SUPERLUMINAL REFERENCE FRAMES AND TACHYONS
RODERICK SUTHERLAND
Master of Science Thesis (Physics)
University of N.S.W.
1974. This is to certify that the work embodied in this thesis has not been previously submitted for the award of a degree in any other institution. A B S T R A C T
A theoretical investigation is made to determine some likely properties that tachyons s~ould have if they exist. Starting from the Minkowski picture of space-time the appropriate generalizations of the Lorentz transformations to transluminal and superluminal transformations are found.
The geometric properties of superluminal 3-spaces are derived for the purpose of studying the characteristics of tachyons relative to their rest frames, and the geometry of such spaces is seen to be hyperbolic. In considering closed surfaces in superluminal spaces it is found that the type of surface which is symmetric under a rotation of superluminal spatial axes
(i.e. which is analogous to a Euclidean sphere) is a hyperboloid, and is therefore not closed. This implies serious difficulties in regard to the structure of tachyons and the form of their fields. The generalized expressions for the energy and momentum of a particle relative to both subluminal and superluminal frames are deduced, and some consideration is given to the dynamics of particle motion (both tachyons and tardyons) relative to superluminal frames. Also, the superluminal version of the Klein-Gordon equation is constructed from the expression for the 4-momentum of a particle relative to a superluminal frame. It is found that the tensor formulation of subluminal electromagnetism is not covariant under a transluminal Lorentz transformation. Therefore in order to obtain the superluminal form Df Maxwell's equations it is necessary to generalize the relativistic electric and magnetic
field transformations and then use these in transforming each of Maxwell's equations separately via a transluminal transformation.
Finally, an attempt is made to deduce the electromagnetic field surrounding a uniformly moving tachyon under the assumption that the field is static in the particle's rest frame (i.e. assuming no emission of radiation). However it is found that no physically realistic solution appears to exist due to the incompatibility of radial symmetry and localization. CONTENTS
SECTION 1 INTRODUCTION 1
SECTION 2 TRANSLUMINAL LORENTZ TRANSFORMATIONS 2
SECTION 3 PROPERTIES OF SUPERLUMINAL REFERENCE FRAMES 7 3.1 SUPERLUMINAL METRIC SIGNATURE 7 3.2 THE ASYMMETRY OF SPACE-TIME 8 3.3 SUPERLUMINAL LORENTZ TRANSFORMATIONS 9 3.4 GEOMETRY OF SUPERLUMINAL SPACES 10 3.5 HYPERBOLIC TRIGONOMETRY 12 3.6 ROTATION OF SUPERLUMINAL SPATIAL AXES 15 3.7 PERPENDICULAR LINES 16 3.8 GENERALIZED HYPERBOLIC ANGLE 17
SECTION 4 FORM OF A TACHYON'S SURFACE 19 4.1 RADIALLY SYMMETRIC SURFACE 19 4.2 APPEARANCE OF A TARDYON RELATIVE TO A SUPERLUMINAL FRAME 20 4.3 APPEARANCE OF A TACHYON IN ITS REST FRAME 22 4.4 AXIAL SYMMETRY ABOUT THE DIRECTION OF MOTION 23 4.5 APPEARANCE OF A TACHYON RELATIVE TO A SUBLUMINAL FRAME 26
SECTION 5 VOLUME AND AREA IN A SUPERLUMINAL FRAME 29
5.1 VOLUME 29 5. 2 AREA 32
SECTION 6 ENERGY AND MOMENTUM 36 6.1 ENERGY AND MOMENTUM RELATIVE TO A SUBLUMINAL FRAME 36 6.2 ENERGY AND MOMENTUM RELATIVE TO A SUPERLUMINAL FRAME 37 6.3 VELOCITY REGIONS IN A SUPERLUMINAL SPACE 40
SECTION 7 THE ELECTROMAGNETIC FIELD OF A TACHYON 44 7.1 PRELIMINARY DISCUSSION 44 7.2 POINT TACHYON SOLUTION VIA THE KLEIN GORDON EQUATION 46 7.3 SUBLUMINAL ELECTROMAGNETIC FORMALISM 49 7.4 GENERALIZATION TO SUPERLUMINAL FRAMES 52 7.5 SUPERLUMINAL MAXWELL EQUATIONS 53 7.6 SUPERLUMINAL 4-POTENTIAL 57 7.7 PHYSICAL SIGNIFICANCE OF E. AND Ei 59 -l 7.8 POINT TACHYON FIELD RELATIVE TO A SUBLUMINAL FRAME 63 7.9 FIELD BOUNDARY 65 7.10 FIELD FOR NON-ZERO RADIUS 66
SECTION 8 DISCUSSION AND CONCLUSIONS 69 8.1 SUMMARY OF FINDINGS 69 ..,, 8.2 CERENKOV RADIATION FROM TACHYONS 72 8.3 CAUSAL LOOP PARADOX 76 8.4 QUANTUM MECHANICAL APPROACH 79
APPENDIX A: SPACE-TIME AND MINKOWSKI DIAGRAMS 80 APPENDIX B: GENERALIZED 4-VECTOR TRANSFORMATIONS 85 APPENDIX C: 88
REFERENCES 104
FIGURES 1-48 106 6.3 VELOCITY REGIONS IN A SUPERLUMINAL SPACE 40
SECTION 7 THE ELECTROMAGNETIC FIELD OF A TACHYON 44 7.1 PRELIMINARY DISCUSSION 44 7.2 POINT TACHYON SOLUTION VIA THE KLEIN GORDON EQUATION 46
7.3 SUBLUMINAL ELECTROMAGNETIC FORMALISM 49 7.4 GENERALIZATION TO SUPERLUMINAL FRAMES 52 7.5 SUPERLUMINAL MAXWELL EQUATIONS 53 7.6 SUPERLUMINAL 4-POTENTIAL 57
7.7 PHYSICAL SIGNIFICANCE OF E. AND Ei 59 -l 7.8 POINT TACHYON FIELD RELATIVE TO A SUBLUMINAL FRAME 63 7.9 FIELD BOUNDARY 65 7.10 FIELD FOR NON-ZERO RADIUS 66
SECTION 8 DISCUSSION AND CONCLUSIONS 69
8.1 SUMMARY OF FINDINGS 69 .., 8.2 CERENKOV RADIATION FROM TACHYONS 72 8.3 CAUSAL LOOP PARADOX 76 8.4 QUANTUM MECHANICAL APPROACH 79
APPENDIX A: SPACE-TIME AND MINKOWSKI DIAGRAMS 80 APPENDIX B: GENERALIZED 4-VECTOR TRANSFORMATIONS 85 APPENDIX C: 88
REFERENCES 104
FIGURES 1-48 106 1
SECTION 1 INTRODUCTION
In recent times a great deal of interest has been shown in the possibility that particles travelling faster than the speed of light may exist (Ref. 1 is a bibliography on this subject). At present the most important task in this field of research is to devise methods of detecting such particles, which have been given the general name of "tachyons". If tachyons exist then presumably they can be detected through their field interactions with normal matter. This means that theories describing the form of their fields must first be considered so that meaningful experiments can be devised. It is always easiest to derive the field of a particle in its rest frame, since in this frame the form of the field is simplest and most symmetrical. In the case of tachyons the rest frames are travelling faster than light and cannot be attained by any observer in our "slower than light" world. (These new frames are called "superluminal" reference frames as opposed to the usual "subluminal" frames.) Nevertheless they still provide the conceptually simplest starting point for deducing the fields of tachyons in their general form, and for this reason the properties of superluminal frames will now be examined in detail. Space-time diagrams of the Minkowski type (see Appendix A) will be used fairly extensively in the following work to illustrate geometrically the aspects considered. 2
SECTION 2 TRANSLUMINAL LORENTZ TRANSFORMATIONS
To transform from subluminal to superluminal frames the Lorentz transformations
vx t - 2 x-vt C x' = Ji-:~ , y I =y, Z I = Z, t I = .... ( 1) will simply be generalized to velocities greater than c. It is important to have a set of unambiguous transformation equations from the outset because confusion can arise from the mixing of real and imaginary quantities which occurs in superluminal transformations. The equations above are satisfactory for transforming between two subluminal or two superluminal frames, but care must be taken when they are employed for transforming across the light barrier. For v>c and x and t real, the equations yield imaginary values for x' and t' since /1-½ becomes C imaginary. This suggests that the x' and t' directions do not lie in the x, t space-time plane. Such a conclusion is wrong however and the transformation simply represents a rotation in this plane. The source of the confusion lies in the assumption that both x and tare real. Whilst it is permissible to adopt this convention, it should be kept in mind that the actual space-time distances, or intervals, along the x and taxes are real and imaginary respectively (or vice versa). The "i" which appears in the equation for x' indicates that if x is real then x' is imaginary, and vice versa, and the same holds true fort' and t. 3
The geometric significance of these transformations becomes clearer if we introduce the new time variables T=ict and T'=ict' which directly represent intervals along the time axes. Equations (1) become
VT ivx x--.- T--- lC C x' = , y I =y' z I= z' TI = •••• ( 2) Ji-~ Since x and tare both real these equations yield x' real and T' imaginary for v
Fig. 1 depicts the rotation of axes for v According to equations (3) there is a discontinuity in the rotation as we pass from v t X ....X -t i/ ---=-zV C x' = t' = ' ~ JI VL. CL. I ITTC Hence for v= 00 we have X x' = - t t' = ' C This shows that for an infinite change in velocity in the. x direction the x and taxes of the two frames are interchanged. However the minus signs indicate that the x' and t' axes have also flipped over to the opposite direction in crossing the light line rather than rotating continuously through it. The rotation of axes for v>c is shown in Fig. 3. There is an inherent ambiguity in sign in the Lorentz transformations due to the presence of the square root. For transformations between subluminal frames we always choose the positive root so that in the limit v+o we have x'+x and t'+t rather than x'+-x and t'+-t. For transformations across the light barrier however, the negative root is preferable since it corresponds to continuous rotation. We will therefore adopt the positive root for subluminal transformations and the negative root for transluminal transformations, so that the rotation of axes is as shown in Fig. 4. Note that it is not correct to represent this rotation as in Fig. 5 (by analogy with Fig. 2). As v passes through c the x't' frame undergoes an absolute change from a subluminal to a superluminal frame, the latter being geometrically different, 5 and hence distinguishable from, subluminal frames. Fig. 5 actually represents the acceleration of the xt frame through -c, it thereby changing to a superluminal frame. Fig. 4 is equivalent to Fig. 2 for v x=+Y(x-vt) , y'=y , z'=z , t'=+Y(t-~) , .... ( 4) C and the transformations for subluminal to superluminal are x=-Y(x-vt)-- , y=y , z=z , t=-Y(t---r- vx) , .... ( 5) C V2, -½ where Y = + I1 - 2 . It is more convenient to insert the C bars on the subluminal variables rather than vice versa since the following work is mainly concerned with superluminal variables. The inverse transformations for the two cases above can be derived from equations (4) and (5). They are x=+ y (x' +vt') ' y=y' ' z=z' ' t=+y (t' +~) ..•. ( 6) C and x=+Y(x+vt) , y=y , z=z , t=+Y(t+~) , •••• ( 7) C respectively. The x and taxes of a superluminal frame moving in the x 6 direction are time-like and space-like respectively relative to any subluminal frame whilst they and z axes are unchanged. The foregoing transluminal Lorentz transformations are derived and discussed in more detail in Appendix C. 7 SECTION 3 PROPERTIES OF SUPERLUMINAL REFERENCE FRAMES 3.1 SUPERLUMINAL METRIC SIGNATURE For the subluminal Lorentz transformations with ----x,y,z,t all real the invariant interval ds is given by (see Appendix A) 2 I 2 -2 -2 -2 -2 ds = c dt - dx - dy - dz 1 , and hence the metric signature is (+---) or(-+++). Transforming this expression using equations (7) we obtain ds 2 = lc2Y2 (dt + ;- dx) 2-Y 2 (dx + vdt) 2 - dy2 - dz 2 1 C 2 2 2 v2 ~2 ;:i.,_ ~2 .J. 2 2 v2 2 2 = IY c dt (1-2 )+2v,~dt-2v,~xdt+Y dx (2 -1)-dy -dz I C C Now 2 -1 V2 ,-1 V I1----z = cz-1) C C Hence we have The superluminal metric signature must therefore be taken to be (-+--) or(+-++). This means that in general, if (a0 ,a1 , a2 ,a3) are the components of a 4-vector A relative to a superluminal frame, the magnitude of the vector is 2 2 2 IAI = lao - al + a2 + a321½ and the magnitude of the 3-vector ij = Ca1,a2,a3) is 2 2 - - a 21~2 I ~I = I a1 a2 3 In particular, the distance between two points in a superluminal space is given by 8 3.2 THE ASYMMETRY OF SPACE-TIME If space-time consisted of only two dimensions, there would be a perfect symmetry between subluminal and superluminal frames of reference. In fact the distinction between such frames would only be a relative one. With the introduction of the other two dimensions, however, the symmetry is broken. Space-time consists of four dimensions, one of which (the subluminal time dimension) is of a different type from the other three. Although this different dimension is the time dimension of subluminal frames, the time direction of a superluminal frame must necessarily be in one of the other three dimensions. (The time direction is always along the world line of a particle at rest in the frame concerned.) Thus the three spatial dimensions of a subluminal frame are all similar whereas a superluminal frame has two similar spatial dimensions and one different. This means that subluminal and superluminal 3-spaces are intrinsically different. The former are Euclidean whilst the geometry of the latter is hyperbolic. It follows from this that tachyons and tardyons are expected to be structurally dissimilar in their rest frames. The asymmetry of space-time is therefore the source of a number of interesting differences between sub and superluminal particles. The properties of superluminal spaces are quite strange in comparison with Euclidean spaces. For example, as will be shown, a straight line does not represent the shortest distance between two points. Also the total circular angle around any point is 00 rather than 2n. The geometric properties of these new spaces will be examined in more detail, but first we will 9 obtain the transformation equations for transforming between two superluminal frames. 3.3 SUPERLUMINAL LORENTZ TRANSFORMATIONS Since one superluminal spatial dimension is always distinguishable from the other two, we will adopt the convention of always calling it the X direction. This is consistent with equations (5), and the equations for transforming between two superluminal frames with v in the common x direction will be analogous to (5) :- x'=+Y(x-vt) , y'=y , z'=z , t'=+Y(t-v~) . . ... ( 8) C These equations will not hold however when vis in they or z direction. The transformation is of a different form for these cases. To determine the correct equations we consider a subluminal coordinate system with axes x,y,z,t. These axes correspond to those of a superluminal frame with the same y and z axes but with the other two axes interchanged (i.e. the x axis is the taxis and the taxis is the x axis). A transformation between two superluminal frames for which vis in they direction corresponds to a rotation of axes in the xy plane (see Fig. 7). Such a rotation is described by the equations x' = xcos~ + ysine, y' = ycose xsin0 ...