arc not invariant for u2 > c2, which is one of several points of difference between his work and the present formulation. The cause of the noninvariance in the Rccami formulation arc the factors of ±i in his electromagnetic field transformations, which result in minus signs appearing on the right hand side of equations (18.5) to (18.10) when transforming from bradyonic to tachyonic frames and vice versa. Using the above definitions of F^ and G^ allows Maxwell's equations to be written in a much more compact form. As the transformations of E, B, D, H and the four-tensors arc the same for u2 < c2 and u2 > c2, then it is obvious that the compact form of Maxwell's equations will also be valid for - oo < u < °°. The first pair of Maxwell's equations, given by equations (3.1) and (3.2), can be combined into a single tensor equation aFV0 3FOT 9Fav where v, a and a run from 1 to 4 and Xa = (x, y, z, ict). As Fap is an antisymmetric tensor and Faa = 0, there arc only four combinations of values for v, a and a which give physically significant equations: (1, 2, 3), (4, 2, 3), (4, 3, 1) and (4, 1, 2). The first combination (1, 2, 3) corresponds to cq. (3.1), while the other three combinations correspond to each component of eq. (3.2). The second pair of Maxwell's equations, given by equations (15.1) and (15.2), can be combined into a single equation »C = K' (,8-12) 78 where v = 1, 2 and 3 corresponds to the components of eq. (15.1) and v = 1 corresponds to eq. (15.2). The two compact equations (18.11) and (18.12) can be written in frame X' simply by replacing unprimed quantities with primed quantities. By substituting the S.L.T.s and the appropriate transformations of E, B, D and H into each component, it can be verified that these compact forms apply when u2 > c2, as well as for the standard S.R. case of u2 < c2. 19: CONSTITUTIVE EQUATIONS. Now consider a material medium for which the relative permeability and permittivity are not necessarily equal to one. In such a medium the values of the field quantities become dependent upon the medium's properties, while Maxwell's equations have to be supplemented by the constitutive equations in order to solve problems. In the discussion that follows, which has been adapted 4 from Rosser^**), ft WJH ^ assumed that Km, K and o can be considered as constants. Here Km is the relative permeability, K is the dielectric coefficient and a is the electrical conductivity. While this assumption is not appropriate for ferromagnetic materials, in which Km is not a constant but instead depends on H, there is a wide range of materials for which Km, K and o arc virtually constant. When the material is at rest in tachyonic frame X' the constitutive equations take the form: B'= Kmu0H' . (19.1) D' = Ke0E' , (19.2) j' = GE' . (19.3) Equations (3.17) and (15.10) give E' and D' in terms of components parallel (II) and perpendicular (1) to the boost. These allow cq. (19.2) to be written in component form as D',| = KE0E'H and D'^ = KE0E'J_ . (19.4) 79 These can be combined into a single equation as 2 D + u x H/c = KC0(E + u x B) , (19.5) which of course applies for - « < u < <». For u = 0 this reduces to the standard relation D = KE0E . (19.6) The parallel and perpendicular components of B' and H' are given by equations (3.18) and (15.11), so that for - °° < u < <*> the constitutive equation (19.1) becomes 2 B - u x E/c = Kmu0(H - u x D) . (19.7) The tachyonic transformations of j' and E' can be used to transform the third constitutive equation from the tachyonic frame Z' to the bradyonic frame Z. Using equations (3.12) and (4.5) gives each of the three components of cq. (19.3) as - iYuOx " up) = <*EX , (19.8a) - ijy = o-Yu(Ey - uBz) . (19.8b) - ij, = ayu(Ez + uBy) . (19.8b) These expressions can be written in vector form as - iyu(j - up),, = - i(j - up)x = o-Yu(E + u x B)x . (19.9b) Note that the tachyonic transformations cause the appearance of a factor of -i here, so that eq. (19.9) only applies for u2 > c2. The S.R. version is similar to eq. (19.9) except that the factor of -i docs not appear. It can be seen once again how the imaginary factors cancel out to give real quantities as measured in bradyonic frame Z. Note that as j is the total current density, it includes the convection current density up. Thus the term j - up is equal to the conduction current density. 80 The scalar and vector potentials are also modified in media for which KTO and K are not unity. Using the constitutive equations in cq. (3.8) gives 1 VxB =-$-%r + Kmnj' , (19.10) Using equations (8.3) and (8.4) and the identity Vx(VxA) = V(V.A) -V*A . (19.11) in eq. (19.10) leads to K K 2 m 3 A* f KmK 3d>'\ V'2A' " 1i" afr - V'(v'-A'+ "ST fv) = " w1 • (19.12) The Lorcntz condition is V.A'+ -~-^r = 0 , (19.13) so that eq. (19.12) becomes V'2A* " "S" 3^" = " K^o>' • (19M) From Maxwell's equations, if p' is the proper charge density at a point then V'.E'=-£-. (19.15) Substituting for E' from eq. (8.3) and using (19.13) leads to . w - ^ -••£• <'"-'« Equations (19.13), (19.14) and (19.16) apply for both bradyons and tachyons. The corresponding equations in bradyonic frame X can be found by uppriming all of the variables in the above expressions. 