An Introduction to the Theory of Tachyons Ricardo S. Vieira∗ Departamento de Física, Universidade Federal de São Carlos (UFSCar), São Carlos, Brasil 2 Abstract these works). Among the formulations proposed, we 1 0 highlight the works of Recami and collaborators [4], 2 The theoryofrelativity,whichwasproposedinthe be- whose results coincide mostly with those who will be n ginning of the 20th century, applies to particles and presented here. Reference [4] is a review on the sub- a frames of reference whose velocity is less than the ve- ject,wheretheinterestedreaderwillfindavastamount J locity of light. In this paper we shall show how this of references and may also get more details about the 2 theory can be extended to particles and frames of ref- theory, besides topics which will not be discussed here 1 erence which move faster than light. (e.g., the tachyon electrodynamics). For an extended theory of relativity, we mean a the- ] h 1 The need for a theory of ory which applies to particles and frames of reference p moving with a velocity greater than c – the speed of - tachyons n light in vacuum –, and also to particles and reference e frames “moving” back in time. In particular it is nec- g Scientists at the European Organization for Nuclear essary to extend the Lorentz transformations for such . s Research (CERN) recently reported the results of an frames. Although we have no problem at all to ex- c experiment[1]onwhichfasterthanlightneutrinoswere tend the Lorentztransformationsina two-dimensional i s probablydetected. Coincidentally,aboutaweekbefore universe, (x,t), we meet with certain difficulties when y thedivulgationofthisresult,Iwasfortunatetopresent h we try to extend them in a universe of four dimen- atthe7thUFSCarPhysicsWeek[2]alectureprecisely p sions, (x,y,z,t). The reasons why this happens will [ onthe subjectoftachyons(the namegivenintheoreti- be commented in section 10. Finally, in section 11 we calphysicstofasterthanlightparticles). Studentsand shallshowthatinasix-dimensionalworld(3space-like 2 Professorswereveryinterestedonthislecture,specially v dimensions and other 3 time-like ones), we can make 7 afterthenewsdiscussedabove. Ihavebeenencouraged those difficulties to disappear and therefore that be- 8 since then to write a paper about the ideas presented comes possible, in this six-dimensionaluniverse, to ex- 1 on that occasion, which constitute the present text. tendtheLorentztransformationsinagreementwiththe 4 Regardless of the results presented in [1] are correct usual principles of relativity. . 2 or not, we point out that there are other experimen- One of the reasons which makes the theory of 1 tal evidences for existence of superluminal phenomena tachyons almost unknown is the (equivocated) belief 1 in nature [3]. Moreover, the theory of tachyons could 1 that the theory of relativity forbids the existence of provide a better understanding of the theory of rel- : faster than light particles, and that the light speed v ativity as well as some issues of quantum mechanics represents an upper limit for the propagation of any i X and we believe that these arguments are enough for phenomena in nature. The argument commonly used onetotrytoformulateanextended theory of relativity, r to prove this statement is, as stated by the theory of a whichappliestofasterthanlightphenomena,particles relativity, that no particle whatsoever can be acceler- and reference frames. Attempts to extend the theory ated to reachor to exceed the speed of light, since this of relativity have been, of course, proposed by many would require spending an infinite amount of energy. scientists, although the original sources are not eas- This is not wrong, but is also not completely correct. ily accessible (in fact, only recently I had knew about Indeed, this argument ignores the possibility of these ∗E-mail: [email protected]. particles have been created at the same time as the 1 Big-Bang. In this way nobody had to speed them up inertial coordinate system. Themovementofaparticle – they just werebornalreadywitha greaterthanlight can be represented by a continuous curve – a straight speed. line in the case of a free particle –, which we shall call Moreover, we can not rule out the possibility that theworld-line oftheparticle. Inparticular,thevelocity such particles might be created through some quan- ofa particlewith respectto a givenframe ofreference, tum process, analogous for instance to the process of say R, is determined by the direction of the particle’s creation of particle-antiparticle pairs. world-line with respect to the time-axis of the inertial Finally, if we associate a complete isotropy and ho- coordinate system