• ( 9) Replacing x and y with et and y respectively, and noting that tane = L = V we obtain et c' V t+~ y - 2 et C C et' = + = C /1+:~ /1+:~ j 1+:~ 10 and Ct V C y-vt y' = y = /1+~ Thus the equations for transforming between two superluminal frames whose relative motion is along their y axes are t+:q. -vt C x' = x, y' = -Y , z'=z, t' = •••• ( 10) /1+~ /1+~ C 3.4 GEOMETRY OF SUPERLUMINAL SPACES The three dimensional space specified by the y,z and t axes is the spatial part of the superluminal frame. The basic geometric difference between this and a subluminal space is that the distance dr between two points is given by 2 dr2 = ldx .- dy2 - dz21 •••• ( 11) in contrast to the Pythagorean formula dr-2 = dx -2 + dy -2 + dz2 for a Euclidean space. This means that if dx2 = dy2 + dz 2 then the distance between the two points is zero. In particular, all points on the surface x2 = y 2 + z2 are zero distance from the origin. (This surface is simply a three dimensional cross- section of the familiar light-cone.) In a Euclidean space, points which are an equal distance from the origin form a sphere, whereas the equation x 2 y 2 - z 2 = constant describing such points in a hyperbolic space is the equation of a hyperboloid. Fig. 8 gives a two dimensional illustration of points equidistant from the origin, the dashed lines denoting the intersection of the null surface x 2 = y2 + z2 with the xy 11 plane. (The z axis will frequently be disregarded in these diagrams because the geometry of any "x=constant" plane is Euclidean. This means that the properties of superluminal spaces are axially symmetric about the x axis and there is a symmetry between they and z direction~) Another consequence of the new distance formula is that straight lines in the space fall into two distinct classes. The distance between any two points on a line satisfies either 2 2 2 2 2 2 . dx > dy + dz or dx < dy + dz depending on whether the absolute value of the line's gradient is greater or smaller than that of the dashed lines in Fig. 8. The two different types of line thus defined correspond to the usual time-like and space-like lines of space-time. Furthermore a straight line represents the longest path which can be constructed between two point using segments of the same type. For example the two curved paths between points A and Bin Fig. 9 are both actually shorter than the direct route. This is because they contain segments with gradients very close to the gradient of the null lines, so that these segments have almost zero length. The fourth path ACB contains a segment of the opposite type and is longer than the straight route. This last path consists of the x and y components of the straight line segment AB and leads us to another point worth noting. Since the length of this line segment is /dx2-dy2 in accordance with (11), we see that its x component is actually longer than the segment itself. It is a general property of any line in a superluminal space that there will be one component which is greater than its resultant length. In contrast to this, any component of a length Lin Euclidean space is of the form L.cose where we have cos0~1 always. The usual trigonometry 12 of a Euclidean space is therefore not valid in superluminal spaces. In particular, the concept of angle must be reformulated for hyperbolic geometry. 3.5 HYPERBOLIC TRIGONOMETRY The angle between two lines in a subluminal space may be defined in terms of the corresponding arc length of a circle of unit radius centred on the intersection of the lines, as shown in Fig. l0(a). This gives a satisfactory measure of angle because of the constant radius of the arc. It follows that the total circular angle is 2n. By analogy we may describe an angle in superluminal space in terms of the unit hyperbola corresponding to the point in question. The size of the angle in Fig. lO(b) is given by the arc length enclosed between the two lines, the distance from the origin of all points on the hyperbola being unity. The angle that a line through the origin makes with the x axis will now be obtained in terms of the coordinates of the line's intersection with a hyperbola whose points are a constant distance h from the origin (see Fig. 11). The equation of the hyperbola is seen to be 2 X .... ( 12) L d and the angle e will be equal to h" Taking dx of equation (12) yields 2x-2y* = 0, ~ = X .... ( 13) i.e. dx y Because of the hyperbolic geometry involved, the length of an increment dL of the hyperbola is given by 13 Inserting e13) gives 2 1 dL = exz-1)~. dx, y and inserting e12) gives 2 X ½ dL = e 2 2 -1) . dx X -h L = Ix . dL = Ihx ----,.2:----:hz..---,,-1 . dx x=h ex -h )~ X = h arcoshe~) efi ~ 1) L X e - h = arcosheh) X coshe = h .... e14) Also we may write 2 k sinhe = ecosh 0-1) 2 2 1 = ex -1)~ ~ ex2-h2/-z = h , so that inserting e12) gives sinhe = f .... e15) 14 Finally using sinh8 tanh8 = cosh8 we have tanh8 = :l. .... ( 16) X Equations (14), (15) and (16) describe the trigonometric properties of the triangle Pxo. Its side lengths are x,y and hand it is analogous to a right angle triangle in Euclidean geometry since the x and y axes are independent of each other. Clearly the circular functions sin, cos and tan must simply be replaced by the hyperbolic functions sinh, cosh and tanh in hyperbolic geometry. It is evident that this conclusion could have been reached mathematically by maintaining the usual trigonometric equations but considering time-like lengths as real and space-like lengths as imaginary (or vice versa). By this we mean that y and L must be replaced in our equations by the imaginary quantities iy and iL, whilst x and h remain unchanged. It follows that e must then be imaginary since its definition becomes 8 = !ff. Thus we have . iL iL ~ cos X iL =~ sin n = h' n = h' tan n h" Using the standard identities sin ia=i sinha, cos ia = cosha, tan ia = i tanha, these become i sinh ½=if, cosh = ~ i tanh h1: = . :l. k h' 1 X ' and cancelling the i's we obtain equations (14), (15) and (16) again. The limit L+1 will now be considered. In this case 8 X 15 becomes the angle between the x axis and the null line. Equation (16) may be written as 8 = artanh ~, and in the limit X f+-1 we have artanh f+- 00 • Hence the angle in question is infinite, which is in accordance with the fact that the hyperbola is of infinite extent. Thus each of the four angles defined by the intersection of the two null lines is infinite, in strong contrast to the finite total circular angle of 2TI in Euclidean space. 3.6 ROTATION OF SUPERLUMINAL SPATIAL AXES It is of interest to derive the equations for a rotation of the spatial axes of a superluminal frame. Considering the arrangement of axes in Fig. 7 again we see that a rotation in the xy plane corresponds from a subluminal point of view to a Lorentz transformation in they direction. Fig. 12 illustrates such a rotation of spatial axes. The Lorentz transformation is of the form - Yr. t- 2 Y-vt C y I = R'- ft = and the substitutions y+y, t+~ and~= tanh8 convert et X these equations to y - c tanh8~ C y' = seche = y cosh8 - x sinhe, and x - tanhe~ x' C C - = = !ex cosh8 - y sinhe). C sech8 C Thus if the x and y axes are rotated through an angle e in the xy plane, the new x,y and z coordinates a~e given by 16 x' = xcoshe - y sinhe, y' = y coshe - x sinhe, z'=z . . . . . ( 17) These equations bear some similarity to equations (9), their their subluminal counterparts. 3.7 PERPENDICULAR LINES We will now consider in more detail what is meant by perpendicular lines in a superluminal space. In Euclidean space, two lines are perpendicular if the four angles at their intersection are equal. Therefore since the angle between any x axis and a null line is infinite, it would appear that the angle between any time-like and any space-like line is infinite and that every possible y axis is therefore equally independent of any x axis. However this is not the case since one choice always appears to be more symmetric than the others. For example we know that in the subluminal ty plane there is a unique y axis associated with each possible taxis. However, this plane is precisely the superluminal xy plane. Thus for each possible superluminal x axis, one space-like line is favoured as they axis. We can find the particular direction perpendicular to a line in the following way (Fig. 13) :- Consider three points on the line such that the middle point O is halfway between the other two (A and B). If we introduce another point C such that AC= BC then, by symmetry, the line passing through O and C must be perpendicular to our first line. Furthermore, two such perpendicular lines will remain perpendicular if they are rotated as shown in Fig. 12. Hence it is meaningful to talk about perpendicular lines in a superluminal space despite the difficulty with the corresponding angles. 17 3.8 GENERALIZED HYPERBOLIC ANGLE This suggests that we should try to extend our angle definitions to the case of intersecting space-like and time like lines. We see that such angles will need to be finite even though the (apparently smaller) angles between time-like and null-lines are infinite. The hyperbolic functions of equations (14), (15) and (16) are not defined in the region f>l, so we must proceed in a different way. Writing equation (16) in the form e = artanh ~ and using the identity artanh a=½ tn(i:~) we obtain 1 + y e = ½ tn ( x) .... ( 18) 1 - :r X This will now simply be generalized to :l.>1: x e = ½ tn(x+y) = ½[tn(-1) + tn(x+y)J x-y y-x = 2!.i + tn(x+y)½ 2 y-x .... ( 19) Hence 0 is now complex, comprising a variable real part and a constant imaginary part. (It is also possible to obtain (19) by generalizing equations (14) and (15) after using the identities 2 k 2 k arsinha = tn[a+(a +1) 2J and arcosha =. tn[a+(a -1) 2 ]. Furthermore, expressions for 0 involving h may be derived. However extra care must be taken since, in passing through the null line, has defined changes from real to imaginary). Taking the perpendicular limit x+O we obtain 0 TT. + zl • •.• ( 2 0) which resembles the Euclidean result. We see that the angle 18 between any two lines of'~pposit;'type contains an imaginary term of magnitude r in addition to the real part, and the two lines are perpendicular if the real term is zero. 19 SECTION 4 FORM OF A TACHYON'S SURFACE We will now consider the possible structure of a classical tachyon. It will be assumed that the tachyon is at the origin of its rest frame and has a definite non-zero volume. The question then arises as to what shape its surface should be. The following discussion of this question is of some importance since it also applies to the shape of equipotential surfaces in the fields surrounding a tachyon. 4.1 RADIALLY SYMMETRIC SURFACE We would expect the surface of a tardyon in its rest frame to be spherical, i.e. constant radius, since this is the most symmetric possible shape. However the corresponding constant radius surface in a superluminal space is a hyperboloid. Although this is the most symmetric shape (the equation of a hyperboloid remains unchanged under a rotation of superluminal spatial axes) it is unlikely to be the right one for a tachyon since a hyperboloid is of infinite extent. The equation of the hyperboloid in its rest frame is x2 - y 2 - z2 = constant. This equation becomes 2 - - 2 -2 -2 Y (x-ut) - y - z = constant when transformed to a subluminal frame moving in the negative x direction. At the instant t=O this reduces to -2 -2 y z = constant, 20 i.e. the surface also manifests itself as a hyperboloid relative to any subluminal frame. The xy cross-section of this is illustrated in Fig. 14, showing the way in which the surface extends to infinity. The whole surface is moving along the x axis with velocity+ u. It is clear that this solution is not suitable since such a tachyon would not be properly localized. 4.2 APPEARANCE OF A TARDYON RELATIVE TO A SUPERLUMINAL FRAME For a tachyon to appear as a normal particle its surface must be closed and of finite area in its rest frame. There are no other requirements to assist us in determining the correct shape except that it be reasonably uniform and symmetric. However by considering the analogous cross-section of a tardyon we are led to one choice as the most probable. Figs. 15 and 16 give the xt and xy cross-sections of a tardyon moving with velocity u along the x axis of a subluminal frame. We assume that it is spherical in its rest frame and that its radius is greater than zero. We wish to find the tyz cross-section of the tardyon. (This is its xyz cross-section relative to the superluminal frame whose axes coincide with those of the subluminal frame.) The equation of the particle's surface in its rest frame is -2 -2 -2 2 x + y + z = R where R is its radius. For the frame in which its velocity is u this equation becomes -2 + z -2 -½ where Y = 1 - u Hence in the tyz hyperplane (i.e. the ~ 21 "x=zero" hyperplane) we have -2 2-2 -2 -2 2 u Y t + y + z = R or uY)2 2-2 -2 -2 R2 ( C C t + y + Z = .... ( 21) This is the equation of a spheroid. (Its ty cross-section is shown in Fig. 17). By putting y = z = 0 the intercepts on the et axis are found to be± cR_ There are three special cases. uY For u=O the intercepts become ±00 , i.e. the spheroid becomes infinitely elongated in the et direction. For u=c the intercepts on the et axis both reduce to zero and the spheroid is flattened into the yz plane. Finally, the spheroid becomes a sphere for k -2u ;z CR =R::::;>u= C , C 1-z ,=}u= uY C By making the replacements ct+x, y+y, z+z, the equation for the tardyon's spatial cross-section in the corresponding super luminal frame is found to be .•.• ( 2 2) The coefficient (uY) 2 may be rewritten as C -2u y 2 -2 = u = 1 -2- -2 2 C c211 - ;-1 I~ - 11 C u so that equation (22) becomes ••.• ( 2 3) 22 4.3 APPEARANCE OF A TACHYON IN ITS REST FRAME It would seem reasonable to assume that equations (21) and (23) also apply to the case of a tachyon. The range of -u is then simply generalized to include u>c. For the situation described above, the particle will be at rest in the super luminal frame if its velocity is infinite relative to the subluminal frame. Therefore inserting u= 00 in equation (23) we obtain .••. ( 2 4) for the surface equation of a tachyon in its rest frame. The xy cross-section of this surface is shown in Fig. 18. Although (24) is the equation of a sphere, it may be misleading to refer to the surface as spherical since its properties are different from those of a sphere in Euclidean space. In particular its radius is not constant. The distance from the origin to a point (x,y,z) on the surface is, according to equation (11), 2 2 2 ½ r = Ix -y - z I Hence the radius decreases to zero as r approaches the null 2 2 cone x = y2 + z • The hyperboloidal surface mentioned initially is the only one which is syimmetric under a rotation of spatial axes in the . . . 