20: THE VELOCITY OF LIGHT IN A TACHYONIC MEDIUM. 81 In free space the velocity of light has been postulated to be equal to c for all incrtial reference frames, regardless of whether the frame is bradyonic or tachyonic. However, the speed of light in a material is modified by the properties of that material and is no longer equal to c. It is known that for a bradyonic medium with refractive index n greater than 1, the apparent speed of light is less than c, and as n increases the speed of light in that medium decreases. In this section the speed of light in a tachyonic medium will be investigated for a simple material in which Km = 1 but K * 1. The magnetization and polarization of such a medium will will also be investigated, using a discussion adapted from the rclativistic case given by Rosscr(49). Suppose that light is travelling through a material medium whose speed is such that u2 > c2 relative to the observer. In the tachyonic incrtial reference frame X' the medium is at rest and has a dielectric coefficient K * 1. It is assumed that the material is non-magnetic, so that the relative permeability is Km - 1. Hence M' = 0 and so the magnetic field is given by B' = n0H'. In bradyonic frame X the material has velocity u such that u2 > c2. The polarization in frame X is given by P„ = P'„ , (20.1a) Px = YuP'x - (20.1b) while the magnetization is M„ = M'H = 0 , (20.2a) . Mi = Yu(M' - u x P\ = - Yu(" " P')i = - (u x P)x . (20.2b) Note that equations (20.1) and (20.2) would also apply if u2 < c2. As (u x P')N =0 and P„ = P'„ in this example, then M(l can be written as Mn = - (u x P)„ = 0, which allows the magnetization vector to be written in a more general form as M = - u x P . (20.3) In frame X this gives 82 H=;r-M = ?•+uxP • <204> so that one of Maxwell's equations becomes V xH = V xfjj- + u x P ") = ^ + j . (20.5) Rearranging this expression leads to a d VxB = u/j+^x(Pxu)+ eo |" + -£\ . (20.6) where Pxu = -uxP and D = e„E + P. As a check it can be seen that setting P = 0 recovers one of Maxwell's equations. In tachyonic frame Z' the constitutive equation D' = KC0E' means that P' = D' - e0E' = E0(K -1)E' . (20.7) Combining this expression with the components of E' given by cq. (3.17) leads to P'„ = e0(K - 1)(E + u x B)„ , (20.8a) P\ = E0(K - DYU(E + u x B)x . (20.8b) Combining equations (20.1) and (20.8) shows that P„ = e0(K - 1)(E + u x B)„ , (20.9a) 2 P, = E0(K - l)yu (E + u x B)± . (20.9b) Equations (20.9a) and (20.9b) apply for - °° < u < «, as the equations and transformations used thus far in this discussion are the same for S.R. and E.R. The perpendicular component of the polarization has an extra 2 factor of YU compared to the parallel component, and so the 2 2 2 polarization is anisotropic. For u > c the factor (K - l)yu becomes negative, whereas it is positive for u2 < c2. Thus there is a clear difference in the possible values measured for Px for the different speed ranges u2 < c2 and u2 > c2. 83 For large relative speeds such that u2 » c2 the polarization becomes heavily anisotropic and greatly favours the direction parallel to the + boost. In the limit as u -» c the transverse polarization Px differs by a sign from the S.R. case in which u -» c-, while for u -> c+ the parallel polarization PM becomes virtually the same as for u -> c\ Note that the magnitude of P± is extremely large in the limits u -» c" and u ->c+. Most textbooks on Special Relativity which treat the problem of the speed of light in a material medium on'y present a first order 2 2 2 theory, for which u « c . In that case yu = 1 and the polarization 2 becomes virtually isotropic. However, the approximation yu = 1 is generally invalid in E.R. unless u2 = 2c2, and so in the present work the solution for terms in u2/c2 will be calculated. In order to simplify the following derivation, it is now assumed that the medium is uncharged so that p' = 0. The medium is nonmagnetic so that Km = 1 and the current in L' is j' = 0. Furthermore the medium, which is at rest in tachyonic frame £', is moving in the positive x-dircction with uniform velocity u relative to a bradyonic frame X. It is now assumed that there is a planc-polarizcd plane wave moving in the positive x-dircction parallel to u. The electric and magnetic vectors of the plane wave arc in the y- and z-dircction respectively. Hence for this case vx = u , vy = vz = 0 , B = Bz , E = Ey and so Pxu = - kpyu . (20.10) where k is a unit vector in the z-dircction. The only component of P which is nonzero is 2 Py= C0(K- l)yu (Ey - uBt) . (20.11) The y-componcnt of cq. (20.6) is 3BZ 3E^ 3P dPj = e + • 9x" <^,lt ^3t +H.u3x • (20.12) The only nonzero vomponcnt of cq. (3.2) is 84 3EV 3B, ftf = -lf • (2013) so that taking the partial derivative with respect to x gives 2 a Ev = ^B, 2 .