An Introduction to the Theory of Tachyons

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by  Ricardo S. Vieira

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