2 2 2 . res t f rame ( 1.e. its equation x - y - z = constant 1s covariant under the transformation given by equations (17]. A spherical surface as chosen above necessarily favours one particular orientation of the axes. Taking equation (24) and transforming it according to equations (17) we obtain 23 . 2 2 2 2 (x'cosh0+y'sinh0) ~y'cosh0+x'cosh0) +z' = R, "* x 12 (cosh2e+sinh2e)+y 12 (sinh2e+cosh2e)+4x'y'cosh0sinh0 +z'z •••• ( 2 5) Hence the surface only takes on the simple spherical form of equation (24) in the frame for which 0=0. The distortion corresponding to a rotation of axes is illustrated in Figs. 19(a) and 19(b). Thus it appears that, except in the limiting case of a point particle (i.e. zero radius), a localized tachyon cannot be fully symmetric in its rest frame. 4.4 AXIAL SYMMETRY ABOUT THE DIRECTION OF MOTION Let us consider the appearance of a tachyon when viewed from a subluminal frame. Moving tardyons, or more specifically their fields, are normally axially symmetric about their direction of motion. This holds true simultaneously in every possible subluminal frame. It would seem reasonable to require this also of tachyons. The most general shape satisfying this condition will now be found for both tachyons and tardyons. Although the tardyon calculation is somewhat unnecessary since the answer is fairly obvious, it will be done as a test run for the tachyon case. Consider first a tardyon at the origin of its rest frame. We may transform to any other frame by rotating the x,y and z axes appropriately then performing a Lorentz transformation in the resulting x direction. It will be sufficient to consider those frames which may be reached by successive application of equations (9) and (6). Calling the coordinates of the rest frame x1 , y1 , z1 , t 1 , and the coordinates of the final frame Xz, Yz, Zz, tz, we have 24 Combining these gives xl = y cos8(x2+ut2) y2sin0 .••. ( 2 6) Y1 = y"2cose + y sin0(x2+ut2) l Since the space-time origins of these two frames coincide, the particle will be centred on the origin of the second frame at time t 2=o. We require that the cross-section of the surface in the x 2=O plane be axially symmetric (i.e. circular) about the x2 axis. This means that the equation of the cross-section must be of the form -2 -2 R2 . Y2 + z2 = Another way of saying this is that the points cz2~R) must lie on the surface. Substituting this into (26) we obtain or j 2 -2 . - - j 2 -2 - R -z1 .s1ne, y1 = R -z1 .cos0. Squaring and adding we find -2 -2 = (R 2 -z-2 )(sin 2-B+cos 2-0) xl + Y1 1 ' -2 -2 -2 ~ xl + Y1 + zl = R2 ••.. ( 2 7) 25 (27) is thus the only possible tardyon surface which satisfies the axial symmetry condition. The essence of this derivation is that the yz cross-section of a particle is unchanged by a Lorentz transformation in the x direction. Therefore since any direction may be chosen as the x direction by arbitrarily rotating our spatial axes, the requirement of axial symmetry about the direction of motion simply reduces to requiring that the particle be axially symmetric in its rest frame about any line through its centre. The analogous surface for a tachyon at the origin of its rest frame may be determined by the same general method. The appropriate transformations are those of equations (17) and (7). We perform a rotation of superluminal spatial axes:- followed by a transluminal Lorentz transformation back to a subluminal frame:- The overall transformation is .••• ( 2 8) As in the tardyon case we require the yz cross-section of the surface to be circular in the resulting subluminal frame. Hence inserting 26 in (28) we obtain Finally, squaring these gives 2 2 2 2 .2 2 x 1 - y 1 = (R -z1)(s1nn6-cosh e) 2 X - •.•• ( 2 9) 1 which is the hyperboloidal surface discussed earlier. We conclude that a hyperboloid is the only possible surface for a tachyon in its rest frame which will appear axially symmetric about the particle's direction of motion relative to any subluminal frame. Thus no closed, finite surface will satisfy this condition. 4.5 APPEARANCE OF A TACHYON RELATIVE TO A SUBLUMINAL FRAME Let us examine the appearance of the spherical surface given by equation (24) relative to a subluminal observer. Applying the transformation (28) to equation (24) yields 2 2 - - 2 -2 2 + Y sinh e(x2+ut2) +z 2 = R , and for the instant of time t 2=O this reduces to -2 2 2 2 - - y x2Y (cosh e+sinh e)+4x2y 2 sinhe.coshe -2 2 2 -2 2 + y2 (cosh +sinh 0)+z2 = R , -2 2 - - -2 -2 2 x2Y cosh20+2x2y 2Ysinh20+y2cosh20+z2 = R ..•. ( 30) 27 This may be simplified somewhat by a rotation of our subluminal spatial axes. Putting x2 = xcos~-ysin~, Y2 = ycos~+xsin~, z2 = z, in equation (30) we obtain (xcosl-ysinl) 2Y2cosh20+2(xcosl-ysinl) (ycosl+xsinl)Ysinh20 - - - - 2 -2 2 + (ycos~+xsin~) cosh20+z = R -2 2- 2 2- -- =:'> x (cos ~y cosh20+sin ~cosh20+2sin~cos~Ysinh20) -2 2- 2 2- -- + y (sin ~Y cosh20+cos ~cosh20-2sin~cos~ysinh20) - - - 2 2- 2- + xy(-2sin~cos~y cosh20+[2cos ~-2sin ~Jysinh20 -- -2 2 + 2sin~cos~cosh20)+z = R This reduces to the standard ellipsoidal form when the rotation is such that the last bracket is zero:- - - 2 -- -2sin~cos~y cosh20+2sin~cos~cosh2e 2- 2- + 2[cos ~-sin ~Jysinh20 = 0 sin2l(l-y2)cosh20+2ycos2lsinh20 = 0 2y tan2~ = - 2-.tanh20 .... ( 31) y -1 The ellipsoid will therefore be oblique to its direction of motion at an angle~ given by equation (31). Its xy cross section is shown in Fig. 20. From equation (30) we see that it reduces to a spheroid 28 Y2-x2 + -y2 + -z2 = R2 2 2 2 (fig.16) in those subluminal frames corresponding to 0=0. In general, any finite tachyon surface which is axially symmetric about the x axis of its rest frame (or can be made so by a rotation of superluminal spatial axes) will be seen to favour certain particular subluminal frames by appearing as axially symmetric about its direction of motion in those frames. 29 SECTION 5 VOLUME AND AREA IN A SUPERLUMINAL SPACE 5.1 VOLUME Looking at Figs. 19(a) and (b) again it is not clear whether the volume of a superluminal object in its rest frame is the same relative to any set of spatial axes. In other words the concept of volume may not have any absolute significance in a superluminal space. To deal with this question we must have a precise definition of volume. The volume of a body is the number of unit cubic elements (i.e. cubes with unit side length) contained within it. If the shape of the body is irregular, its volume is found by dividing it up into infinites mal cubic increments and then integrating to find the total. In a subluminal (Euclidean) space the stacking orientation of the volume elements is arbitrary, having no effect on the volume obtained (see Figs. 21(a) and (b)). However the unit sides of each separate element must always be perpendicular to one another (i.e. the elements must be cubic) because the volume of a unit rhombohedron is different from that of a unit cube. Since the sides of the elements must be mutually perpendicular, there is a particular orientation of spatial axes corresponding to each allowed element, and vice versa. This applies in both subluminal and superluminal spaces. However for a superluminal space the unit cube associated with one set of axes takes on an apparent rhombohedral shape in our diagrams relative to another set of axes (see Figs. 22(a) and (b)). (Since the rotation of axes is taken to be in the xy plane, the extension of the unit elements in the z direction is unaffected and only their xy 30 cross-section will be represented in the following diagrams.) The question is whether the volume of a body in a super luminal space is independent of the orientation of the volume elements used in calculating it. For example referring to Figs. 19(a) and (b) again we want to know whether we will obtain the same volume if we choose the sides of our volume elements to be parallel to the x' ,y' and z' axes instead of the x,y and z axes. Consider a body of arbitrary shape which has been divided up into unit volumes relative to the coordinate system S' as in Fig. 23(a). Relative to a different set of spatial axes S this will appear as in Fig. 23(b). Obviously the number of S' unit elements in the volume is not changed by the rotation. Therefore if the volume of an S' unit element as measured relative to S is equal to the volume of an S unit element (i.e. unity), the same overall volume will be found for the body relative to either set of axes. Thus the problem reduces to finding the volume of the S' unit element relative to Sas shown in Fig. 24. It is not necessary to perform an integration to determine this. By symmetry it is clear that the volume of the element with xy cross-section OABC in Fig. 25 will be the same as for OADE (assuming no change in the z dimension). Also, it is evident that the method of measuring volume shown in Fig. 24 will yield the same result as if the space were Euclidean, since the type of geometry is irrelevant to the procedure. Therefore the volume corresponding to OADE is given by the product OA. OE . 1 where OA and OE are the apparent lengths on our diagram rather than the actual lengths in the space, and the 1 represents the unit length in the z direction. The actual length of the line 31 between O and A is unity, so A must lie on the hyperbola x 2 -y 2 =1. Therefore we may write We now proceed using normal Euclidean trigonometry. From (32) we have OA = (OF 2+AF 2)½ = (x2+x 2-1)½ = (2x2-1)½ Also, tanqi = AF- OF 2 = (x -1)½ X ' 2 -k cos qi = (l+tan qi) 2 Using the identity cos2a = 2cos 2a-1 we obtain 2 COS2~ = 2 X -1 'f 2 2x -1 Now, OE = OC. cos2qi = OA.cos2qi = (Zx2 -1)-½ 32 Hence we conclude that the volume of the OADE element (and therefore the OABC element) is given by = 1 The volume is equal to that of an S unit element. Thus we see that the concept of volume in a superluminal space has absolute significance independent of the way we choose to measure it, just as for a Euclidean space. 5. 2 AREA In the above discussion we used the fact that the method of calculating the volume of OADE was independent of the type of geometry involved. This is true in general. For example 2 2 2 2 . the volume enclosed by the surface x +y +z =R 1n Fig. 18 will be the same as for a Euclidean sphere, i.e. }TTR3 The situation is not quite so simple for area, because for the elements of area to lie on the surface concerned it is usually necessary for them to be oblique relative to one another. This leads to different values of area from the corresponding Euclidean results. To illustrate this. the area of the surface x 2 +y 2 +z 2 =R 2 will now be calculated. We divide the surface up into circular elements of area dA as shown in Fig. 26. We may write dA = 2TTD.ds .••. ( 3 3) 2 2 k where D=(y +z) 2 is the radius of the particular element of area in question and ds is its infinitesmal width. For convenience we choose ds to lie in the z=0 plane so that (using equation (11)) .... ( 3 4) 33 The equation of the cross-section of the surface in this plane is Taking dxd of this gives 2x + 2y *= 0 ~ X .... ( 3 5) dx y Now (34) may be written in the form Therefore inserting (35) gives 21!.:2 ds = I 1 - x2 .dx y 2 ¾ X 2 = I1 - 2 2 I . dx R -x - jR2-zx21½ - 2 2 • dx R -x Substituting this into (33) we obtain 2 2 ½ j R2 - 2x2 j ½ clA = 2 TT (y + z ) • 2 2 • dx R -x 2 2 ½ j R2 - 2x2 I½ = 2TT (R -x ) • z 2 . dx R -x 2 2 !.: = 21rlR -2x I 2 • dx Hence the total surface area of the sphere is A= f dA 2 2 !.: 2 1T I R - 2 x I 2 • dx 34 This integral may be solved by dividing it into two parts: R Substitute x = ~sina in the first part:- 12 2 k R R(l-sin a) 2 .-cos a. da f 12 = - R2f cos 2 a.da 12 Substitute x = ~cosha in the second part:- 12 2 2 k 2 k R f ( 2 X - R ) 2 • dx = fR(cosh a-1) 2 • 12sinha.da __ R2[/Zx(2x2 l)½ 1 h(./Zx)J 12 2R 7- -zarcos -r A = 4TTR 2[12x(l 2x2)½+1 . (/Zx) ]R/12 12 2R -7 2ars1n -r 0 R2 TT R2 ll 1 = 41r-(--O) +41r-(---arcosh ( 12) -0) 124 l2 22 = 1rR2 (-2!.+2-l2arcosh(ll)) n 2 ~ A ~ 2. 98 TTR .... ( 36) 35 which is smaller than the value 4TTR2 found in Euclidean space. 36 SECTION 6 ENERGY AND MOMENTUM 6.1 ENERGY AND MOMENTUM RELATIVE TO A SUBLUMINAL FRAME It is usually argued that the rest mass of a tachyon must be imaginary. This conclusion is reached by taking the subluminal energy and momentum expressions m - u 0 E = p = .•.• ( 3 7) and generalizing them to u>c to obtain the energy and momentum of a tachyon. Since becomes imaginary, m must also be Fl 0 imaginary for E and r to be real. However this is only true if energy and momentum are defined as in (37). To be consistent with our earlier transformation equations, the general expressions for E and E should be taken as 2 m C E = o F7f .••• ( 3 8) and these require m0 to be always real. These expressions can be deduced as follows:- 2 The conservation of m0 and of the rest energy m0 c of a particle follow from the homogeneity of space-time. In addition, isotropy of space-time requires that the four components 2~ Zdz m C d , m C -:'l'7 0 S 0 us be conserved, ds being an infinitesmal element of interval along the world-line of the parti~le. These lead to (38) if we put 37 2 d(ct) m C E 0 ds - ' and dr 2 z cdi dz) = C - m0 C ds'S * S 'as S mo ds re ' since it is easy to show that d(ct) = ~,-½ and dr = ~, 1 ;-1-½ ds 11 ds C C C dr- dr e.g. ds = lc2dt2 - dr2 1½ = u = -le 6.2 ENERGY AND MOMENTUM RELATIVE TO A SUPERLUMINAL FRAME We now wish to find the superluminal expressions for the energy and momentum of a particle. As stated earlier in §3.1, if (a0 ,a1 ,a2 ,a3) are the components of a 4-vector A relative to a superluminal frame then .... ( 39) and where a is the 3-vector (a1 ,a2 ,a3). In particular we have 38 .... ( 41) = I u 2 -u 2 -u 21½ .... ( 4 2) X y Z and 2 2 2 k IP I = IP -p -p I 2 •.•• ( 4 3) - X y Z where U=(u ,u ,u) is the velocity of a particle relative to the - X y Z superluminal frame and p=(p ,p ,p) is the particle's momentum. - X y Z Consider the 4-momentum vector pointing along the particle's world-line. Its components along the four superluminal axes are m c2dx 2~ 2dz m C d m C - ' o ds 0 S o ds Identifying these with the superluminal energy E and momentum 2 C dt = m C 0 2 m C = 0 d X 2 - d :l 2 - d Z 21½ 11 - c 2dt2 2 m C '9" E = 0 ux-u2 2 -uz21½ ' 11 - 1 C 2 dx • C m C Px - 0 ds 2 dx = m C 0 lc2-dt2-dx2+dy 2+dz 2 1½ dx m C = 0 at _ d X 2 - d:l 2 - d Z 21½ !1 c 2 dt2 39 mu 0 X u 2 -u 2 -u 2 ½ l _ X ~ Z C and similarly m u mu 0 y 0 Z 2 u 2 -u 2 -u 2 ½ u 2 -u -u 2 ½ l _ X y Z l _ X ~ z C C Hence using (42) these expressions may be written in the form m c 2 m u E = 0 0- •••• ( 4 4) UZI½ I1 - ~ 2 m C mu 0 0- E = .