(20.14) 3x " 3t3x ' The expression for Py given by cq. (20.11) can be substituted into (20.12) to give 2 1 B (K 1)Yu u(K 1)Yu v JL J_^ " /3E/dEj, 3BdB±\Z\ ' '•" /3E(dtyV 3BdB, z\ 3x ~ c2 3t + c3 'U " u irj + c^ [dx - u 3x~J ' 2 where e0u.0 = 1/c . Using eq. (20.13) to collect terms and then taking the partial derivative with respect to t gives 2 'YY " c2 J9x3t ~ c2 3t2 + c2 ^3t2 + Zu3x3t/1'. Combining this expression with cq. (20.14) and then rearranging terms leads to ( v± \ 2 K 2 2 3 EV _ " c ^ /2U(K - l)>a E 2 3x " KU' £ * (*ft^ • Equation (20.15) is valid under the above assumptions for - ~ < u < °°. In the limiting case of u -» 0 it reduces to 3xf = ^"3^' (2016) which is the ordinary wave equation for the propagation of light in a stationary medium having dielectric coefficient K. In the opposite limit of u -» oo cq. (20.15) instead reduces to 2 2 3 EV i 3 EV 3^=^-3/' (20,7) which only gives the same result as for the u -> 0 limit when the dielectric is actually the vacuum, for which K = 1. In the limit as in c or c+ cq. (20.15) reduces to 2 7 3 EV _ ^3^ 2 a -Ev 3x2 ~ ' c2 3l2 " c3t3x ' (2018) 85 As the electric and magnetic fields form a plane wave in this example, it is therefore assumed that the solution of cq. (20.15) is of the form Ey = E^xp^ni[£ - v t]V (20.19) with v = Xv being the velocity of the plane wave. The idea of a tachyonic plane wave has already been discussed in section 7(d), where it was pointed out that a plane wave in a tachyonic frame is also a plane wave when viewed from a bradyonic frame. Hence the solution (20.19) is expected to be valid for the full range of relative speeds - «> 2 2 2 2 2 2 2 3 EV 4TC V EV 9 EV 9 EV 4n v Ev 3x2 " v2 ' dt2 ~ 4TC v fcy ' 3t9x " v Substituting these expressions into eq. (20.15) and cancelling common factors leads to an expression which is a quadratic in v: >tD 2UV(K-1) C* - KU" C' - KU' The solutions of cq. (20.20) arc U(K - 1) ± cV^fl - ^f) _ _A .(20.21) * uz K - 77 where the upper signs apply for light travelling in the positive x- dircction and the lower signs apply for light travelling in the negative x-direction. The speeds v+ and v. are the speed of light inside a dielectric material as measured by a bradyonic observer £. The material itself travels with speed u relative to X such that 0 < u < °° for the upper signs and - °° < u < 0 for the lower signs. In the limit as u -* c* then the measured speed of light in the dielectric is v± -> + c, and in the limit as u -> - c* then v± -» - c as expected. When u is small such that u2 « c2, equation (20.21) reduces to the standard first order theory result for S.R.: 86 ••-*£••(«-;)• .(20.22) and as u -» 0 this gives v± -» ± c/v K . For a vacuum K = 1 and so eq. (20.15) gives the standard result v± = ± c. In the limit as u -* ±« the; V± -» ± C\ K . The dielectric coefficient K is related to the refractive index n via the relation n = ^K . ' (20.23) Here n is defined to be the refractive index of the medium when it is at rest, i.e. n is its "proper refractive index". Equation (20.21) can therefore be written as u(n2- 1)± ncfl - ^j\ v± = ^T l • (20-24) 2 n - ~z2 c where - <» < u < °°. This expression can be simplified still further by factorizing the denominator, so that = c(nu±c) t nc x u Note that a different speed range applies for each of these two equations, and that reversing the sign of u gives v+ = - v. Thus it can be seen that reversing the direction of motion of the medium has no effect on the magnitude of the speed of light in that material. This result is known to apply in S.R., and has now been extended to apply in E.R. When n = 1, i.e. the medium is a vacuum, equation (20.24) reduces to the required result v± = ± c. Eq. (20.25) agrees with the S.R. expression given by Bohm(50) for the observed phase velocity when light is travelling through a moving fluid of refractive index n and speed u. Figures 15(a), (b) and (c) arc plots of v+/c against u/c for three different values of the refractive index: (a) n = 1.33, (b) n = 1.60, (c) n = 2.00 . It can be clearly seen in each case that the speed of light in the material is a monotonically increasing function as u/c increases over the plotted range from 0 to 5. Each of the three curves passes 87 through the required point (1, 1) and has v+/c = 1/n for u/c = 0. As n increases the speed v+/c decreases for u/c < 1 but increases for u/c > 1. In each case the asymptotic limit is v+/c -» n for u/c -» ». 