associated with R. Similarly, the ′ mogeneity to the space-time, then it follows that none relative velocity v between two frames R and R is de- ′ of their directions should be privileged with respect to termined by the direction of the time-axis of R with the others and thus the existence of faster than light respecttothatofRandachangeofreferenceturnsout particles would naturally be expected instead of be re- to be, in this geometric description, a mere hyperbolic garded as surprising. It is not the possibility of exis- rotation2 of the coordinate axes. tence of tachyons which requires explanation, on the Fromtheseassumptionspresentedabove,thetheory contrary, an explanation must to be given in the case of relativity can be fully formulated. In particular, we of these particles do not exist. point out that from these assumptions we can deduce theprincipleofinvarianceofthespeedoflight(atleast in two dimensions). Indeed, the simple fact that the 2 On the space-time structure geometry of space-time is hyperbolic implies the exis- tence of a special value of velocity which appears to Asisknown,theformulationofthetheoryofrelativity be the same to all inertial frames, that is, whose value wasduetotheeffortsofseveralscientists(e.g.,Lorentz, does not change when one go from a inertial frame to Poincaré,Einstein,Minkowskietc.). Thegeometricde- another. We canconvince ourselvesof this noting that scription of the relativity theory – the so-called space- in a hyperbolic geometry there must existcertainlines timetheory –initsturn,wasfirstproposedbyPoincaré (the asymptotes) which do not change when a hyper- [5]in1905andafterindependentlybyMinkowski[6]in bolic rotationis implemented. Therefore,if there exist 1909,whereamoreaccessibleanddetailedpresentation a particle whose world-line lies on those asymptotes, of the subject was presented. the direction of the word-line of that particle should This geometric description, which contains the very notchange by a hyperbolic rotationand, hence, its ve- essenceofthe theoryofrelativity,maybe groundedon locitymustalwaysbethesameforanyinertialframeof the following statements, or postulates1: reference. Theexperimentalfactthatthespeedoflight 1. The universe is a four-dimensional continuum – is the same in any inertial frame provides, therefore, a threeofthemareassociatedwiththeusualspatial strong argument in favor of the hyperbolic nature of dimensions X, Y and Z, while the other one it is space-time. associated with the time dimension; For future use, we shall make some definitions and conventions which will be used throughout the text. 2. The space-time is homogeneous and isotropic; Since we intend to deal with particles “moving” in any direction of space-time, it is convenient to employ a 3. The geometry of the universe is hyperbolic- metric which is always real and non-negative. Let we circular. In the purely spatial plans, XY, YZ and define, therefore, the metric by the expression ZX thegeometryiscircular(i.e.,euclidean),while intheplansinvolvingthetimedimension, namely, ds= c2dt2 dx2 dy2 dz2 . (1) | − − − | TX, TY andTZ, the geometryis hyperbolic (i.e., The choice of thpe metric, of course, does not affect the pseudo-euclidean). final results of the theory, since we always have a cer- In terms of the Poincaré-Minkowski description of tain freedom in defining it. space-time, any inertial frame can be represented by Intermsofthe metric(1), wecanclassifythe events an appropriate coordinate system, which we shall call as time-like, light-like and space-like as the quantity 1The influence of gravity will be explicitly neglected in this 2In the case of |v|>c we should consider a extended hyper- text. bolicrotation asdiscussedinthesection4. 2 c2dt2 dx2 dy2 dz2 be positive, zero or negative, what is called the switching principle3. This principle − − − respectively. A similar classification can be attributed is based on the fact that any observer regard the time toparticlesandframesofreference. Thus,forinstance, as flowing from the past to the future and that any slower than light particles (bradyons) will be classi- measurement of the energy (associated to a free parti- fied as time-like particles and faster than light parti- cle) results in a positive quantity. Thus, the switching cles (tachyons) as space-like particles. Particles mov- principlestatesthatabackwardparticle(whoseenergy ing with the speed of light (luxons) will be classified, is negative) should always be physically observed as a of course, as light-like particles. typical forward particle (whose energy is positive). We can also classify the particles according to their “direction of movement” in time. A particle