••• ( 4 5) 1 + u 21½ I ~ C 2 2 2 2 2 Zk:. for u (44) or (45). m0 1s always real. The 4-momentum of a particle relative to a superluminal frame is P=(~,Px,Py,Pz). It follows from equation (39) that the magnitude of P is E2 2 2 2 1 I p I = I2 - Px +Py+ p I "z .... ( 46) C Z Therefore inserting the energy and momentum expressions derived above we find 40 2 2 2 2 2 2 mu + m u + m u ½ 0 X 0 y O Z IPI = 2 2 2 u - u - u X z ~ C = = I+- m2 C 21½ 0 = m C 0 which is consistent with the subluminal result. All this is similar to the usual subluminal formalism. The main difference is that the change in metric signature requires that 1~1 and l~I be redefined as in equations (42) and (43). Some of the consequences of these expressions will now be considered. 6.3 VELOCITY REGIONS IN A SUPERLUMINAL SPACE Equation (42) stems from equation (11), since u must by definition equal drdt" It is clear that the magnitude of a particle's velocity may be zero even though none of its components are zero. This corresponds to motion along a null line in the superluminal space. Since the distance between any two points on such a line is zero the velocity must also be zero, even though the particle may be moving relative to the xyz coordinate grid. (Note that this does not mean the particle's world-line lies along the surface of a light cone in space-time. In fact relative to other superluminal frames the particle will not be moving along a null line.) u=O is the boundary between 41 two different velocity regions corresponding to the two distinct classes of straight lines in a superluminal space 2 2 2 mentioned in § 3. 4. These two regions are u >u +u and X y Z u 2 c, and one might expect the reverse to be true for superluminal frames. However due to the asymmetry between subluminal and superluminal frames introduced by the two transverse spatial dimensions this is not the case. A particle with velocity u>c relative to subluminal frames (i.e. a tachyon) may also have u>c in a super 1 umina. 1 f rame prov1"d e d u 2 is. 1 ess tan h u z+u 2 in. tath f rame. X y Z The velocity regions available to tachyons and tardyons in a superluminal space are as follows:- u 2 >u 2 +u 2 X y Z tachyon O~u Tardyons are always restricted to trajectories satisfying 2 2 2 u >u +u. However a tachyon may be freely accelerated through X y Z t h e u 2 =u 2 +u 2 b arrier.. X y Z Using (44) and (45) the energy and momentum regions for a particle in a superluminal space may be classified as follows:- 42 u2 O~E< 00 tardyon m c~p<00 0 For u 2 >u 2 +u 2 the ranges of u,E and pare the same as for a X y Z subluminal frame with tachyons and tardyons interchanged. However there is no subluminal analogue to the u2 m c 2 E = 0 1 + u 21"2 I 2 C and mu m 0 0 + 2 2 11!.: C 2 Hence for u ranging from Oto 00 we see that E ranges from m c 0 to O and p ranges from O to m c. For u=c we find 0 2 m C m C 0 0 E = and p = It is curious that p increases when E decreases in this velocity region. Fig. 28 gives a three dimensional representation of the 43 u >u and u uy, u>c region and the ux SECTION 7 THE ELECTROMAGNETIC FIELD OF A TACHYON 7.1 PRELIMINARY DISCUSSION The electromagnetic field surrounding a tachyon will now be considered. The easiest way to find the general solution for the field of a tachyon relative to a subluminal frame of reference is to deduce first the tachyon's field in its super luminal rest frame then transform the result to the general subluminal frame. Therefore pursuing this approach we will consider a charged point tachyon at rest at the origin of an inertial superluminal frame. There are a few observations that can be made about its rest frame field at the outset:- (a) Since the tachyon is at rest we expect its field to be static and have no magnetic components. (b) We hope the solution to be such that the field intensity falls to zero in each of the limits x-+oo, y+00 , z+00 , and is not infinite anywhere except at the origin. (c) We expect the intensity to be zero in certain of the regions in Fig. 29:- (i) For any field to exist in region II it would have to be propagated from the tachyon at a speed faster than light relative to any subluminal frame. Since electromagnetic fields travel at speed c the intensity must be zero in this region. This can be seen by considering the appearance according to a superluminal observer of an expanding light pulse emitted from the tachyon at time t=O. The wave front will be spherical in any subluminal frame:- 45 -2 -2 -2 2-2 X + y + Z = C t Transforming to a superluminal frame by way of equations (7) we obtain 2 2 2 ~ Y2x2 c;--1) y - z = Y2t2(v2-c2) . C Now 2 1 1 y = = 2 V j1 - Cz - 1) ~I C Hence we have 2 2 2 X y - z = c2t2 ' showing that the subluminal light sphere becomes a hyperboloid in a superluminal frame. Furthermore since. c 22 t must b e positive.. t h e propagating. wave f ront is confined to the x 2>y 2+z 2 region. Therefore there can be no field in region II because none of the points in this region lie on light cones originating at the tachyon. All this is illustrated in Fig. 30 for the case of v= 00 • (ii) It is doubtful whether there will be any field in region III since this corresponds to the negative time direction of subluminal frames. Such a field could influence events earlier in time and thereby lead to causality paradoxes. For the reasons above only region I will be considered in the following discussion. 46 7.2 POINT TACHYON SOLUTION VIA THE KLEIN-GORDON EQUATION To deduce the electric field surrounding a point tachyon in its rest frame we will derive the superluminal version of the Klein-Gordon equation and solve it for virtual photons. Taking the covariant equation (46), squaring it and making the replacements fl -+ a i a°z' we obtain the operator equation We want a static solution with virtual particles of zero rest mass. Operating with both sides on some function~ and setting = 0 and m = 0 we obtain ~at o a2 z - az z - ~ = o . [ax ay ½]az By analogy with the well-known tardyon solution,~ will be interpreted as the electric scalar potential V. Thus we have •••• ( 4 7) This is Laplace's equation for a superluminal space. It has an infinity of solutions. To obtain a unique solution for V we need to include some extra restriction, and the most obvious additional assumption is that of radial symmetry. To introduce this into (47) we need to rewrite the equation in terms of the polar variables r, e and~- For region I in the superluminal space these variables will be related to the cartesian coordinates x,y and z in the following manner (Fig. 31) :- 47 r = ( X 2 -y 2 -z 2)½ 2 2 1' e = artanh (y + z ) 2 •••• ( 4 8) X qi = artan (~) y (The equation for qi is the same as for Euclidean space since qi lies in the yz plane.) The quickest way of achieving the change to polar variables is by considering the normal sub luminal Laplace equation:- .•.• ( 4 9) This equation may be converted to (47) by making the alterations X + X y + 1y .••• ( 5 0) z + iz V + V The form of (49) in spherical polar coordinates is known to be 1 [-a (r2 1 a . -av 1 a2 - av) + -- -(sine-) + ----=-- ~vd = 0 ' .... ( 51) r 2 ar ar sine ae ae sin2e a~ where r = c-2X +y-2 +z-2)½ Cy2+z22½ e = artan - .... ( 5 2) X ~ = artan (~) y Hence by changing (52) and (51) in accordance with (50) we will obtain the polar form of (47). Thus, equations (52) are altered to 48 - r :;::: ( X 2 -y 2 -z 2/:z 1. ( }'.: 2 +z 2) ½ e :;::: artan X Crz+z2)½ :;::: i artanh X 4i" a::: artan so that, comparing these with equations (48), the required changes in the polar variables are seen to be r+r, 0+i0, 4i+~. The second of these makes the following change also necessary:- sine+ sinie a::: i sinh0 . Thus by making the substitutions r + r e + ie sine+ i sinhe V+V in equation (51), the superluminal Laplace equation in polar coordinates is found to be (More orthodox methods also lead to (53) but are considerably more tedious.) Now, we are looking for a radially symmetric solution for V (i.e. V independent of both e and~). Under this assumption equation (53) reduces to 49 1 [~( z av) J = 0 ;-z arr ar ' 2 av ~ r = - K (K = constant) ar ' K - + constant. ~v = r Assuming V+O for larger, K V = -r •••. ( 5 4) Introducing the electric field intensity E = - arav r- we have E = K r .••. ( 5 5) -rr - ' where r is the radial unit vector. Equations (54) and (55) resemble the ordinary subluminal results. Nevertheless there are several undesirable features about this solution. We see that its equipotential surfaces are the familiar "r•constant" hyperboloids discussed earlier in §4.1. Thus the field intensity will not become negligible at large x,y and z for x 2 ~y 2 +z 2 In 2 2 2 fact~ is infinite at any point on the null cone x =y +z Although it is easiest to deduce the field in the tachyon's rest frame, it is preferable to examine the field relative to a subluminal frame, since these are the frames which concern us in practice. Consequently the generalized field transformations for v>c will be deduced at this point. 7.3 SUBLUMINAL ELECTROMAGNETIC FORMALISM In the usual subluminal formalism of special relativity, the electromagnetic 4-tensor F is defined by the equation µv F .... ( 56) µv 50 where Aµ are the covariant components of the 4-potential Aµ= (V,A ,A ,A). The transformation equations for Aµ are X y Z A'z =A z .••. ( 5 7) Using Maxwell's equations may be written in the tensor Fµ\/ form •••• ( 5 8) where Jl-1 is the current density 4-vector. By definition Aµ is related to the electric and magnetic field intensities E and H by the equations H = 'iJ X A .••• ( 5 9) 1 'aA E = 'iJ V - C a'f l where d v - Ca_, - E._). - ax ay ' az Combining (59) with (56) gives 0 -E -E -E X y z E 0 H -H X z y = F µv •••• ( 6 0) E -H 0 H y z X E H -H 0 z y X so that substituting (60) into (58) we obtain Maxwell's equations in the form 51 v E = 4'1Tp v H = o .... ( 61) 1 ag V X E = C at 1 a~ 4'1Tj V X = + H --c C at Written out in full these become aE z az = 4'1Tp aH z = 0 - - -- c at a'.E a'.E att X Z 1 -- - = - __J_ C at ••.. ( 6 2) = - - -- ay c af aH" 4'1TJ z - -- --X c at c aH aH" 1 aE 4'1Tj X z= _ __J_ + __:_.:t__ az ax c at C 4'1Tj - -- --z c af c according to By transforming Fµv -\) F' ax F .... (63) po ax-0 I• µv 52 and evaluating in turn the components we find the relations v- E' = E E' = y(E --H E' = y(E +~) X X y y C z) z z C y .... ( 6 4) v- H' = H H' = y(H +VE z) H' = y(H --E ) X X y y C ' z z C y 7.4 GENERALIZATION TO SUPERLUMINAL FRAMES The tensor form of equations (56) and (58) expresses the fact that these equations are covariant under a subluminal Lorentz transformation. It is tempting to assume that they will also be covariant under a transluminal transformation. The superluminal Maxwell equations would then have the same form as (61). The fact, however, that this is not so can be shown by finding the wave equation corresponding to the third of equations (61) in the superluminal case. Taking the curl of that equation we get 1 a = - c at (.Y_x!!_) ' and inserting the fourth equation yields 1 " 1 aE 4 . = a + ~) - Cat c·-....::::.Cat C Using the identity .Y_x(v'xa) = y(y.a) this becomes 41T ~ -Z- at C 53 so that for i=2 and p=0 (i.e. ~-~=0) we obtain which may be written as 1 2 a ••.• ( 6 5) ~ atz) E = o This equation, however, is not covariant under the transformation of equations (10). By returning to the superluminal Klein Gordon equation discussed in §7.2 the proper superluminal wave equation is seen to be 1 a2 = 0 .•.. ( 6 6) ~ at 2 which is covariant under (10). Equation (66) also changes to the form of the subluminal wave equation when transformed via equations (7), whereas (65) does not. Therefore since (65) must be considered untenable we are led to the conclusion that the electromagnetic 4-tensor Fµv cannot be covariant under equations (5) and (7). This may be traced in turn to the fact that the 4- potential (in terms of which Fµv is defined in equation (56)) is not a 4-vector with respect to a transluminal Lorentz transformation. As a result there is no straight-forward procedure that will lead to the superluminal form of Maxwell's equations. 7.5 SUPERLUMINAL MAXWELL EQUATIONS The approach we will adopt is to generalize equations (64) to v>c by way of the arguments used in deriving (5) and (7). We take the pairs Ey,Hz and Ez,Hy to behave like the pair x, et under a rotation in space-time, and E and H to behave like y X X 54 and z. Thus generalizing (64) by analogy with (5) we obtain •••. ( 6 7) Some support for this procedure may be found by comparing equation (47) with the first equation of (62). Substituting E av E av X ay' z az into (47) yields aE aE aE X _L z = 0 , •••. ( 6 8) ~ ay ~ and putting p=0 in (62) we have a'E a'E clE X z + _L + = 0 •••• ( 6 9) ax ay az The correct transformations relating~ and~ to~ and H must transform (68) into (69). Let us assume that the correct equations for Ex, Ey, E2 are of the form E = E ,E =y(a. E +a. ~ ) ,E =y(a. E +a. ~ ) , ••.. ( 7 0) X X y 1 y 2C Z Z 3 Z 4 C y where each of the unknowns a. 1 ,a.2 ,a.3 ,a.4 , equals either +1 or -1. Transforming (68) by using (70) and (7) we find (-ax -+-cl clt -)Ecl - --(y[a.lEcl - +a.2.:..tt_;l)--(y[a.3EV- cl - +a,4-nVrr ]) = 0 ax clX clX clt X cly y C clZ z C y aH a'E clH V Z Z a -y-- - a y-- a "Y_y_L = 0 2 c ay 3 az 4c az aH ail v( z _L 1 a'Ex C a.2-----= + a 4 - - -) = 0. ay az C clt .... ( 71) 55 Now, for r=o the sixth equation in (62) becomes aH aFfY aE z = 1 X .... ( 7 2) ay az C at By comparing equations (69) and (72) with the two brackets in equation (71), we see that the left hand side of (71) will equal zero if we put Inserting these values in (70) we obtain agreement with the first three equations of (67). The inverse transformations corresponding to (67) are Ex=Ex, E =+y(E +~ ), E =+y(E -~) y y C Z Z Z C y .••• ( 7 3) -H =H H- =+y(H --E)V -H =+y(H +-E)V X X' y y C Z ' Z Z C y The superluminal Maxwell equations may now be found by using (73) to transform equations (62) to superluminal frames. We assume that Jµ = (cp,i) transforms as a contravariant 4-vector under a transluminal Lorentz transformation so that we have (from equation (B7) in Appendix B) :- cp = + y(cp+vj ) = + y(j +~cp), jy = jy,jz = jz. C X ,J X X C •••. ( 7 4) Using equations (5), (73) and (74) to transform the first of equations (62) we obtain ax a at a a v a v (- -+- -) E +-( +y[E +..:+I ]) +-( +y[E -..:+I ]) ax ax ax at X ay y C z az z C y = 4'ITy ( p+:;-j ) C X 56 •••. ( 7 5) Transforming the sixth equation in the same way leads to •.