21; CONCLUSION. The new formulation of the theory of tachyons wc proposed in Paper I has now been extended to cover electrodynamics for tachyons. Throughout this work it has been our aim to develop a theory of tachyons which is logical and internally consistent, by extending and slightly modifying Special Relativity to cover a new range of allowed particle speeds. Simply by assuming that the two postulates of Special Relativity apply to all reference frames, regardless of whether they arc bradyonic or tachyonic frames, it was shown by rigorous derivations that the tachyonic transformations of E, B, D, H, P and M arc the same for u2 < c2 and u2 > c2. Not only arc the permittivity and permeability constants of free space the same for tachyonic and bradyonic observers, but the usual invarianccs for the electromagnetic field vectors also apply for the full speed range - °o < u < ». The tachyonic transformations of the electromagnetic field tensor and the stress-energy tensor were also shown to apply for - °o A derivation of the transformation of tachyonic charge and current densities showed that there arc two types of tachyonic transformations: those with +i when transforming from bradyonic frame X to tachyonic frame X' and those with -i instead. This point was developed further in section 11, wherein the generalized four- vector transformations were discussed and tachyonic transformations were divided into their generic types. The derivation of the transformation of charge and current densities also highlighted another feature of this formulation. It turns out that there is never any choice of signs in the transformations, even though the choice of whether to use +i in preference to -i at first appears to be quite arbitrary. The high degree of internal consistency required in this theory precludes the use of certain signs in many cases, so that ihcrc is no real choice in the signs used in various tachyonic transformations. As an example, using the other signs for the 88 tachyonic transformations of the charge and current densities would have resulted in the tachyonic permeability and permittivity constants of free space being negative. The correct choice of signs means that these constants arc the same for both bradyonic and tachyonic observers. A complete and rigorous derivation in section 5 showed that the y- rulc automatically leads to the conservation of total electric charge in each and every incrtial reference frame, regardless of whether some tachyons appeared to be switched. However, it turns out that the total electric charge measured in two different bradyonic reference frames is not the same when there arc several switched tachyons involved in the system. This presents no real problem, as it is always possible to transform to another reference frame in which all of the particles arc cither bradyons or unswitched tachyons, for which the total electric charge is an invariant. A detailed investigation of the electric and magnetic fields produced by an electrically charged tachyon travelling through a vacuum showed that the fields produced by the tachyon were real inside a Mach cone and imaginary outside the cone. As the edge of the cone measured the furthest extent that the field generated by the tachyon had reached, then it was clear that the imaginary field outside the cone was undetectable. This same example showed two more effects which arc solely due to tachyonic speeds. The edge of the cone is the Hlial point of contact between the observer and the tachyon's electromagnetic field, and at this contact the observer's instruments arc subject to an "optic boom". At the instant of contact the fields arc instantaneously infinite, but thereafter they arc finite and detectable. This means that the instruments register a sudden jump, analogous to the "sonic boom" effect produced by aircraft travelling at supersonic speeds. The second effect produced by lachyons is called the "two source effect", wherein the tachyon appears to the observer to be in two scpaiaic locations which arc moving directly awiy from each other. The second image is in fact a delayed image due to the finite transmission time of the electric and magnetic fields, which of course propagate with speed c according to the second postulate. Both the "two source effect" and the "optic boom" have heen theoretically demonstrated by RccamK8.9.10), and 89 they arc a feature of all formulations of the theory of tachyons. It was also shown that a test charge cannot tell the difference between a charged tachyon going in one direction and its switched counterpart travelling in the opposite direction. Of course, allowance must be made for the relative speed of the bradyonic observers. Section 7 covered some topics in tachyonic optics, including the Dopplcr effect, aberration and plane waves. It was shown that tachyons exhibit a blucshift for approach and rcdshift for recession, just like bradyons. Negative frequencies indicated that the light was being received in reverse order to the order of emission. The transverse Dopplcr effect for tachyons was shown to be a rcdshift for u The scalar and vector potentials for tachyons were shown to transform in a similar manner to the S.L.T.s. The potentials produced by an electrically charged tachyon were shown to have the same behaviour as for the fields: they were real and detectable inside a Mach cone, but were imaginary and undetectable outside the cone. The retarded potentials also had to be modified to allow for superposition due to the two source effect. Sections 9 and 10 developed Lagrange's equations and Hamilton's equations respectively for a charged tachyon. The classical tachyonic Hamiltonian discussed here has given an indication