which moves to the future will be called a forward particle and a particle moving to the past, backward particle. A particle with has an infinite velocity exists only in theverypresentinstantandwemightcallitamomen- tary particle. A similar classification can be employed for reference frames as well. 3 The switching principle and an- tiparticles Figure1: AforwardparticleP interactswithaphoton γ and becomes a backwardparticle Q. Inthe previoussectionwehaveintroducedtheconcept Therefore, one might think that there are no differ- of backward particles as particles which travel back in ences at all between forward and backward particles, time. Now we shall clarify how we can interpret them since apparentlythe latter arealwaysseeninthe same froma physicalpoint ofview. However,for the discus- way as the first ones. However, this is not so because sion becomes simpler we shall concern ourselves just whenever a backwardparticle is observedas a forward with time-like particles. particle,someofitspropertiesturnsouttobeswitched Let us begin by analyzing what must be the energy in the process of observation. For instance, if a back- ofabackwardparticle. Weknowfromtheusualtheory ward particle has a positive electric charge, say +e, of relativity that the energy of a (time-like) particle is then, due to the conservation of electric charge prin- related to its mass and its momentum through the ex- ciple, we must actually observe a “switched” particle pression E2 =p2c2+m2c4. This quadratic expression carrying the negative charge e. Let us clarify this fortheenergyhastwosolutions: oneofthemrepresents − point through the following experiment. thepositiverootofthatequationandtheothertheneg- Consider the phenomenon described in the Figure ative root(geometrically,this equationdescribes,for a 1, which describes a forward particle P with electric given m, the surface of a two-sheet hyperboloid). In charge +e who interact at some time with a photon the theory of relativity we usually interpret the states γ and, by virtue of this interaction, becomes a back- of positive energy as states which are accessible to any ward particle, Q. Note that the particle Q actually is forward particle or, which is the same, that a forward the same particle P, but now it is traveling back in particle always has a positive energy. Because of this time. Therefore, the actual electric charge of Q is still association, we must for consistency regard negative +e. Nevertheless, when this process is physically ob- states of energy as accessible only to backward par- served, the observer should use (even unconsciously) ticles, so that any backward particle has a negative the switching principle and the phenomenon is to be energy. interpreted as follows: two particles of equal mass ap- These two separate concepts which not have a phys- proach each to the other and, at some point, collide ical sense by themselves – namely, particles traveling backward in time and negative states of energy (asso- 3Sometimes the terminology “reinterpretation principle” is ciated to free particles) – can be reconciled through employed 3 and annihilate themselves, which gives rise to a pho- infourdimensions(thesedifficultieswillbediscussedin ton. Since the photon has no electric charge and the section 10). We shall give here two deductions for the observedchargeof the forwardparticle is +e it follows Extended Lorentz Transformations (ELT), a algebraic that the observed charge of the backward particle has deduction and a geometric one. to be e. The conclusion which follows from this is Algebraic Deduction: sinceintwodimensionsthe − that the sign of the electric charge of a backward par- postulates presented on the previous sections are suffi- ticle must be reversed in the process of observation. cient to proof that light propagates at the same speed Thus, a backward particle of mass m and electric c for any inertial frame, we can take this result as our charge+eshouldalwaysbe observedasaforwardpar- starting point. ticle with the same mass and opposite electric charge. So, consider a certain event whose coordinates are Butthesepropertiesarepreciselythesameasexpected (ct,x) with respect to an inertial frame R and (ct′,x′) to antiparticles. Therefore the switching principle en- withrespecttoanotherinertialframeR′ (whichmoves ableustointerpretabackwardparticleasanantiparti- withthevelocityv withrespecttoR). Supposefurther cle. The conceptofantiparticlescanbe seen,hence, as that the coordinate axes of these frames are always a purely relativisticconcept: itis notnecessaryto talk likelyorientedandthattheoriginofRandR′ coincide about quantum mechanics to introduce the concept of when t′ =t=05. antiparticles4. Under these conditions, if a ray of light is emitted Finally, let us remark that these arguments are also from the origin of these frames at t = 0, then this valid in the case ofspace-likeparticles, i.e., in the case ray will propagates with respect to R according to the of tachyons. In the section 8 we shall see that the en- equation ergyofatachyonisrelatedtoitsmomentumandmass 2 2 2 x c t =0, (2) through the relation E2 = p2c2 m2c4, an equation − which describes now a single-shee−t hyperboloid. From and for R′, by the principle of invariance of the speed this we cansee that tachyonsmust have an interesting of light, also by property: they can change its status of a forward par- ′2 2 ′2 ticle to the status of a backward one (and vice-versa) x c t =0. (3) − through a simple continuous motion. In other words, by accelerating a tachyon we can make it turn into (2) and (3) implies therefore, anantitachyonandvice-versa(noticemoreoverthatat ′2 2 ′2 2 2 2 somemomentthe tachyonmustbecomesamomentary x c t =λ(v) x c t , (4) − − particle, i.e., an particle with infinity velocity). This, (cid:0) (cid:1) of course, it is only possible to space-like particles. whereλ(v)doesnotdependonthecoordinatesandthe time, but may depend on v. In the other hand, since the frame R moves with 4 Deduction of the extended ′ respect to R with the velocity v, follows also that − Lorentz transformations (in 2 2 2 ′2 2 ′2 x c t =λ( v) x c t . (5) two dimensions) − − − (cid:0) (cid:1) Therefore, from (4) and (5), we get λ(v)λ( v) = 1. Let us show now how the Lorentz transformations can Besides,the hypothesisofhomogeneityandis−otropyof begeneralized,orextended,toframesofreferencemov- space-time demands that λ(v) does not depend on the ing with a velocity greater than that of light (as well velocitydirection6 andthenweareledtothecondition astoframesofreferencewhichtravelbackintime). In thissectionweshalldiscusshoweverthetheoryonlyin λ(v)2 =1 λ(v)= 1. (6) ⇒ ± two dimensions. As commented before, we meet with several difficulties to formulating a theory of tachyons 5Hereafter, whenever we speak in the frames of reference R and R′ should be implicitly assumed that the relative velocity 4Theconnectionbetweenbackwardparticlesandantiparticles betweenthemisvandthattheconditionsabovearealwayssat- was,ofcourse,proposedalreadybyseveralscientists(e.g.,Dirac isfied. [7],Stückelberg[8,9],Feynman[10,11],Sudarshan[12],Recami 6Indeed, only in this case the transformations do form a [4]etc.). group,see[5]. 4 Thus we have two cases to work on. Let us first space-likereferenceframes,thatis,thetransformations analyze the first case, namely, λ(v)=+1. In this case associatedto v >c, where the correctsigndepends if ′ | | the equation (4) becomes the frame R is a forwardor backward reference frame withrespecttoRandcanbedeterminedbytheFigure ′2 2 ′2 2 2 2 x c t =x c t , (7) 2. − − Geometrical Deduction: from the geometrical whose solution, as it is known, is given by the usual point of view, the ELT can be regarded as a (hyper- Lorentz transformations, bolic) rotation defined on the curve7 ct xv/c x vt ′ ′ 2 2 2 2 ct = − , x = − . (8) c t x =ρ . (13) 1 v2/c2 1 v2/c2 − − − Note that thepse transformations cpontains the identity We can call such a(cid:12)(cid:12)transform(cid:12)(cid:12)ation as an extended hy- perbolic rotation. Notethat(13)representsasetoftwo (for v =0) and they are discontinuous only at v = c. ± orthogonalandequilateralhyperbolæ. Theasymptotes Consequently, these transformations must be valid on the range v <c<v as well, but nothing can be said − for v out from this range. Inthisway,wehopethatintheothercase,i.e.,when λ(v) = 1, the respective transformations should be − relatedto velocitiesgreaterthan thatoflight. Nowwe will show that this is indeed what happens. To λ(v)= 1, equation (4) becomes − ′2 2 ′2 2 2 2 x c t = x c t , (9) − − − Through the formal substitu(cid:0)tions x (cid:1) iξ and ct → ± → icτ, we can rewrite (9) as ± ′2 2 ′2 2 2 2 x c t =ξ c τ . (10) − − Equation(10)hasthe sameformasthe equation(7) and therefore has the same solution: cτ ξv/c ξ vτ ′ ′ ct = − , x = − . (11) 1 v2/c2 1 v2/c2 − − Figure 2: The Curve c2t2 x2 =ρ2. Expressing tphem back in terms opf x and t, we get − (cid:12) (cid:12) ofthiscurvedividetheplaneo(cid:12)ntofourd(cid:12)isjointregions, ct xv/c x vt ′ ′ ct = − , x = − , (12) namely,theregionsI,II,IIIandIV,asshowintheFig- ± v2/c2 1 ± v2/c2 1 ure 2. − − andwehavepjusttoremovethesignpalsambiguityonthe Toexpresssucharotationisconvenienttointroduce theextendedhyperbolic functions,coshθandsinhθ,de- expressions above. The correctsign, however,depends on the relatively direction of the frames R and R′ in 7Such a rotation can be more elegantly described through their “moviment” on the space-time, and can be seen the concept of hyperbolic-numbers [13]. A hyperbolic number in the Figure (2). In the case of a forward space-like is number of the form z = a + hb, where {a,b} ∈ R and h : {h2 = +1, h ∈/ R}. By defining the conjugate z¯= a−hb, transformation,it is easyto show that the correctsign it follows that |z¯z|= |a2−b2|= ρ2, which represents an equa- is the negative one. tion like (13). This leads to a complete analogy with complex Notethattheseequations,aswellasthoseofthepre- numbers, but with the difference that now these numbers de- scribe a hyperbolic geometry. We alsostress that the samecan viouscase,arediscontinuousonlyforv = c. Butnow ± bedonethroughtheelegantgeometricalgebraofspace-time[14], theyarerealonlywhen v >c. Theequations(12)rep- with the advantage that this formalism could allow, perhaps, a | | resent thus the Lorentz transformations between two generalization oftheseconcepts tohigherdimensions. 5 fined by the following relations or, in terms of the tangent, ct=ρcoshθ, x=ρsinhθ, (14) σ(θ) σ(θ)tanθ coshθ = , sinhθ = , whereθ istheusualcircularparametersothat0 θ < 1 tan2θ 1 tan2θ ≤ − − 2π and ρ is given by (13). Note that in this geometric q q (19) (cid:12) (cid:12) (cid:12) (cid:12) description the velocity v is given by where (cid:12) (cid:12) (cid:12) (cid:12) +1, π/2<θ <π/2 v/c=tanhθ. (15) σ(θ)= − . (20) ( 1, π/2<θ <3π/2 − Expressions for coshθ and sinhθ can be determined in several ways. For instance, we can use the usual The equivalence between (16) and (18) or (19) is hyperbolic functions coshϕ and sinhϕ (where ϕ is the found when one takes into account (17). usual hyperbolic parameter so that < ϕ < + ) −∞ ∞ Once defined the extended hyperbolic functions it is to define them. Effectively, by introducing in eachdis- easy to obtain expressions describing an extended hy- jointregionofthespace-timeahyperbolicparameterϕ, which must be measured as shown in Figure 2, we can see thatthe functions coshϕandsinhϕcanbe usedto parametrize each one of the four branches of the curve (13). Once specified the region which θ belongs, ρ and ϕ determine in a unique way any point of the curve (13) and, therefore, they also determine the extended hyperbolic functions. With these conventions, we find out that +coshϕ, θ I ∈ sinhϕ, θ II coshθ − ∈ , ≡−coshϕ, θ ∈III +sinhϕ, θ IV ∈  (16) Figure3: Graphicforthehyperbolicextendedfunction +sinhϕ, θ I coshθ. Thegraphicofsinhθ issimilartothis,butwith ∈ +coshϕ, θ II a phase difference of π/2 rad. sinhθ  ∈ , ≡−sinhϕ, θ ∈III coshϕ, θ IV perbolic rotation. Let (ct,x) = (ρcoshθ1,ρsinhθ1) be − ∈ wheretheparametersθ andϕshouldbe relatedbythe tthoeacionoerrdtiianlactoesorodfinaatpeosinytstoemn tRh,ewphlaicnheiwtiitshasrseuspmeecdt formula to belongs to the sector I of space-time. If we take a passive hyperbolic rotation (i.e., if we rotate the coor- +tanhϕ, θ (I, III) tanθ =tanhθ ∈ . (17) dinateaxes),saybyanangleθ12,weshallobtainanew ≡( cothϕ, θ (II, IV) inertial coordinate system R′ and the coordinates of − ∈ ′ ′ that same point become (ct,x)=(ρcoshθ2,ρsinhθ2) Expressions for the extended hyperbolic functions can ′ with respect to R. Since θ12 =θ1 θ2, it follows that also be found without making use of the usual hy- − perbolic functions. To do this, we parametrize (13) ′ ′ ct =ρcosh(θ1 θ12), x =ρsinh(θ1 θ12). (21) through the circular functions instead, putting ct = − − rcosθ and x = rsinθ, with r = √c2t2+x2. This al- lows us to write directly, By replacing the expressions of cosh(θ1 θ12) and − sinh(θ1 θ12)byanyoneoftheaboveexpressionsand − cosθ sinθ by simplifying the resulting expressions, taking also coshθ = , sinhθ = . (18) ′ ′ cos2θ cos2θ intoaccount(17),wefindoutthat(ct,x)isrelatedto | | | | p p 6 (ct,x) through the equations given by an analogous expression: ct′ =δ(θ12)(ctcoshθ12−xsinhθ12), ct′′ =ε′(θ23) ct′−x′tanhθ23 , (22) 2 ′ 1 tanh θ23 x =δ(θ12)(xcoshθ12 ctsinhθ12), − − q (26) (cid:12) (cid:12) where x′′ =ε′(θ23) x′(cid:12)−ct′tanhθ23(cid:12). δ(θ)=(+11,, ttaann22θθ ><11. (23) q 1−tanh2θ23 − ′ (cid:12) (cid:12) Intheseequationsε (θ23)d(cid:12)eterminesthe(cid:12)signalscorre- Finally, using (19) and putting tanθ12 = v/c , we ob- sponding to the transformation from R′ to R′′. These tain directly the required transformations, which are signals, however, do not need to equal necessarily the identical to those obtained before, namely, signalsrelatedtothe transformationfromR to R′. In- deed, while in the definition of ε(θ12) the frame of ref- ct xv/c x vt ct′ =ε(θ12) − , x′ =ε(θ12) − , erence R was supposed to belong to the region I of 1 v2/c2 1 v2/c2 space-time, the