•. ( 7 6) so that combining (75) and (76) gives and aH aH 1 aEX 4nj z ___y = + __x ay az C~ C We have thus deduced two of the eight superluminal Maxwell equations. By progressively transforming each of the other six equations of (62), the complete set is found to be aE aE aE X + --'-V + Z = 47T p - ~ ay a"'z" aE z - --ay aE ••.• ( 7 7) z l aH = - _J_ ~ Cat aE ___y ax aH aH __Z _ _:j_ = ay az --c aH aH 1 aE 4njy -----X Z =+---L+ az ax C at C 57 aH aH 41Tj X z _L = + -- ax ~ c j If we introduce the notation E. - (+E ,+E ,+E ) -1 X y Z Ei - (+E -E -E) X' y' Z ..•• ( 7 8) H- - (+H ,+H ,+H) -1 X y Z Hi= (+H -H -H) X' y' Z then equations (77) may be written in the more compact form V.Ei = - 4,rp V.H 1 = 0 ..•. ( 79) VxE. - -1 VxH. - -1 -C where a . a a ) v - Cax'ay'az · These equations are the superluminal analogue of the ordinary Maxwell equations in (61). ~i and ~i may be considered the covariant and contravariant forms respectively of the electric field 3-vector ~' and H- and Hi may be construed in the same ·- -1 - way. The physical significance of E. and Ei will be dealt with -1 shortly. 7.6 SUPERLUMINAL 4-POTENTIAL To be consistent with the second and third of equations (79), the scalar and vector potentials V and A must be defined by the equations 58 .... ( 80) 1 a~ E. - -l vv +cat Inserting these equations into (67), the following transformation equations for Aµ are obtained:- V = - y(V--A)- v- A =+y(A- --V)v- A =A-- A =A . . ... ( 81) C X 'X X C 'y y' Z Z The inverse transformations are -V = + y(V--A)V -A =-y(A --V)V A- =A -A =A . . ••• ( 8 2) C X 'X X C 'y y' Z Z In §7.3 we took V,Ax,Ay,Az to be the components of a 4-vector Aµ. By examining (81) we see that V,Ax,Ay,Az cannot be the superluminal components of this same 4-vector because (81) are not of the same form as either (BS) or (B6). This result is not unexpected since it was pointed out in §7.4 that the electro magnetic 4-potential could not be a 4-vector with respect to a transluminal Lorentz transformation. However V,A ,A ,A will X y Z transform as the components of a 4-vector from one superluminal frame to another. Hence the superluminal potential 4-vector is a different 4-vector from the subluminal potential 4-vector. For two superluminal frames whose relative motion is in the x direction, the transformations for E- and H- will be of the same form as (64). Using these and (80) the following transformations for the 4-potential are obtained:- V V V' = + y(V+-A) A'=+y(A +-V) A'=A A'=A. . •.• ( 8 3) C X 'X X C 'y y' Z Z By comparing (83) with equations (B2) we conclude that V,A ,A ,A must be the covariant components of the superluminal X y Z potential 4-vector:- 59 Using this result it is now possible to write equations (81) and (82) in tensor form as follows:- (Aµ is equal to (-V,+Ax,-Ay,-A2 )in accordance with (BIO).) 7.7 PHYSICAL SIGNIFICANCE OF E. AND E1 -l -- - The physical significance of the two different electric field vectors defined in equations (78) will now be found by examining the force on a test particle with unit charge in a static potential field. Although the potential in equation (54) has physically undesirable features, it will be useful as an example in the discussion which follows. The equipotential surfaces of this potential function in the xy plane are shown in Fig. 32. We assume that the potential is positive and that the force Eon the particle is repulsive. (The field intensity and the force are equivalent here since the test particle has unit charge.) Now consider the three components of this force:- av av •••• ( 8 4) -ax ' E y az For V in equation (54) we find Kx Kz E = + 3 , E = EI_ E = X - r3 r y r 3 ' z Taking K as positive we see that Ey and E2 are inwards towards the x axis. The resultant of these three components at any point P (see Fig. 32) will also point towards the x axis rather 60 than away from it. It would seem that we should consider this as the direction of the force on the test particle, but this is not the case. The force is actually radially outward from the origin. This must be true for the field to be symmetric under a rotation of spatial axes. For any point on the x axis, we conclude from symmetry that the force must be directed along the axis in the positive x direction. However the line joining the origin with the point Pin Fig. 32 may equally well be chosen as our x axis. Hence the force direction at P must be along this new x axis, i.e. radially outward from O, in order that no set of axes be favoured. The force on the test particle is therefore not in the direction defined by the vector sum of the force components in (84). If we consider the vector sums corresponding to two different sets of spatial axes we find Et+ E f.+ E i # E i' + E -O'+ E t'. x- y- z- x'- y'l z'- (~,f and~ are the superluminal cartesian unit vectors.) Thus the direction defined in this way is not independent of the choice of axes. The actual force direction is given by •••• ( 8 5) and this equation holds true for any choice of spatial axes. This suggests that we should consider both the covariant and the contravariant components of the 3-vector (E ,E ,E) in X y Z describing the force field. Taking E ,E ,E to be the covariant X y Z components, the contravariant components are Ex,-Ey,-Ez in accordance with (BlO). We use these to define two different field vectors as follows:- 61 E. - (E , E ,E ) -l X y Z av = - --. (i=l,2,3) .... ( 86) ax l Ei (E -E -E ) - X' y' Z = (- :~, + :~, + ~~) •••• ( 8 7) E. simply consists of the vector components of the potential -l gradient. ~i specifies the direction of the force on a particle and is equal to the rate of change of the particle's momentum:- d1/ Ei = •••• ( 8 8) ~ dpx dpy dpz = ( dt ' dt ' dt ) dp dpz Hence we see that rt and cit" are equal to the positive gradient of the potential in their respective directions. Thus if we consider a particle which starts from rest and moves in the direction of~ l , they and z components of its velocity will be seen to be "uphill". It should be noted that the magnitude of the force in any direction is equal to minus the gradient of the potential in that direction. In particular we may write av f' -ar'- •••• ( 8 9) This equation shows that the method of obtaining (55) from (54) was valid. It is interesting to examine the motion of a particle constrained to move only on a line in the xy plane parallel to they axis. If the geometry were Euclidean, such a particle starting from rest would simply move in the direction of they component of the net force vector. However for hyperbolic 62 geometry the particle will accelerate in the opposite direction, i.e. in the direction of E = -Y For the potential we are considering, this means it will move towards the x axis as shown in Fig. 33, rather than away from it. Although this result seems surprising at first, it is seen to be reasonable by viewing the motion relative to the x' and y' axes as in Fig. 34. Equations (84) to (89) are valid in general for any static potential. It is apparent that the net force on a test particle is always in the direction perpendicular to the equipotential surfaces, just as in the subluminal case. However this is not the direction of maximum potential gradient. For any point P, the force component in an arbitrary direction e (see Fig. 35) rises from a minimum value for the radial direction (0=artanh~) X to infinity in either of the null directions (0=±00 ). Continuing the rotation, it then falls to zero for the direction perpendicular to the radial direction. The net force is thus in the direction of a local minimum in the potential gradient. The magnitude of this force is given by .... (90) This definition is necessary in order that the relationship be consistent with equation (43). The significance of the quantities E. and Ei in equations -1 (79) is now clear: E. consists of the vector components of the -1 potential gradient whilst Ei gives the direction of the field. 63 The magnitude of the field intensity in this direction is given i !,: by (E .. E ) 2 • -l - 7.8 POINT TACHYON FIELD RELATIVE TO A SUBLUMINAL FRAME We are now in a position to transform the point tachyon field solution in equations (54) and (55) to a subluminal frame. Using (82) we obtain = + y"Y_y = A = 0 C ' z Hence expressing V in terms of subluminal variables we have yK V = (Y 2 [x-vt] - - 2 -y-2 -z) -2 ½ .... ( 91) y V K = C ( Y2[- x-vt ]2 -y-2 -z-2)½ For simplicity we will consider the instant of time t=O, so that these equations reduce to V = YK (y2-2X -y-2 -z-2)½ y V K C = 2-2 -2 -2) ½ ( Y X -y -z Thus in a subluminal frame the equipotential surfaces of a tachyon's field correspond in shape to the hyperboloid 2-2 -2 -2 Y x -y -z = constant discussed in §4.1 and shown in Fig. 14. From equation (54), the rest frame electric field components are found to be 64 av Kx E = = + , X ax 3 r av E = - = !r , y ay r3 av Kz E 3 z az r By applying the transformations (73) to these expressions, the subluminal electric and magnetic field components are obtained:- Kx KY(x-vt) E = - - X r3 (Y 2 [x-vt] - - 2 -y-2 -z) -2 % E + KYz: y = y(-~) = - 2 - - 2 r (Y [x-vt] -y-2 -z) -2 % Kz KYz E = + y(-3) = - z r (Y 2 [-x-vt-]2 -y-2 -z) -2 ~ H = o .... ( 9 2) X -YKzV - vy(-Kz) = + C 2 - - 2 -2 -2 \ C r 3 (Y [x-vt] -y -z) -YKyV - Hz=+ vy(-!l.) = - C C · r3 (Y 2 [-x-vt-] 2 -y-2 -z)-2 % Taking the instant ~=O again we see that, even at large distances from the particle, the field intensity is strong near the conical surface 2-2 -2 -2 y X -y -z = 0 The intensity is actually infinite at any point on this surface. Furthermore, returning to (91) it is seen that the potential is infinite at any point satisfying 65 2 - - 2 -2 -2 Y (x-vt) -y -z = 0 If a test particle located at such a point is allowed to move under the action of the field to a region of small V, the particle must acquire an infinite amount of energy from the field. This field solution obviously must be considered unphysical. 7.9 FIELD BOUNDARY Another feature that should be noted about the field solution in equations (91) and (92) is that it becomes imaginary in the region 2 - - 2 -2 -2 Y (x-vt) < y +z .... (93) The physical significance of this becomes apparent by transforming the inequality back to the superluminal rest frame. We obtain This is the equation corresponding to region II in Fig. 29 and it was shown in §7.1 that, in general, no field which propagates at speed c can exist in this region. Hence the field intensity will be zero for the subluminal region given by (93). A subluminal observer will initially detect no field as the tachyon approaches him. With the arrival, however, of the moving conical surface 2 - - 2 -2 -2 Y (x-vt) = y + z there is a sudden jump in the intensity as shown in Fig. 36. (The forward part of the cone is excluded since it corresponds to region III in Fig. 29.) 66 The existence of such a boundary is a general characteristic of any tachyon field whose propagation speed is c. Although it has the appearance of a shock wave, it should not be interpreted as a radiation wavefront because the field is static in the tachyon's rest frame. For the field of equations (92) the instantaneous jump is from zero to infinite intensity. This must obviously be avoided in any realistic electromagnetic field solution. 7.10 FIELD FOR NON-ZERO RADIUS The field solution in equations (54) and (55) can be avoided by considering the more general case of a tachyon with non-zero radius. The surface of the tachyon may then have any shape we wish. Hence in solving equation (47) we are no longer restricted to radial symmetry and other solutions are possible. (In fact, as pointed out in §4.3, there is no finite closed tachyon surface which is radially symmetric in its rest frame.) The appropriate solution for a particular tachyon is now dependent on the shape of the tachyon and on the distribution of its charge density. Nevertheless the rest frame potential V must still satisfy (47) at any point outside the tachyon's surface (i.e. where the charge density is zero) and we will therefore consider what form solutions of (47) may have. By transforming the usual subluminal potential K v = -r to a superluminal frame using (81), the equipotential surfaces of this field relative to the superluminal frame are found to be of the form of equation (22), i.e. spheroidal (see Fig. 17). One might expect static solutions of this form to exist for 67 tachyons with appropriate shapes and charge distributions. In particular one might hope for a rest frame potential solution with spherical equipotential surfaces, as in Fig. 18, for some reasonably simple tachyon charge distribution. However it is possible to show that these and similar solutions do not exist. Consider a static potential solution which is positive, continuous, smooth and monotonically decreasing towards zero at infinity. We will examine two infinitesmally separated equi potential lines of this potential in the z=O plane. If this potential is to bear any resemblance to the solutions suggested above then at some point Pin this plane the equipotential lines must be as shown in Fig. 37. The two lines are parallel to the y axis at P and bend down towards this axis as y increases. (For the simple potentials suggested in the last paragraph, the point P for any equipotential line will be on the x axis.) Since the potential is positive and asymptotes to zero as x . a2v increases, --2 must be positive at any point. Furthermore, from ax 2 the diagram it is not difficult to see that -.:-7a v must be negative ay at P. Therefore the combination of the first two terms in a2v equation (47) must be positive and hence --2 must also be az positive for the left hand side to equal zero. The equipotential lines in question are the z=O cross section of the field's equipotential surfaces. Let us consider the cross-section of these surfaces in the plane of constant y 2 which passes through P. Since we have found that -.:-7a v must be az positive, the curvature of these "constant y" equipotential lines must be away from the x=O plane. In particular if the lines are parallel to this plane at P (as were the "z=O" lines) then they must curve upwards as shown in Fig. 38. This is obviously incompatible with the ellipsoidal surfaces mentioned 68 above, or any similar surface which curves downwards to close on itself as in Fig. 39. This "curving upward" of the equipotential surfaces is characteristic of solutions to equation (47). The surfaces curve away from the tachyon rather than around it. To see the full significance of this we will now consider another restriction that must be placed on possible field solutions. As mentioned above the equipotential surfaces of a tardyon's field relative to a superluminal frame are spheroidal, as represented in Fig. 17. This field extends into region II of Fig. 29. However the field of a tachyon is necessarily restricted to region I. This means that any tachyon field similar in form to the tardyon solution would have to terminate abruptly at the boundary between the two regions. To avoid the situation of the equipotential surfaces ending discontinuously at this boundary, the field must be such that the surfaces curve back to the particle as in Fig. 40. The potential would then drop continuously to zero as the boundary was approached from the region I side. However the analysis above excludes the existence of such a solution since the xz cross-section of the potential cannot curve back if the xy cross-section does, and vice versa. The only other way whereby the surfaces of constant potential may avoid the boundary is by curving off to infinity as in the radially symmetric solution (Figs. 32 and 36). We are thus restricted to an unpromising range of field solutions. 