of the correct Hamiltonian to be used in a development of quantum mechanics for charged spin zero tachyons, which is presently the subject of an investigation by the authors^'). The discussion in these sections also served to demonstrate the high degree of internal consistency in this formulation. Whereas the material in sections 3 to 12 applied for tachyons in a vacuum, the material in sections 15 to 20 extended the development of this theory to cover tachyons moving in media, and in some circumstances for media themselves to have tachyonic speeds relative 90 to the observer. This included the proof that the tachyonic transformations of the electromagnetic field vectors D, H, P and M arc the same as their rclativistic transformations. In order to show how various tachyonic transformations combine to give the correct results, section 17 comprised an investigation into the electric dipolc moment of a tachyonic current loop. Even when the loop appeared to be switched the correct transformation of the electric dipolc moment was obtained. A discussion of the constitutive equations for tachyonic materials has also been given for the simple case in which the relative permeability tcm, the dielectric coefficient K and the electrical conductivity a were all constant. The discussion in section 20 consisted of an investigation of the polarization of a tachyonic dielectric medium, followed by a derivation of the speed of light in that same material. It was shown that the speed of light in such a dielectric is a well-behaved function with the asymptotic limit v+/c -» n for u/c -» °°, where n = rK is the refractive index. Throughout this work it has been shown that Special Relativity can be extended in a logical and consistent manner to include particles having speeds greater than c, simply by using the postulates of Special Relativity. Tachyons obey conservation of energy, momentum and electric charge as well as Maxwell's equations and other clcctrodynamical relations. In many cases the formula describing a tachyonic quantity is the same as the rclativistic equivalent: only the numerical values due to the high speeds involved distinguish between a tachyonic and a bradyonic system. Rules for tachyonic transformations have been given and various examples of four-vectors have been worked through, frequently confirming results obtained using other methods. At many stages in both this paper and in Paper I it has been mentioned that various results arc logical, consistent and well- behaved. It was our intention to develop a detailed formulation of the theory of tachyons which was logical and internally consistent, yet still allowed tachyons to interact with ordinary matter. While these aims have certainly been achieved, it must be pointed out that this formulation is subtly different to others, notably the one proposed by 91 Rccami ct al/7). If allowance is made for the different metrics used, the form of the transformations of spacctimc coordinates, velocities, momenta, energies and forces arc all similar in the two formulations, only differing in signs on some components. The differences have arisen due to the way these two formulations have been set up: in the present work these transformations arc derived from the fundamental postulates, whereas Recami considers symmetry arguments. However, when dealing with electrodynamics the differences between the two formulations become more pronounced. For example, the transformations of the electromagnetic field vectors E, B, D and H arc different in the two formulations, which of course leads to considerably different conclusions. What has been described here as an electrically charged tachyon is a tachyonic magnetic monopolc in Rccami's work^2'1^). This difference in ideas as to the nature of charged tachyons also carries over into how tachyonic scalar and vector potentials arc handled in the two formulations. The present formulation maintains that Maxwell's equations arc the same for bradyons and tachyons, which has resulted in a rigorously developed theory of tachyonic clcctromagnctism. In Rccami's formulation, terms involving tachyonic charge and current densities arc added to Maxwell's equations in order to symmetrize them. There arc other differences between the two formulations, for example in the interpretation of the switched particle. Rccami icgards switched tachyons as being "antitachyons", but in the present formulation they have been shown to have subtly different properties to those one would expect of a true antiparticlc. These different interpretations arc merely separate solutions to the problem of allowing tachyons to obey the laws of conservation of energy, momentum and electric charge in all incrtial reference frames. Another consequence of there being different interpretations on the apparent behaviour of tachyons involves the question of whether tachyons arc localized particles relative to bradyonic observers. They arc certainly localized relative to tachyonic observers, but the two source effect