frame R′ might belong to any region | − | | − (24|) of space-time. Thus, ε′(θ23) should be regarded as a p p where we put ε(θ12) = σ(θ12)δ(θ12). ε(θ12) deter- function to be yet determined. mines the correct sign which must appear in front of Substitution of (25) into (26) gives us the law of these transformations, as can be visualized in the Fig- transformationbetween R and R′′. After some simpli- ure 2. fications, one can verify that the resulting expressions have the same form of the ELT, namely 5 The composition law of veloc- ct′′ =ε′′(θ13) ct−xtanh(θ13) ities and inverse transforma- 1 tanh2(θ13) − tions q(cid:12) (cid:12) (27) x′′ =ε′′(θ13) x(cid:12)−cttanh(θ13) (cid:12). The transformations deduced in the previous sections 2 1 tanh (θ13) doformagroup. TheordinaryLorentztransformations − q is just a subgroup of it. Let us demonstrate now this where (cid:12)(cid:12) (cid:12)(cid:12) ′ group structure. ′′ ε(θ12)ε (θ23) ε (θ13)= (28) First, note that the identity is obtained with v = 0. δ(θ12,θ23) We shall show now that the composition of two ELT with still results in another ELT. For this we introduce a ′′ ′ thirdinertialframeR ,whichmoveswithrespecttoR +1, tanhθ12tanhθ23 <1 with velocity u=ctanhθ23. R′ by its turn is assumed δ(θ12,θ23)= , (29) ( 1, tanhθ12tanhθ23 >1 to moves with the velocity u = ctanhθ12 with respect − toR. Wealreadyknowthetransformationlawbetween and ′ R and R, and we can write it down: tanhθ12+tanhθ23 tanh(θ13)= =tanh(θ12+θ23). ct′ =ε(θ12) ct−xtanhθ12 , 1+tanhθ12tanhθ23 (30) 2 1−tanh θ12 From(17)wecanseethat(30)consistsofageneraliza- q(cid:12) (cid:12) (25) tionoftheadditionformulaforthehyperbolictangent, x′ =ε(θ12) x(cid:12)−cttanhθ12 (cid:12). which reveals its geometric meaning. In terms of the 2 velocity v, equation (30) can be rewritten as 1 tanh θ12 − q u+v In its turn, the law of trans(cid:12)(cid:12)formationfro(cid:12)(cid:12)m R′ to R′′ is w = 1+uv/c2. (31) 7 Equation(31)expressesthecompositionlawofveloc- The definition is the following: two frames ofreference ities in this extended theory of relativity. It is exactly are said to be conjugate if their relative velocity is in- ′ the same aspredictedby the usualtheory ofrelativity, finite. Thus, if v is the velocity of the frame R with but now it applies to any value of velocity. respect to the frame R, the conjugate frame of refer- ′ ∗ Let we show also that the inverse transformation ence associated to R is another reference frame R , does exist. To do this, we impose onto (27) the condi- whose velocity is w = c2/v when measured by R. In ′′ ′′ tionsct =ctandx =xandrequirethattheresulting fact, we obtain from (31), transformation be the identity. For this it is necessary ′′ u+v c2 to have θ23 = θ12 and ε (θ13) = 1, which enable lim = . (34) us to evaluate ε−′(θ23) from the resulting expression of u→∞(cid:18)1+uv/c2(cid:19) v ′′ ε (θ13). In fact, we find that Conjugate frames of reference are important be- cause a space-likeLorentz transformationbetweentwo ′ ε (θ23)=δ(θ12) ε(θ12)=σ(θ12), (32) frames, say, from R to R′, can be obtained also by a ∗ (cid:14) usual Lorentz transformation between R and R . For since δ(θ12,−θ12)=δ(θ12), with δ(θ12) given (23). this, we simply have to replace v ⇋c2/2, ct∗ ⇋x and Substituting this result into (26) we obtain the x∗ ⇋ ct. In fact, since w = c2/v is less than c for required expressions for the inverse transformations, ∗ v >c, it follows that the transformation from R to R which when expressed in terms of the velocity v are is given by given by ct xw/c x wt ct′+x′v/c ct∗ = − , x∗ = − . (35) ct=ε−1(θ12) , 1 w2/c2 1 w2/c2 1 v2/c2 − − | − | (33) Makingthepreplacementsindicatepdabovewecanseein p ′ ′ this way that we shall get the correct transformations x +vt x=ε−1(θ12) , between R and R′. 1 v2/c2 | − | From a geometrical point of view the passage of a and where we put ε−1(θ12)p=σ(θ12). givenframeofreferencetoitsconjugateconsistsofare- flection ofthe coordinateaxes relativelyto the asymp- Note that the signs appearing on the inverse trans- totesofthecurve(13),sincethisreflectionpreciselyhas formation are different from that present on the direct the the effect of changing ct by x and vice-versa (and transformations. This difference is a consequence of thus the effect of replacing v by c2/v). Therefore, we what was commented before, that in the transforma- ′ canseethatanextendedLorentztransformationcanbe tion from R to R we had assumed the starting frame reducedtoausualLorentztransformationbyperform- Ralwaysbelongingtothe regionIofspace-time,while ′ ′ ing appropriatereflections relativelyto asymptotes (in in the transformation from R to R it is the frame R caseofaspace-liketransformation)andaroundtheori- which