69 SECTION 8 DISCUSSION AND CONCLUSIONS 8.1 SUMMARY OF FINDINGS Most of the conclusions in this paper ultimately stem from the transformations in equations (5), (8) and (10). Although equations (10) have not previously been explicitly stated in the literature on tachyons, the adoption of (5) and (8) has been advocated before by R. Mignani and E. Recami (see Ref. 5) .* The general approach of those authors with respect to transluminal Lorentz transformations is essentially the same as presented here, except that they insist on maintaining the same metric signature for both subluminal and superluminal frames so that it is necessary to introduce the extra transformations y = iy z = iz The method chosen in §7.5 for generalizing Maxwell's equations was also suggested by them in the brief discussion of electro magnetism in Ref. 5. In §3 the properties of superluminal spaces were discussed and these were found to be very strange in comparison with the familiar properties of Euclidean spaces. For example the existence of null lines was found to be characteristic of such spaces, and the angle between any two intersecting null lines was shown to be infinite. Another peculiarity follows from the known macroscopic causality of tardyons. From the fact that effects are always later in time than their causes relative to a subluminal frame, we conclude that a superluminal observer must see "positive x direction" causality. In other words since the There are, however, some aspects of this paper with which the present author does not agree. 70 positive time direction of a subluminal frame corresponds to the positive x direction of a superluminal frame, tardyon events as seen by a superluminal observer will always be at points of larger x than their causes. A more precise way of saying this is that the entropy of an isolated tardyon system will be seen to increase in the positive x direction of a superluminal frame. It is most unlikely that any sort of observer could actually exist in a superluminal frame because of the peculiar geometry involved. Nevertheless such frames serve as a convenient mathematical structure relative to which the properties of tachyons may be investigated. To this end, an important property of superluminal spaces is that the concepts of volume and area continue to have an absolute meaning, as shown in §5. This being the case, the general ideas employed in that section may be used in calculations such as those concerning surface and volume charge densities. A consistent though somewhat different theory of dynamics may be developed for describing the motion of particles relative to a superluminal frame. The expressions for energy, momentum, etc. are similar in form to their familiar subluminal counter parts. However it is necessary to take account of both the contravariant and covariant components of spatial vectors when forces are introduced, and hence certain new concepts such as the two different electric field vectors E. and Ei need to be -l employed. Due to the definition for the magnitude of a 3 vector, care must be taken in deciding whether or not to describe the motion of a particle in terms of components. For instance the net direction of the force on a 71 particle in a static potential is not given by the sum of the three force components. On the other hand, the motion of a particle along a null line must be described in terms of its momentum components p ,P ,p , since the net magnitude of its X y Z momentum along such a line is always zero. Some interesting general conclusions can be drawn from the work on tachyon surfaces in §4. The main point is that there is no closed surface which is symmetric under a rotation of superluminal spatial axes (i.e. radially symmetric), since the general shape corresponding to radial symmetry is that of a hyperboloid. Furthermore, if an enclosed constant volume is deformed progressively to approximate closer to radial symmetry, it will become less and less localized. This incompatibility between complete symmetry and localization is inherent in the question of the form of tachyon fields, and lies at the root of the difficulties encountered in the study of such fields. In order that the intensity of a tachyon's field be negligible at large distances from the particle, it is necessary that the field show a strong departure from radial symmetry in its rest frame. This lack of symmetry will also be evident relative to sub luminal frames. In particular, any tachyon field which is axially symmetric about an x axis of its rest frame will be axially symmetric about the particle's direction of motion in certain preferred subluminal frames, but not in others. It appears to be impossible to derive the electromagnetic field of a tachyon classically. The solution found for a point tachyon must be considered physically unrealistic and, as shown in §7.10, the introduction of extended charge distributions does not help to resolve the problem. The difficulties faced are intimately connected with the existence of the cut-off boundary illustrated in Fig. 36, which is peculiar to tachyons. 72 In addition to the problems encountered in the preceding work, there are two well known difficulties confronting a theory of faster-than-light particles. These will now be briefly described. V 8.2 CERENKOV RADIATION FROM TACHYONS From a quantum mechanical point of view, spontaneous emission of photons will occur from a particle provided the laws of conservation of energy and momentum are not violated in the process. There will be a definite non-zero probability of a transition from the state "tachyon" to the state "tachyon plus emitted photon". We will now determine under what circumstances this process is permitted. Consider a particle with energy 2 m C 0 E = -2 ½ 1 - u --z-c and momentum m u = o- £ -2 1 - --z-u ½ C We assume that the particle is moving in a medium of refractive index n, and that it emits a photon of energy hw and momentum --n"' C A hK=hw-k, - being the speed of light in the medium and k the - C- n unit vector in the direction of K. Taking u to be the particle's initial velocity and v to be its final velocity, conservation of energy tells us 2 2 m C m c 0 0 = + •••. ( 9 4) -2 -2 nw u ½ V ½ 1 - 2 1 - 2 C C 73 whilst conservation of momentum gives mu m V o- = o- + ii.le .... ( 9 5) -2 v2 ½ 1 - u ½ 1 - -z- -z-c c Squaring (95) we find 2-2 2-2 mu m V 2-2 2 2m vnwn 0 0 w n 0 = n + cosa , •••. ( 9 6) -2 -2 2 -2 u V C V k2 1 - --z- 1 - --z- C 1 - --z- C C C where a is the angle between the vectors v and k. 2 Squaring (94) and dividing by c we obtain 2 2 m2c2 m C 2-2 2m i'iw 0 0 ii w 0 = + -2 -2 -2- -2 u V C V ½ 1 - --z- 1 - --z- 1 - --z- C C C 2 2 2 2 m C m C 2m nw 0 + 2 2 0 + 2 2 0 ---_---=-2- mO C m C 0 -2 u v2 V ½ 1 - 2 1 - 2 1 - 2 C C C 2-2 2-2 mu m V i'i2w2 2m i'iw 0 0 = + 0 •.•• ( 9 7) -2 -2 -2- -2 u V C V ½ 1 - ----r 1 - ----r - --z- C C 11 C The upper signs apply for u,v •••• ( 9 8) In the case of a vacuum (i.e. n=l), this equation only yields real (physical) values for a for the velocity range v>c. In other words, a tachyon is free to emit photons in a vacuum whereas such emission is forbidden for tardyons due to violation 74 of the conservation laws. Equation (98) also predicts the well V known Cerenkov radiation of a tardyon moving faster than the speed of light~ in a refractive medium. The problem is now to derive the probability for photon emission per unit time and hence the power emitted. This calculation has been done for an electron in a medium (Ref. 6) and the result may be generalized to the tachyon case. The total radiated power is given by we 2- 2 -p e WU (l C ) d- = ! 2 -----z=z . w •••• ( 9 9) 0 41rc e: 0 nu where 2E(vn_l) C 2 fl (n -1) we is the cut-off frequency imposed by the condition that cosa in equation (98) must be ~1. Unfortunately the cut-off frequency becomes infinity for n=l, and consequently equation (99) predicts the power emitted by a tachyon in a vacuum to be infinite. To overcome this difficulty it was suggested (Refs. 7, 8 and 9) that -wc=fi"E be adopted as the cut-off on the grounds that a tachyon could not emit more energy than it had. However Wimmel (Ref. 10 and 11) pointed out that this limit is dependent on one's choice of reference frame, since the tachyon will have a different energy in each frame. Thus in the absence of a medium (i.e. a preferred frame) no value of energy should be favoured as the cut-off limit. The problem therefore remains unsolved. If tachyons do emit energy at a finite rate in a vacuum then, owing to the increase of a tachyon's velocity with decreasing energy, their velocity must be seen to approach infinity. Since the state of infinite velocity is dependent on 75 our choice of reference frame (see Fig. 41), we are led to the conclusion that a radiating tachyon must approach infinite velocity relative to every frame in turn. This means that it must eventually "exceed" infinite velocity relative to each frame and turn backwards in time. This will appear to a subluminal observer as the collision of a tachyon with an antitachyon, so that every radiating tachyon will have an associated antitachyon. Returning to the question of the power emitted, it is reasonable to expect that no more energy can be emitted than is available at the time. We can, however, always choose our frame so that the tachyon's velocity is infinite at the instant concerned, and from equation (38) we see that for u= 00 the tachyon's energy will be zero. Therefore it seems to follow that no energy can ever be emitted. This argument appears to preclude the possibility of VCerenkov emission from a tachyon in a vacuum on the grounds of energy non-conservation. The flaw in the above argument is that, relative to the u= 00 frame, a tachyon which emits a photon will be seen to annihilate with an antitachyon, and the energy of the antitachyon must be taken into account. Since the energy is emitted in discrete quanta rather than continuously, there must be a sharp bend in a tachyon's world-line associated with the emission of a photon as shown in Fig. 42. Thus if the velocity of the tachyon is infinite then the velocity of the antitachyon cannot be infinite, and vice versa. It is impossible to choose our reference frame so that the energies of both the tachyon and the antitachyon are zero. Hence there is always energy available for radiation and it is not difficult to show that there is still no cut-off restriction. 76 Electromagnetic CerenkovV radiation only occurs with charged 2 particles as indicated by the factor e in equation (99). Therefore even assuming that the infinite radiation difficulty cannot be overcome by theoretical advances, it can of course be avoided simply by requiring that tachyons be uncharged. V However the rate of emission of gravitational Cerenkov radiation is also found to be infinite (Ref. 9), and this cannot be by passed so simply. In addition to the references already quoted, V further discussion of the question of Cerenkov radiation may be found in Refs. 12, 13 and 14. 8.3 CAUSAL LOOP PARADOX The following paradox constitutes a serious objection against the existence of tachyons. Consider three devices A,B,C for emitting and receiving tachyons. A andC are standing side by side at the origin of the reference frame S, whilst Bis in the frame S' moving with velocity v( We now assume that A is switched on at time t=O. The problem is "What happens?" If a is emitted then bis emitted and hence C stops A from emitting a. If a is not emitted then C is not activated by band there is no reason why A should not have emitted a. We conclude that if a is emitted then it is not emitted, and vice versa. This is a genuine paradox. One way of attempting to resolve the paradox is by high lighting the fact that, relative to the frame S', pulse a is seen to travel from B to A rather than vice versa. An observer at rest in S' and standing beside B will see the pulse emitted rather than received by B. One thus argues that a cannot be regarded as a proper signal for this reason and that this is where the loop is broken. It is possible to avoid this objection quite simply however by introducing another device D at rest in the frame S. We assume that pulse a is now sent from A to D, and that Dis in close proximity to B when the pulse arrives. Dis then programmed to send an ordinary electromagnetic signal to B to reform the causal chain. Two observers at rest beside A and D respectively will agree that A sent the pulse and D received it, so that relative to A and D there is no ambiguity in the direction of the pulse. Since no other device is directly involved with pulse a, the order of emission and reception relative to other frames is irrelevant, and the improper signal doubt has thus been dispelled. Another device E may be introduced at rest in S' to remove the same objection in relation to the reception of pulse b. Thus the paradox remains unresolved. If we assume that the theoretical framework of relativity is correct and add the extra postulate that signals may be sent at speeds faster than light, we are faced with a logical contradiction. For the arrangement described above we find that 78 a signal is sent if and only if it is not sent, and vice versa. The only way out of this quandary is to restrict the independence of the components involved in the experiment. This means assuming that although the three devices A, Band C operate satisfactorily on their own or in pairs, the incorporation of them into one experimental arrangement is necessarily accompanied by the emergence of some interconnection between the parts which prevents the desired mode of operation. If this were the case then the overall set-up would have some extra degree of unity and hence could not be considered as equivalent to the sum of separate parts. One can, of course, escape from the paradox by adopting a different underlying space-time structure. For example we may postulate that there is a particular frame of reference relative to which tachyonic signals must always travel forward in time. Introducing such a preferred frame solves the paradox by violating the principle of relativity. However it seems more reasonable to conclude that faster-than-light signalling is impossible. This places severe restrictions on the possible properties of tachyons if they exist. Any experiment aimed at producing and then detecting tachyons could in principle be used to create a causal loop as described. Therefore we conclude that such an experiment must yield a null result. The existence of virtual tachyons is not ruled out however, and Refs. 15 and 16 consider this possibility. A large number of papers have been written on the causality difficulties associated with tachyons. Ref. 17 contains an interesting discussion of this topic. 