means that tachyons appear to be in two mutually receding places at the same time relative to a bradyonic observer. It is the view of the authors that the second image is an artefact caused by the speed of the source being greater than the speed of the information as it travels to the observer, in much the same way 92 as an aircraft travelling at a supersonic speed can sound to the observer as though it is in two places. Yet another difference of interpretation between the two formulations concerns the apparent shape of a tachyon which is a sphere when viewed in its own rest frame. When viewed by a bradyonic observer, Rccami considers such a sphere to appear to occupy the whole space bounded by a double, unlimited cone and a two-shected hypcrboloid connected at a point (42). In the formulation proposed here the tachyon would instead appear to the bradyonic observer to be an ellipsoid which undergoes various degrees of elongation according to the relative speed of the observer and the tachyon. The technique adopted here of making minimal extensions of Special Relativity into Extended Relativity has been successful in developing a logical and internally consistent theory of tachyons. The authors plan to continue this approach in future papers on tachyons, which will include discussions of such topics as fluid dynamics, shock waves, quantum mechanics and various aspects of plasma physics. ACKNOWLEDGEMENTS: One of us (RLD) wishes to acknowledge the receipt of a University of Melbourne Postgraduate Research Scholarship. REFERENCES. (1) R. L. Dawc and K. C. Mines: University of Melbourne Preprint UM- P-90/35, submi'led to Nuovo Citicnto. (2) E. Rccami: Riv. Nuovo Cimcnto, 9(4), (1986); quote taken from p. 159. This paper is the second of Rccami's seminal reviews on his theory of lachyons, and will henceforth be referred to here as Review II. (3) 0. M. P. Bilaniuk and E. C. G. Sudarshan: Phys. Today, 22(5), 43 (1969). (4) II. C. Corbcn: Intern. J. Thcor. Phys., 15, 703 (1976). 93 (5) H. C. Corbcn: Nuovo Cimento. 29A, 415 (1975). (6) H. C. Corbcn: in "Erice-1976: Tachyons. Monopoles and Related Topics", edited by E. Rccami, (North-Holland, Amsterdam), 31 (1978). (7) Recami has written an extensive number of papers detailing his formulation of the theory of tachyons, many with coauthors such as R. Mignani, G. D. Maccarronc, M. Pavsic, P. Caldirola, A. Castellino, V. Dc Sabbata and W. A. Rodrigucs Jr. The material in Rccami's papers is summarized in his Review II (sec rcf. (2) above), and also in E. Rccami and R. Mignani: Riv. Nuovo Cimento, 4, 209 (1974): Erratum, 4, 398 (1974). These review articles also contain an extensive list of references in their bibliographies. (8) E. Rccami: sec rcf. (2), section 6.15. (9) E. Rccami and R. Mignani: Lett. Nuovo Cimento, 4, 144 (1972). (10) R. Mignani and E. Rccami: Nuovo Cimento A, 14, 169 (1973). (11) E. Rccami: sec rcf. (2), section 5.12. (12) E. Rccami and R. Mignani: Riv. Nuovo Cimento, 4, 209 (1974). This paper is the first of Rccami's two review articles, and will henceforth be referred to as Review I. (13) R. Mignani and E. Rccami: Nuovo Cimento, 30A, 533 (1975). (14) E. Rccami: sec rcf. (2), section 15.1. (15) E. Rccami and R. Mignani: Phys. Lett. B, 62, 41 (1976). (16) E. Rccami and R. Mignani: in "The Uncertainty Principle and Foundations of Quantum Mechanics", cd. by W. C. Price and S. S. Chissick, (John Wiley, London), (1977), chapter 4, p. 321. (17) P. Lorrain and D. R. Corson: "Electromagnetic Fields and Waves", 2nd. edition, (W. H. Freeman & Co., San Francisco), (1970), section 5.19. (18) E. Rccami and R. Mignani: sec rcf. (12), section 10.3. 94 (19) R. L. Dawc, K. C. Hincs and S. J. Robinson: Nuovo Cimcnto, 101 A, 163 (1989). (20) R. Rcsnick: "Introduction to Special Relativity", (John Wiley & Sons, Inc.). section 4.4 (1968). (21) W. G. V. Rosscr: "An Introduction to the Theory of Relativity", (Buttcrworths, London), section 7.4 (1964). (22) E. Rccami: sec rcf. (2); quote taken from p. 155. (23) E. Rccami: sec rcf. (2); figure 15 on p. 53. (24) R. Mignani and E. Rccami: Nuovo Cimcnto A, 14, 169 (1973) and Erratum, 16, 208 (1973). (25) T. M. Hclliwcll: "Introduction to Special Relativity", (Allyn and Bacon, Inc., Boston), (1966); section XII B. (26) E. Rccami and R. Mignani: sec rcf. (12), section 17. (27) E. Rccami and R. Mignani: sec rcf. (12), figure 23. (28) H. Muirhcad: "The Special Theory of Relativity", (MacMillan), (1973), p. 72 - 73. (29) H. Muirhcad: ibid, p. 70-71. (30) C. Miller: "The Theory of Relativity", 2nd. edition, (Clarendon Press, Oxford), (1972). section 2.9. (31) R. Rcsnick: ibid, p. 84 - 87. (32) W. K. H. Panofsky and M. Phillips: "Classical Electricity and Magnetism", 2nd. edition, (Addison-Wcslcy, Inc.), (1962), section 11.5. (33) E. Rccami: sec rcf. (2), sections 10.5 and 15.2. (34) W. G. V. Rosscr: ibid, section 9.4. (35) W. G. V. Rosscr: ibid, section 6.4. (36) D. F. Lawdcn: "An Introduction to Tensor Calculus, Relativity and Cosmology", 3rd. edition, (John Wiley & Sons, Inc.), (1975), chapter 4. 