is fixed onthe regionI.When v <c this asym- | | gin (for a backward time-like transformation). This metry has no effect at all, since in this case the signals gives us a new way to derive the ELT. are always the same in both expressions. But when It is interesting to notice also that, if a particle has v >c however,we shouldalertthatthe inversetrans- | | velocityu=c2/vwithrespecttotheframeR,thenthe formations can not be simply obtained by replacing v ′ velocityofthisparticlefortheframeR willbeinfinite. by v. It is still necessary multiply them by 1. − − Inotherwordsthisparticlebecomesamomentarypar- Finally,wementionthattheassociativityofELTcan ′ ticle to the frame R. More important than that, if be shown in a similar way, which completes the group 2 the particle velocity u is greater than c /v, and v <c, structure of the ELT. ′ this particle becomes a backward particle to R and it should be observed as an antiparticle by this refer- 6 Conjugate Frames of Reference ence frame. On the other hand, if v > c the frame of ′ reference R should observe an antiparticle whenever We shall introduce now an important concept which u<c2/v. ′ canonlybe contemplatedinanextendedtheoryofrel- In an analogous way, since the frame R moves with ativity: the concept of conjugate frames of reference. the velocity v with respect to R , it follows also that − 8 aparticlewithvelocityu′ =c2/v musthaveaninfinite complete a full oscillation. Now, the signal appearing velocity with respect to R. So, in the case of v < c, on(37)isdeterminedaccordingtotheFigure2andthe | | the reference frame R should observe an antiparticle if analysis becomes more or less complicated. Of course, u′ <c2/vand,inthecaseof v >c,onlyifu′ > c2/v. we have no problems at all when v <c, then we shall | | − | | These relationships might be, of course, more easily analyze only the case where v > c. Suppose first ′ | | found by analyzing (30) or (31). thatthe referenceframe R is a forwardframe withre- spect to R. In this case we find that ε(θ) = 1 and − the moving clockwill workin the opposite directionas 7 Rulers and clocks ′ compared to the clock fixed at R. This means that ′ the clock at R is a backward-clock with respect to R. Consider two identical clocks, one of them fixed in the We can convince ourselves of this from what was com- ′ frame R and the other fixed to the frame R. Further, mentedintheprevioussection,whereitisnecessaryto ′ consider thatthese clocksaresynchronizedon t=t = put there u=0 and v >c (and therefore u<c2/v). ′ 0, where R and R are in the same position. We wish | | This is an interesting situation indeed, because we tocomparethetimingrateoftheseclocks,asmeasured have just seen that for the frame R, both clocks work by one of those frames. For example, suppose we want clockwise (if the clocks are forward ones) or coun- to compare the rhythm of these clocks when the time terclockwise (if the clocks are backward ones) when intervals are always measured by R. For this, suppose ′ v > c. In the other hand, for the frame R if its ′ ′ thattheclockfixedonR takesthetimeτ tocomplete | | own clock is working clockwise, then the moving clock a full period of oscillation. The time T corresponding should work counterclockwise and vice-versa. In the to this period of time, but now measuredby R, can be ′ casewherethe referenceframeR is backwardwithre- found through the inverse transformations (33). Since spect to R these asymmetries persists yet, but now is ′ this clock is at rest with respect to R, we should put the frame R which will see both clocks working differ- ′ x =0onthe firstofthe formulæ(33)andweshallget ′ ently, while for R they will work accordingly. These ′ asymmetries, of course, just express the fact that the τ −1 T =ε (θ) . (36) extended Lorentz transformations, the direct and in- 1 v2/c2 | − | verse one, are asymmetric by themselves in the case of Then, we can verify thpat a forward moving clock v >c. | | Letweconsidernowtwoidenticalrulers,oneofthem (with respect to R) becomes slower in measuring time placed at rest in the frame R and the other placed at than an identical clock at rest, when the speed of the ′ restwithrespecttoR. Wewishtocomparethelength clock is less than that of light (as it is well-known). of these rulers, when analyzed by one of these frames. But for a faster than light clock we get that it contin- ′ ′ ues to be slower for v/c < √2 and it becomes faster If l0 is the length of the ruler at R, when measured when v/c > √2. I|t is| interesting to note that for by this frame, the respective length L, as measured v/c =| √2| both clocks go back to work at the same by R, is obtained by determining where the