79 8.4 QUANTUM MECHANICAL APPROACH This paper has been concerned with a classical rather than a quantum theory of faster-than-light particles. A number of papers have been published however on the development of a quantum field theory of tachyons. In several of these (Refs. 18 to 21) the conclusion was reached that no propagation of signals faster than light was possible within the theoretical framework developed, so that no violation of causality could occur. Since this restriction is clearly not inherent in the classical framework, the possibility suggests itself that it may be necessary to adopt a quantum mechanical approach from the outset to obtain meaningful results. It could be that, unlike tardyon mechanics, a quantum theory of tachyons does not reduce back to classical point particle theory in the macroscopic limit. The fact that tachyons cannot be completely localized in the theory lends some support to this notion. (Since the momentum of a tachyon is restricted to the range m c~p<00 , the 0 eigenfunctions belonging to the range 0~p APPENDIX A SPACE-TIME AND MINKOWSKI DIAGRAMS The fundamental physical reality underlying the mathematics of Special Relativity is that the universe consists of a four dimensional continuum whose geometry is hyperbolic. This continuum is called space-time and the distance (or "interval") between any two points (or "events") within it is given by 2 2 2 2 2 k ds = (dx +dy +dz -c dt) 2 .... (Al) The velocity of light c is the scaling factor relating our unit of time to our unit of spatial length, i.e. 1 second= 3xl0 8 metres. Equation (Al) is obviously a generalization of Pythagoras' Theorem to four dimensions, but the minus sign indicates that one of the dimensions is of a different type to the other three (i.e. space-time is not isotropic). The significance of the minus sign is that a new type of geometry ("hyperbolic geometry") is involved, rather than a simple extension of Euclidean geometry from three to four dimensions. Although this different geometry of space-time is non Euclidean it is nevertheless flat, as opposed to the curved non-Euclidean geometry of General Relativity. It is hard to illustrate four dimensional hyperbolic geometry on a two dimensional Euclidean piece of paper. However a representation known as the Minkowski diagram can be constructed. The main point to keep in mind when dealing with space-time diagrams is that the apparent distance between two points on the diagram is not necessarily in proportion to the 81 corresponding distance in space-time. For example, 11 is greater than 12 in Fig. 44 although it does not look it. This can be seen by using equation (Al):- (Usually only the xt plane is represented in a Minkowski diagram. However if necessary one of the other two spatial axes (y and z) may be included projectively on the diagram as in Fig. 45.) Equation (Al) indicates that it is possible for two events to be separated by zero distance even if the components dx,dy, dz, cdt, of the separation distance are not all zero. It follows that there is a hypercone (the four dimensional analogue of a cone) associated with every event in space-time consisting of all those events which are zero distance from the event in question (Fig. 45) . The equation of the hypercone associated with the origin O is seen to be From this equation it is apparent that the hypercone associated with any particular event is the four dimensional representation of a spherical light pulse emitted at the event. For this reason these hypercones are commonly referred to as "light- cones". Particles appear in space-time as static trajectories, or "world-lines". Every known particle is such that an increment of interval ds along its world-line at any point is imaginary under the definition (Al) fords. This means that the 82 world-line of such a particle (a "tardyon") always lies inside the light-cones associated with events along it (i.e. the world line must always be more vertical than the sides of any light cone, as in Fig. 45). In general, straight lines in space-time fall into two distinct categories, space-like and time-like, according to whether the spatial component of their length is greater or less than the time component. Thus any segment of a tardyon's world-line is always time-like. In graph work it is equally permissible to use either oblique or rectangular sets of axes (see Fig. 46). However rectangular axes are usually employed because they are more convenient (since Pythagoras' theorem and the simple laws of right-angled triangles may be applied.) In Fig. 47 the space and time-coordinate grids for two different reference frames are shown together. The slanting lines correspond to constant position in a frame S' moving to the right relative to the vertical grid frame S. We choose the x=O and x'=O lines to be the t and t' axes respectively. (The time axis of a particular reference frame corresponds to the world-line of a particle at rest at the origin of that frame.) In this diagram we have two different position grids but only one time grid. The same set of constant time lines is used for both frames. Whilst this is consistent with Newtonian physics, we know in Special Relativity that two spatially separated events which are simultaneous for one observer will not be simultaneous relative to another observer in a different frame. Fig. 47 must therefore be non-relativistic. Taking the lines of constant time shown to be those of frame S, we want to find the time grid corresponding to S' for the relativistic case. It is not difficult to show (see for 83 example Ref. 2) that the lines of constant t' must be inclined as in Fig. 48, and that the angle between the x and x' axes must be equal to the angle between the t and t' axes. (The scales have been chosen so that the world-line of a photon makes an angle of 45° with the x and taxes.) Thus we have two different coordinate grids for describing the same underlying space-time continuum. All the usual predictions of Special Relativity now follow from this. We see that the underlying reality suggested by the Special Relativity theory is that the universe is a four dimensional continuum possessing hyperbolic geometry. The fact that different observers obtain different results for length and time measurements is due to them using different coordinate grids for describing the events existing in space-time. In particular, the normal three dimensional space seen by an observer is a cross-section of the overall four dimensional continuum, and there is a different three dimensional cross section corresponding to each reference frame. We could, of course, have combined space and time in pre-relativity physics to form a four dimensional space as in Fig. 47. However this would have been artificial since time was absolutely separate from the space dimensions in Newtonian physics - there was no mixing of space and time. If ds is defined as in equation Q\l) then it must be imaginary for time-like intervals. However it is equally permissible to adopt the convention of defining ds by 2 2 2 2 2 ½ ds = (c dt -dx -dy -dz) in which case space-like intervals are the ones which are imaginary. Since it is usual to take the coordinates x,y,z,t to 84 be all real, it seems inconsistent to define lengths along either the time axis or the spatial axes to be imaginary. Therefore it is better to define ds by 2 2 2 2 2 k ds = I c dt -dx -dy -dz I 2 and this definition will be adopted in this paper. A more detailed coverage of the material in this appendix may be found in Refs. 2, 3 and 4. 85 APPENDIX B GENERALIZED 4-VECTOR TRANSFORMATIONS Th e contravariant components o f a 4-vector,a-µ_ =(a -0 ,a-1 ,a-2 ,a),-3 transform from one subluminal frame to another according to •••• (B 1) whereµ and v here denote two different frames, and -µ _ -0 -1 -2 -3 ---- x = (x ,x ,x ,x) - (ct,x,y,z) Also, the covariant components of the 4 vector, aµ=Cao,a1,a2,a3), transform via the equation .••• (B 2) We will generalize this formalism to v>c so that for contravariant components we have ax µ -v aµ = --a .••• (B 3) ax:v and for covariant components we have •••• (B 4) Equations (B3) and (B4) will now be written out in full under the assumption that the superluminal frame is moving with velocity +v in the x direction relative to the subluminal frame. Using equations (5) to calculate the partial derivatives, equation (B3) becomes a 0 =- Y(-0a - cav-1) , a 1 =- Y(-1a - cav-0 ) , a 2 =-2a , a 3 =-3a , •••• (B 5) 86 and using equations (7), (B4) becomes .... (B6) The inverse transformations are -µ ax" a and a = --a µ axµ \) Expanding these gives -0 0 V 1 a =+Y (a +-a ) , -1 a =+Ya( 1 +-aV 0) , -t a =a 2 , -3 a =a 3 .•.. (B 7) C C and -a = - Y( a --aV ) - a = - Y( a --aV ) .•. (B 8) 0 0 C 1 ' 1 1 C 0 As in the usual subluminal case, the contravariant transformation is of the same form as the inverse covariant transformation and vice versa. The subluminal contravariant and covariant components of a 4-vector are related in the following manner:- •.•• (B9) This follows from the equation since the subluminal metric signature is (+---) and hence the subluminal metric tensor is given by From equations (BS) and (B6) we see that the relationship between the superluminal contravariant and covariant components of a 4-vector must be 87 •..• (BlO) This is in accordance with the transformation since it follows from the superluminal metric signature (-+--) that the superluminal metric tensor is 88 APPENDIX C The aim of this appendix is to discuss the ideas of superluminal reference frames and transluminal Lorentz transformations in more detail than in the body of the thesis. In particular, consideration will be given to exactly what assumptions were made in deriving the transformations stated. The question could be asked, for example, if they are the only transformations possible. Also, the notion of a preferred superluminal spatial direction could be confusing and so the necessity of it must be satisfactorily demonstrated. In relativity we are concerned with the coordinates of events occurring at various points in space at particular times. The Lorentz transformations relate the coordinates (x,y,z,t) of an event relative to one inertial frame to the coordinates (x' ,y' ,z',t') of the same event relative to another frame. Each event may be considered as a point in a four dimensional space spanned by four mutually perpendicular axes labelled x,y,z and t. The three dimensional regions corresponding to constant values of, say, yin this picture will be those planes (or more correctly "hyperplanes") which are normal to they axis. We will now consider a particle at rest at the origin of the dashed frame. It is assumed that the dashed frame is moving with constant velocity in the undashed x direction and that the origins of the two frames coincide at t=O. The trajectory of the particle may be represented by a straight line in the xt plane:- 89 This line consists of the set of events (x',y' ,z' ,t') = (0,0,0,t'), and any line parallel to it in the four dimensional space must correspond to constant spatial position in the dashed frame. Clearly the (O,O,O,t') line is the t' axis (i.e. the time axis of the particle's rest frame). The three dimensional planes which are normal to the t axis correspond to the usual three dimensional Euclidean space seen by an undashed observer, each plane corresponding to one instant of time t (i.e. all the events in each plane are simultaneous). The essence of special relativity, however, is that simultaneity is not an absolute concept. Events which are simultaneous relative to one frame of reference need not be simultaneous relative to another frame. Therefore the above spatial planes of constant t cannot also be the planes of constant t' . It is well known that subluminal special relativity can be represented geometrically by way of Minkowski diagrams, and that this geometric representation is entirely equivalent to the algebraic formulation. The Minkowski picture is obtained by imposing a particular type of geometry ("hyperbolic", or "pseudo-Euclidean", geometry) on our four dimensional space. In this picture the three dimensional planes of constant time for any particular reference frame are those planes which are 90 normal to the frame's time axis (although it should be noted that, due to the different geometry, such planes will not appear on a diagram to be normal to their time axis from the point of view of our Euclidean intuition). The two dimensional cross-sections of the planes of constant t and t' are shown below:- We thus have a geometric representation of the subluminal Lorentz transformations whereby they and z axes remain unchanged whilst the x and taxes rotate towards one another. By choosing the scale appropriately it is possible to make an inclination of 45° correspond to motion at the speed of light (see Appendix A). SUPERLUMINAL FRAMES Up to this point, no mention has been made of faster than light frames and particles. Let us now consider a particle moving with constant velocity v>c in the undashed (subluminal) x direction so that it passes through the origin at t=O. Clearly the path of this particle will appear on our diagram 91 as a straight line in the xt plane inclined at an angle greater than 45° relative to the taxis - i.e. a space-like line (see Appendix A):- cli ------..