95 (37) C. Mfllcr: sec ref. (30), section 5.9. (38) D. F. Lawden: sec ref. (36), section 29. (39) J. D. Jackson: "Classical Electrodynamics", 2nd. edition, (John Wiley & Sons, New York), (1975), p. 605. (40) H. Muirhcad: ibid, sections 4.1 and 6.2.2. (41) E. Rccami: see rcf. (2), chapter 8. (42) E. Rccami: sec rcf. (2), section 8.2. (43) E. Rccami: sec rcf. (2), section 8.4. (44) W. G. V. Rosscr: ibid, section 8.5. (45) W. G. V. Rosscr: ibid, section 8.6. (46) W. G. V. Rosscr: ibid, section 8.4. (47) W. G. V. Rosscr: ibid, sec section 10.4 for the definition of tensors and also p. 396 for the list of invariants. (48) W. G. V. Rosscr: ibid, sections 8.6.3 and 9.1. (49) W. G. V. Rosscr: sec rcf. (1), section 8.6.4. (50) D. Bohm: "The Special Theory of Relativity", 2nd. printing, (Addison-Wcslcy), (1989), p. 75 - 76. (51) R. L. Dawc and K. C. Hincs: in preparation. FIGURE 1: The exchange of a tachyon T between two bradyonic objects X and Y, as seen by observers X and X'. Observer X, using axes (x.ict), sees T as an unswitchcd tachyon T+ carrying positive energy, positive momentum and charge +Q from X to Y. Observer X', using axes (x'.ict'), sees T as a switched tachyon T. travelling from Y to X with positive energy (sec text in section 2.2), negative momentum and electric charge -Q. At the X-vcrtex, X sees the reaction X( -> Xf + T+ but X' sees the reaction X, + T. -> Xf. At the Y-vcrtcx, X sees the reaction Y( + T+ -> Yr but X' sees the reaction Y; -» Yf + T_. FIGURE 2: A collision between tachyons as represented by a Minkowski diagram. Observer X0 using axes (x0, ict0) sees the collision T, + T2-» T3++ T4 wilh all tachyons being unswitchcd. Observer X' using axes (x', ict') sees the reaction T, + T2 + T3. -> T4 as tachyon T3 is switched. This means that tachyon particle number is not conserved if one or more tachyons appears to be switched. Observer XT using axes (xT, ictT) sees all of the particles as bradyons in the reaction T, + T2-»T3 + T4. FIGURE 3: An electromagnetic envelope of a charged tachyon traveling with speed u forms a "Mach cone" with scmi-vcrtcx angle given by sin8 = c/u. The circles represent the wavefronts of the field generated by the tachyon T as it passed through the points 1, 2, 3, ... . For all points inside the cone E and B arc real and detectable, whereas for all points outside the cone E and B arc imaginary and undetectable as the field has not yet arrived at that point. The "Mach cone" produced by a charged tachyon gives rise to the electromagnetic "two source effect". An observer at A detects the wavefronts emitted by the tachyon when it passed through positions 1 and 2. An observer at B delects the wavefronts radiated from positions 2 and 3, while. C detects the wavefronts radiated from positions 3 and 4. FIGURE 4; A tachyon T+ carrying charge + Q and with speed 2c approaches a test charge + q, which is at rest relative to the observer. The circles arc electromagnetic wavcfronts emitted by T+ as it passes through each of the numbered points. The wavcfronts will arrive at the test charge after T+ has already gone past, so the test charge is unaffected during the approach of Tt. The field outside the cone is imaginary and undetectable as it has not reached the test charge. FIGURE 5: The wavcfronts emitted by T+ during recession arc received by the test charge in the same order as their order of emission. The test charge continues to experience the two source effect. The field inside the cone is real and detectable. (The deviation in the paths of T+ and the test charge have been ignored for simplicity here.) FIGURE 6: A switched tachyon T. with apparent charge - Q and speed - 5c/2 approaches the test charge from the right. In this new frame the test charge still has charge + q but now has speed - 3c/4 relative to the observer. Again it can be seen that the wavcfronls will arrive at the test charge after T. has gone past, so the test charge is unaffected during the approach of T.. (Note that the horizontal scale has been expanded by the amount necessary to counteract Lorcntz contraction in this and the following figure.) FIGURE 7: Even after T. has gone a long distance past, the test charge still experiences the two source effect, just as it did for the field produced by T+. The wavcfronts emitted by T during recession arc received by the test charge in the same order as their order of emission. (Again the deviation in the paths of T and the test charge have been ignored as an unnecessary complication in these diagrams.) FIGURE 8: A Minkowski diagram illustrating both the optic boom and the two source effect produced by tachyons. The tachyon travels along the worldlinc through the numbered sequence, and at each point it emits a pair of photons. Both bradyonic observers X and X', using axes (x, ict) and (x\ ict') respectively, do not see the tachyon as it approaches but see two apparent images after the initial optical contact as the tachyon recedes. The intersection of the photon worldlincs indicates that both £ and Z' sec two tachyons receding from each other. Observer Z receives photons emitted during the approach of the unswitched tachyon T+ in the order 6, 4, 3 (reverse order) and receives photons emitted during the recession of T+ in the order 9, 10, 12, 13 (same order). Observer Z' receives photons emitted during the recession of the switched tachyon T. in the order 6, 4, 3 (reverse order) and receives photons emitted during the approach of T. in the order 9, 10, 12, 13 (same order). FIGURE 9: A photon is emitted from a source at rest in frame Z'. The photon has energy u', frequency v' and travels at angle 8' with respect to the x'-axis. In frame Z the photon has angle 0 with respect to the x-axis and has energy w and frequency v. FIGURE 10: The Dopplcr effect for bradyons (u/c < 1) aid tachyons (u/c > 1). The "+" or "-" in brackets indicates whether the tachyonic source is switched: + for unswitched, - for switched. For relative approach 6=0 and the observed frequency is always blucshiftcd. For relative recession 9 = n and the observed frequency is always rcdshiflcd. For transverse motion 0 = ± n/2 and the observed frequency is redshifted for 0 < u/c < and blucshiftcd for V 2 < u/c. Negative frequency indicates that the light is received in reverse order to its order of emission. FIGURE 11: The scalar and vector potentials of a tachyonic point charge moving with speed u along the common x«-»x' axes. In (a) the actual position 01 the charge is at G at the time t = 0 when the potentials arc calculated at the origin O in frame S- The information collecting sphere which carries the information from the charge to the origin at time t = 0 leaves the retarded position F at a time t - r/c. During the time interval in which the information regarding the potentials travels from F to 0 the charge has moved from F to G. In (b) the charge is at rest in frame X', which has a uniform speed u relative to X along the common x<->x' axes. The origins of L and X' arc coincident at t = t' = 0. FIGURE 12: The origin of the two source effect for scalar and vector potentials generated by a charged lachyon. The tachyon intersects the information collecting sphere at two points. The earlier intersection is on the left, and in the time it takes the information collecting sphere to partially contract towards O at the speed of light the tachyon has crossed the sphere and intersected it on the right. At the instant the potentials arc measured at the origin at time t = 0 there arc two contributions, one each from the retarded positions, while the tachyon itself has exited the sphere. FIGURE 13: A Minkowski diagram representing a tachyonic electron ejn+ emitting a photon y. Superscripts indicate the apparent charge on the tachyon, while the subscripts indicate whether the tachyon is incoming (in) or outgoing (out) and whether it is unswilchcd (+) or switched (-). Observer Z0 using axes (x0, ict0) sees only unswiiched tachyons, so the observed reaction is Cjn+ -» c„ul+ + y. Observer X' using axes (x', ict') sees part of the lachyon's worldlinc as having undergone switching, so that the observed reaction is Cjn+ + Cjn. -> y. Note that cin. is not a positron as only one photon is emitted in this reaction, not two photons as in a normal electron-positron annihilation. FIGURE 14: The electric dipole moment of a moving coil. In (a) the coil is at rest in frame X' and carries a conduction current i'c. The wire is a rectangle of sides a' and b' and has a uniform cross- sectional area A'. Frame X' has speed u relative to frame X, so that in (b) the coil has speed u relative to X- In this frame the coil has an electric dipole moment due to an apparent separation of charge along the arms EB and CD. The possibiity that the coil is switched in frame X if u > c means that the unit vector normal to the plane of the coil may appear to reverse direction. FIGURE IS: Plots of the observed speed of light in a dielectric v+/c against the relative speed of the observer u/c for three different values of the refractive index n. In each case v+/c < 1 for u/c < 1 and v+/c > 1 for u/c > 1. 'Light Cone >-x Figure 1 Figure 2 Figurr Z • Test Charge Figure 4 Figure 5 v . , • 11 Test Charge Figure 6 Figure 7 • , . V (adjusted) Worldline of tachyon T Figure 8 Figure 9 —I 1 1—1— y i i 1 1 1 1 r - - 3 " \ e = o<-) .y' XL « CD CO 2 ^ e= + -(+) 1 /e=o 2 UJ y' ~ .y ^ e = +^ -"~"'—. 2 /' CO *--. ^ / *--.. \ O ---. \ / e=*(+f V) i o "O (0 t-._ 0 cc DC >. -- u c \ 5 -1 cr *\. CO -2 - a \2 m /e = 0(+) i i i -A-. i ,, i_ .J i i i - \ 1 2 Relative Speed U/c Figure 10 yA (x ,y ,z) (b) Position at time t -- of the c information collecting sphere Figure 11 which reaches 0 at t = 0 u - . • " (Earlier) (Later) Retarded Retarded position position Actual position 1 of charge at t = 0 \ / N / \ -t—>• \6H \ •4—«• ES Information collecting Earlier sphere V sphere contracts toward A ^: 0 with speed c / Later sphere Figure 12 /''Light / Cone Figure 13 " I ^ z u u <• + + ++ + +>• B UNSWfTO) firtSW0s (a) (b) Figure 14 (a) Figure 15 (b) (c)