extreme | | points of the moving ruler is at a given instant t, say, timing rate. Moreover,in the case of a backwardmov- t=0. Making use of the second equation of the direct ing clock, we can see from the switching principle that transformations , we find that this clock should work in the counterclockwise, which is due to the fact that a backward-clock should mark L=ε(θ) l′ 1 v2/c2 , (38) 0 the time from the future to the past. · | − | Let us now verify what we got when the clocks are To v < c we have thepusual Lorentz contraction, ′ | | appreciated by the reference frame R. In this case we but for v > c we get that the moving ruler will be | | shouldusethedirecttransformationsandthusweshall smaller than the ruler at rest when v/c < √2 . The | | get rulersgobackonceagaintohavethesamelengthwhen v/c = √2 and, finally, for v/c > √2 they should τ T′ =ε(θ) , (37) p|res|ent a “Lorentz dilatation.”| Mo|reover,regardingR′ 1 v2/c2 | − | asaforwardframewithrespecttoR,itfollowsthatfor whereτ isthetimespentbpytheclockfixedatR(which theframeRthemovingrulerwillbeorientedcontrarily ′ is moving with speed v with respect to R) for it to with respect to the ruler at rest. − 9 ′ If,ontheotherhand,measurementsaremadebyR, As it is known, the expressions for the energy and then we find that momentum are obtained by the formulas L′ =ε−1(θ) l0 1 v2/c2 , (39) ∂ (u) ∂ (u) · | − | p(u)= L , E(u)=u L (u). (42) and now for the reference frapme R the ruler at motion ∂u (cid:20) ∂u (cid:21)−L (which has the velocity v) point out to the same di- − Applying (42) onto (41) we obtain, thus rection as its ruler at rest. We find again the same asymmetry commentedaboveforthe clocks. These re- αu/c αc sultscan,ofcourse,bemoreeasilyobtained–andfully p(u)= , E(u)= . − 1 u2/c2 − 1 u2/c2 understood – through Minkowski diagrams. − − (43) p p To find α we may use the fact that for low speeds 8 Dynamics these expressions should reduce to that obtained by Newtonian mechanics. Thus, for instance, if we ex- Inthissectionweintendtoanswersomequestionscon- pand the expression for the momentum in a power se- cerning the dynamics of tachyons. The expressions for ries of u/c and we keep only the first term, we should the energy and momentum for a faster than light par- get p αu/c, while Newtonian mechanics provides ≈ − ticlewillbedeductedandweshallshowhowthesepar- p=mu. Thus we get α= mc and then − ticles behave in the presence of a force field. Asastartingpointwemightassumethattheprinci- mu mc2 p(u)= , E(u)= , (44) ple ofstationaryactionalsoappliestofasterthanlight 1 u2/c2 1 u2/c2 − − particles. This, of course, follows directly from the as- p p sumption of homogeneity and isotropy of space-time, which are the same expressions of the usual theory of sinceweknowthatthisprincipleistrueforslowerthan relativity. light for particles. In the case of a forward space-like particle (i.e. in As one knows, the principle of stationary action the case of a forwardtachyon), the LaGrange function statesthatthereexistaquantityS,calledaction,which takes the form assumes anextreme value (maximumorminimum) for any possible movement of a mechanicalsystem (in our (u)=αc u2/c2 1. (45) L − case, a particle). On the other hand, in the absence p and we get by (42), the following expressions for mo- of forces, the motion of a particle corresponds to a mentum and energy, space-time geodesic, which reduces to a straight line byneglectinggravity. Thismeansthatincaseofafree αu/c αc particle the differential of action dS should be propor- p(u)= , E(u)= . (46) u2/c2 1 u2/c2 1 tionalto the line element ds ofthe particleand wecan − − write in this way p p The constantα, however,no longercanbe determined dS =αds, ds= c2dt2 dx2 dy2 dz2 . (40) by comparingthese expressionswith those obtained in | − − − | Newtonianmechanics,sincethevelocityoftheparticle Wemustemphasize,phowever,thattheconstantofpro- is always greater than the speed of light in this case. portionality α can take different values at different re- Butwecansteadevaluatethelimitoftheseexpressions gions of space-time, since these regions are completely as u , which give us disconnected regions. Therefore, it is convenient to →∞ consider each case separately. lim p(u)=α, lim E(u)=0, (47) In the case of a forward and time-like particle, (40) u→∞ u→∞ takes the form by where we can see that α equals the momentum of dS = (u)dt, (u)=αc 1 u2/c2, (41) a momentary particle, that is, the momentum of a in- L L − finitely fast particle. wherewehadintroducedtheLaGprangefunction, (u), Since the mass of a particle must be a universal in- L to express the action in terms of particle velocity. variant, it follows that we can also define a metric in 10