- ..-, - ' ------;-J' ' / ' / u'J\ "' ' / / / ' / ______' '-..;.w;/.'------~ X () It should be noted that the trajectory of a tachyon must lie within the present space-time picture in order that it be observable relative to subluminal frames. We will consider the (superluminal) rest frame corresponding to this particle. The world-line of the particle on our diagram must correspond to constant position in this frame and, taking the particle to be at the origin, the line must comprise the set of events (x' , y' , z' , t ') = ( O, O, O, t ') , i.e. it is the t' axis just as in the earlier discussion concerning a subluminal particle. All lines in space-time which are parallel to this axis must denote constant position in this superluminal frame. Now only the three dimensional planes of constant t' are· required to complete the picture (i.e. the spatial part of the superluminal frame). By analogy with subluminal frames, the planes of simultaneity are taken as normal to the t' axis, each such plane corresponding to one instant oft'. This ensures that the four axes of a superluminal frame are all perpendicular to one another (assuming the three spatial axes are perpendicular). The nature of superluminal inertial frames has now been completely determined:- 92 It is important to note exactly what assumptions have been made in the preceding argument. They are:- 1. that standard relativity theory is valid; 2. that the Minkowski picture is equivalent to this for subZuminaZ frames; 3. that a tachyon corresponds to a space-like line in Minkowski space-time; 4. that the superluminal rest frame of a tachyon is defined in analogy to subluminal frames to have three spatial dimensions and one time dimension, and to be such that the spatial position of the tachyon in the frame is constant; 5. that the spatial plane of such a frame for any instant of time is normal to the time axis, as is the case with subluminal frames. It should be realized that for every superluminal frame there is a corresponding subluminal frame with the same four axes, the only difference being that the labels on the x and t 93 axes are interchanged. Hence the introduction of super luminal frames is a rather tr~vial step and certainly should not be considered an extension of special relativity. The Lorentz transformations describe all possible rotations of four perpendicular axes in space-time, and since one axis must always be inside the light cone there is a subluminal frame corresponding to every possible orientation. Thus if the labelling on the axes is ignored, it is true in one sense to say that there are no frames other than subluminal frames. There are, however, 4!=24 different ways of labelling four perpendicular axes with the letters x, y, z and t, and for only six of these will the axis inside the light cone be labelled t. These are subluminal frames, and the other eighteen are superluminal frames. (The six subluminal permutations actually correspond to only one subluminal frame with various interchanges of the spatial axes. Likewise the eighteen superluminal cases actually correspond to only three superluminal frames with various permutations of the spatial axes.) Because one of the superluminal spatial axes must lie within the light cone (i.e. must have the properties of the subluminal time dimension) it will be different from the other two spatial directions, and hence the spatial part of a superluminal frame cannot be isotropic. (We have chosen to call the preferred spatial direction the x' direction. However, in general we can determine the special direction in a superluminal frame by considering the corresponding subluminal frame with the same four axes. The preferred direction is always along the axis of motion of this subluminal frame relative to the superluminal frame, whichever direction that 94 is.) Clearly the existence of a preferred superluminal spatial direction is nothing more than a restatement of the fact that four dimensional space-time is not isotropic, having a preferred dimension (the subluminal time dimension). TRANSLUMINAL LORENTZ TRANSFORMATIONS The exact form of the transluminal Lorentz transformations can now be deduced. Consider a superluminal frame moving with velocity v>c along the x axis of a subluminal (undashed) frame. (Subluminal coordinates will hereafter be written with bars over them.) We assume that the origins and spatial axes of the two frames coincide at t=t=O. Corresponding to the superluminal frame there will be a subluminal frame with the same four axes but with the x and t labels interchanged. This frame will be denoted by dashed coordinates. It will be seen to be moving with some velocity v -~'- I I /, 'Ca-t. I I I We wish to find the relationship between V and v. From the diagram we have dx1 tancp = = V 1 -C cdt1 cdtz C tan Hence since $1 is always equal to $2 on a Minkowski diagram, we may combine these equations to get - V C - = C V - V = •••• ( C1) V The transluminal transformation from the undashed subluminal frame to the superluminal frame is simply a combination of two steps: a Lorentz transformation from the undashed to the dashed subluminal frame:- x' = y (x-vt) , y'=y , z'=z , --t'=y(t----z-,vx) .... ( C2) C -2 -½ where y = (1-z)V , followed by the interchanges: - C y'+y, z'+z , x'+ct, ct'+x .... ( C3) The overall transformation can be found by using equations (Cl), (C2) and (C3). Inserting (Cl) in (C2) we obtain 2- x' = y(x-cvt) , y'=y, z'=z , ---t'=y(t-"),X c2 where y = (1---z-) -½ , and inserting (C3) gives V - 2- t = -(x--)Y - C t , y=y , z=z , x=cy--x (t--) C V V Now, V y = 1 = C y_y. 2 1 2 I - C (1-½-)'2 ( :;--1) '2 V C Hence we obtain 96 2 t = y_Y (x-~t) C2 V - vx = -y(t----z-) C -x and X = vy(t--) V = -y (x-vt) Thus the transluminal Lorentz transformations have the form:- - vx X = -y (x-vt) , y=y , z=z , t=-y(t----z-) , C 2 1 2 V -½ where y = ( :;--1) -Yz = 1----z- C C These are equations (5) in the thesis. SUPERLUMINAL LORENTZ TRANSFORMATIONS The equations for transforming between two superZuminaZ frames will now be discussed in more depth. These trans formations are designated as equations (8) and (10) in §3.3 of the thesis. It is important to understand why two different sets of transformations are needed. We will examine four cases:- (A) Consider three different reference frames, one subluminal and two superluminal, moving relative to each other along their common x axis. (A) (Here X stands for all the axes x, x and x', and similarly Y stands for y, y and y'.) ....______,..y 97 The first superluminal frame Sis moving with constant velocity v>c along the x axis of the subluminal frame S so that the appropriate transluminal equations are (equations (5) in §2):- - vx X = - y ( X - vt) , y = y , z=z , t=-y(t-2 ), C v2 -½ where y = l-c2 The second superluminal frame S' is moving with velocity v' x' = y'(x-v't), y'=y, z'=z, t'=y'(t-v'~), C -1: VI 2 2 where y' = l--2- . C These last equations are identical with the usual subluminal Lorentz transformations, as one would expect from the symmetry between subluminal and superluminal frames when only two dimensions are considered. (B) If, on the other hand, the collinear motion of the three frames were in, say, their common y direction then the appropriate transformations would be:- -- - vv x=x ' y=-y(y-vt) ' z=z ' t=-y(t-z) ' C v2 -½ where y = 1-z , C and x'=x , y'=y' (y-v't) , z'=z , t'=y' (t-~), C ,2 -½ where y' = 1----y-V c 98 X (B) .______.;~Y The symmetry between the transformations for the two cases (A) and (B) above (i.e. for collinear motion in the x and y directions respectively) is in accordance with the isotropy of subluminal space. (C) So far the special transformation of equations (10) in §3.3 has not been required. However consider now the case where the relative motion of the superluminal frame Sand the subluminal-frame Sis in the x direction whilst the relative motion of the two superluminal frames is in they direction. X (cj ______,.y The transformation between Sand Sis given by equations (5) (see above) once again, but it can be shown that the correct transformation relating the two superluminal frames Sand S' is of the form of equations (10) in §3.3: y_!_y__ t+ --z-- y-v't x'=x, y' ------, z'=z, t'= C ' 2 1 ' 2 1 • (1+;-) ~ (l+v )~ C ~ 99 It should be mentioned that the transformation relating Sand S' will also be of the normal transluminal form given in equations (5) provided a general rotation of spatial axes is first performed to simplify the J- (D) The final case to be considered is where the relative motion of Sand Sis in they direction whilst the relative motion of the two superluminal frames is in the x direction. X (DJ ...... ______y The appropriate transformations are:- x=x, y=-y(y-vt) , z=z , t=-y(t-~) C v2 -½ where y = 1--z- ' C and 100 I t+ V X x-v't c2 x' = 'y'=y' z'=z t' I 2 1 ' (1 +:;-) :>z C Case (D) has been included simply to illustrate (in conjunction with (C)) that the asymmetry between the longitudinal and transverse superluminal transformations (i.e. equations (8) and (10)) is not in conflict with the isotropy of subluminal space. The above discussion has been aimed at indicating precisely when equations (8) and equations (10) should be employed. It now remains to explain more fully why the form of the transverse transformations must necessarily be different. This could be achieved by starting with the transformations S+S and S+S', then deducing what form the transformation S+S' must have. However the problem is more simply and clearly dealt with by changing to the Minkowski description of the situation. The following two space-time diagrams represent cases (A) and (B) respectively (i.e. the two collinear cases):- ------(B) cJ c,I,' / / / / / / / / / / .t -- - / ' ---~ ~------______-~ ,,, ------., In each of these two cases the two consecutive rotations are coplanar. (The rotation of the spatial axis has been deleted in both cases for the sake of simplicity. This rotation will, 101 however, take place in the usual way as described in §2 of the thesis.) Note that if the t~t' rotation in each case is large enough, the time axis will be rotated right around into the bottom half of the light cone. In fact a sufficiently large rotation will take it through the full circle so that it will re-enter the upper part of the cone, thereby becoming a normal subluminal time axis again. We now turn to cases (C) and (D) (i.e. the transverse cases). Only (C) will be explicitly discussed. The transverse situation is harder to represent clearly and will be treated in several steps:- J ~ (c)(i) Fig. (c)(i) illustrates the transluminal transfor~ation s~s. For visual simplicity the diagram has been constructed so that the superluminal x and taxes appear to be at right angles rather than the subluminal axes. It is a property of Minkowski diagrams that any frame may equally well be chosen to be the one whose axes are at right angles in the diagram . ..,:,x' ------(,.,. -,, ,,, _,4'/" ' ...... __ -*' / Cc)(iO ' .------/ ' / ' / ' / ' / / c,I,' 102 Fig. (c)(ii) illustrates the rotation of the time axis for the superluminal transformation S~S', the rotation being in the yt plane. Corresponding to the superluminal frame S there is, of course, a subluminal frame with the same four axes but with the x and t labels interchanged. Since no further use will be made of the S frame in fig. (c)(i), the notation x,y,z,t will now be used for the coordinates of this new subluminal frame (to be consistent with the notation of §3.3 of the thesis):- x,c,l (c) Qii) "f,1 Thus from fig. (c)(ii) we see that the superluminal transformation S+S' simply corresponds to a rotation of axes in the xy plane (i.e. to a Euclidean rotation of the x and y axes):- -x,x;cJ,c4/ ------...... -,,-:;, <:" ..... _ _.,.,/ ---- / ' ' -. -- / / (c)~'v) ' / ' ' / cff,x.,' -, A sufficiently rotate the time axis through a complete 360° circle. However note that, in contrast to the collinear cases (A) and (B), it always remains a superluminal time axis since it never passes inside the light cone. The spatial rotation of the x and y axes is described by the equations 103 x' = xcose + ysine , y' = ycose xsine, and from these the algebraic expressions for the transverse superluminal transformation S+S 1 can be found, as shown in §3.3. 104 REFERENCES 1. 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Fox, R., Kuper, C.G. and Lipson, S.G., Do Faster-than Light Grau Velocities Im 1 Violation of Causality? Nature vol. 223: P.597 1969 . 20. Fox, R., Kuper, C.G. and Lipson, S.G., Faster-than-Light Group Velocities and Causality Violations. Proc. Roy. Soc. vol. A316: P.515 (1970). 21. Schroer, B., Quantization of M2<0 Field Equations. Phys. Rev. vol. D3: P.1764 (1971). l.UO FIG. I (v.:c) J(f) cA-' FIG.2 (y / / / FIG, 3 / 6'->c) / / / 107 ' " FIG. b " " " " " ', -----:~--'----~ ~ X / / / FIG. 'i / / / / ______:/7t- \°" -ir;,-~ r -,c' / / / / / / x/ / / / FIG. t / ', / (v>c) " / ' ___fu' 7 ~/,::::::::=--~-.,i: c,f' / / / / / / 108 FIG.7 o~·,:-+---,----,toy- (ff) t(l) FIG.8' ~/ 1 -- " ' / / /",' ? .-- -It!:-----....,.__'''~ C B / / / FIG,Cf / / ,/ A / / / / 0 110 A F/G.13 FIG. 15 X 109 F / G. IO(a), FIG. JO(t) 'l ------~~ ---·- ~ // FIG. II ;/ /f (e=i) /1 I · I I I I l FIG. 12 111 f/G.16 FIG .11 - ---t--'°o~Ji-ii: ·i('I) X (c1) FIG. IS rR ' ' hl=0 112 x' FIG. /q {e') i 1 __...,__ .... ~~-....,._-~i 't . FIG. 20 .L .L .) FIG :21 (a) Al:f::~~~~~~ x-. -:ic' FIG.22 (a.) F/G.23 x' I (a) s' ( ,. ,~----.... , I \ \ I I I I I I A 8 I I \ ' I ,o C I ...... ------,,; J. J.'i F/G-.24 - - _____ x -. --· x'-- s i 0 1 FIG 25 -x f/G,26 JA t 1 -R 115 (c.i) FIG.2~ 116 FIG. 2'1 EIG.30 X (cJ) .L.l. / FI(;. '31 FIG. 32 X FIG.33 ,,_.,,.....,_~.it " -.LC '------,-.~-,:=--- "" - -- - '.. '\. . "/C '\. l..U:S x' 7· / ----~=--~---,I 'J FIG. '35 FIG, 36 l. l. ~ Fie. '37 '/ ~''v / / / / / / ;;>( /· FIG. 38° / / / / / / / / FIG. 3Cf L~U FIG. J+O c:J FIG. !+ 1 .. FIG.t,.'2 0!------•r X .1. .G .1. cA-- FIG. 1/-'3 t ' / D__.::,r'v Acudc /: / I "f-:-B / : X- FIG. 4:4-. :~:~I I I i I I I I 0 X '_, FIG-. 4-s- ' 122 FIG. t,.6 .. - I - - . F/G. t.,..7 A- A,' ,FIG. ftg-