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Superluminal Reference Frames and Tachyons

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Superluminal Reference Frames and Tachyons 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<c, and x' imaginary and T' real for v>c, showing that the x' and T' axes are rotated into the normal time-like and space-like regions respectively by a "transluminal" transformation (i.e. through the light barrier). Once the proper geometric interpretation of transluminal transformations is understood, it is less confusing to adopt the convention of keeping all variables real in both subluminal and superluminal frames. This can be achieved by choosing a different metric signature for the superluminal frames. To keep x' and t' real for both v<c and v>c the general trans­ formations corresponding to equations (1) must be of the form vx t -~ x-vt C x' = , y I =y' z I = z ' t I = .... ( 3) Fig. 1 depicts the rotation of axes for v<c, the x and taxes being held stationary. As the velocity of the x't' frame increases in the positive x direction the x' and t' axes rotate towards the light line, until in the limit v+c they coincide with it. This rotation may be represented equivalently by Fig.2 in accordance with the equality of all subluminal inertial frames. 4 According to equations (3) there is a discontinuity in the rotation as we pass from v<c to v>c. This can be seen by considering the limit v+00 • The x' and t' equations may be rewritten as t X ....X -t i/ ---=-zV C x' = t' = ' ~ JI VL.
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