~~

Reprinted from AMERICAN]OTJRNAL OF PHY$ICS, Vol. 30, No.10, 718-723, October, 1962Printed in u. s. A.

"Meta" Relativity

0. M. P. BI]:.ANIUK, V. K, DESHPANDE, AND E. C:G. SUDARSHANDepartment of Physics and Astronomy, University of Rochester, Rochester, New York

(Received May 21, 1962)

In pre-relativity times, Thomson, Heaviside, and Sommerfel.d.. among others, had examinedquestions arising from the assumption that a particle may move faster than the velocity oflight in vacuo. Such a hypothesis is reexamined in the framework of classical (nonquantum)theory of special relativity.

"There was a lady named BrightWho traveled faster than Light. .."

see if there could possibly be any physical contentin such a generalization. An attempt is also madeto devise an experiment by which the existence ofsuch a third class could be testeq directly. Itshould be pointed out that in the present dis-cussion only the classical (nonquantum) aspectsof the problem are examined, since they stand forthemselves. In this sense, this paper purpor~ tocontinue the discussion on the "Oberlichtge-schwindigkeitsteilchen " {particles of super-lumi-

nary velocities) elaborated by Sommerfeldl,2 in

INTRODUCTION

O NE of the favorite topics for luncheon con-versations among physicists is the specula-

tion whether the existence of a class of particles,created with a velocity v > c, may be hypothesized.One would then be dealing with three distinctclasses of particles. The first two are conven-tional. Class I includes all particles which travelat velocities smaller than the velocity of light.Class II is made up of particles which can onlyexist wheri traveling with the velocity of light.The third class would then comprise the hypo-thetical particles which are created at super-luminary velocities. In this paper, the implica-tions of such a hypothesis are investigated witha rigour somewhat greater than gastronomic, to

1 A. Sommerfeld, K. Akad. Wet. Amsterdam. Proc. 8,346 (1904) (translated from Verslag v. d. gewone ver-gadering d. Wis-en Natuurkundige Afd., November 26,1904, Dl. XIII). Earlier, less exhaustive, treatmentsinclude: J. J. Thomson, Phil. Mag. 28, 13 (1889) ; 0.Heaviside, Electrical Papers (MacMillan and Company,London, 1892), Vol. 11, Chap. 47, p. 497; Th. Des Coudres,Arch. Neerland. Sci. [II] 5, 652 (1900).

2 A. Sommerfeld, Nachr. Ges. Wiss. GOttingen, pp.

201-235, February 25, 1905.

~

"META" RELATIVITY719

pre-relativistic times, except that in the presentexamination the postulates of special relativityare strictly adhered to. A field-theoretical treat-ment of the hypothesis will be published at alater date.

(0)

Py

( b)

Py

(c)

MASS SHELL CONSIDERATIONS

Let us start by writing down the two criteria,which a 'Consistent relativistic theory should

satisfy.(a) In any frame of reference the energy of a

particle must be positive.(b) Laws of particle dynamics must be in-

dependent of frame of reference.

I t is conventional to satisfy both these de-mands by requiring the particles to be char-acterized by energy-momentum four-vectorslying inside, or on, the forward light cone. Eventsviewed from a different frame are then describedby new four-vectors which are transforms of theoriginal ones.

The energy-momentum four-vectors associatedwith the first two classes satisfy the invariantrelation

FIG. 1. Three-dimensional models of the (E,p) surfacesdescribed by the invariant relation E2-p2c2=mo2c4, (a) forthe class I particles with mo2c4>O, (b) for the class IIparticles with mo2c4=0, and (c) for the class III particleswith mo2c4 <0.

E2-p2c2=mO2c4. (1)

For mo2c4>O, or class I particles; Eq.. (1)represents a two-sheeted hyperboloid of revolu-tion around the E axis. A three-dimensionalmodel of such an (E,p) surface is shown in Fig.1 (a). The criterion (a) above restricts the (E,p)coordinates of a particle to lie on the positiveenergy sheet, but all points on this sheet canbe transformed into one another under properLoreritz transformations. I t should be noted thatwhile there exists a Lorentz frame in which theclass I particle has zero momentum, as a resul1 ofthe nonzero mass there exists no frame in whichsuch a particle has zero energy.

For class II particles, with mo2=O, the (E,p)surface becomes a cone of revolution about theE axis, as shown in Fig. l(b). It may appear atfirst that only the upper cone has physicalsignificance, and that a Lorentz transformationcan take a point on the upper cone only intoanother point on the upper cone. Any trans-formation (e.g., a reflection) into a point on thelower cone appears to introduce a particletravel-ing with negative energy, a situation excludedbv criterion (a). A further examination (see ex-

amples below) reveals, however, that such atransformation implies the photon travelingbackward in time. Taken by itself this conclusionappears nonsensical, but when taken togetherwith the negative energy result, it leads to asimple physical reinterpretation. The photon,which in the first system carried positive energyfrom (x,t) = (X1,tJ to (x,t) = (X2,t2) with t2 > ti,would appear to the other observer not as a weirdnegative energy particle traveling backward intime, but as a positive energy particle travelingforward in time, but going in the opposite direc-tion. Thus, the reinterpretation brings the eventsback into the fold of ordinary phenomena.

The above reinterpretation acquires particularsignificance when a third class of particles, forwhich v>c, is postulated. For such a particle tohave ph'ysical significance its energy

E=moc2/[1- (V/C)2.Jt, (2)

720BILANIUK, DESHPANDE ANDSUDARSHAN

discussed in the third example below. To facili-tate this discussion, two simpler cases are ex-amined first.

and its momentum

p=mov/[1-(v/c)2Jt, (3)

must be real. This implies imaginary "rest mass"for this particle, which may seem to disqualifythe whole idea right from the start. One shouldrecall, however, that in classical mechanics themass mo is a parameter which cannot be measureddirectly even for slow particles. As Max Jammer3puts it, mass "does not do what it doe~ becauseit is what it is, but it is what it is because it doeswhat it does." Only energy and momentum, byvirtue of their conservation in interacti0ns, aremeasurable, therefore must be real. Thus theimaginary result for the rest mass of the hypo-thetical "meta" particles offends only the tradi-tional way of thinking, and not observable

physics.On similar grounds, one can resolve the ques-

tion of proper length Lo and proper time T 0. Onlythose quantities which the observers can measuremust be real. This means that

L=Lo/[l- (v/c)2Jl (4)and

EXAMPLES

Below, three examples are discussed in whichthe reinterpretation of phenomena involvingclass III negative-energy particles is explored indetail. I t is shown that the time reversal whichalways accompanies propagation of negative-energy particles in effect reirrtroduces aboundfor the energy. Before we discuss the specificcases below, let. us recall that, according to theoriginal criteria, various observers must agree onthe identity of the physical laws but not on thedescription of specipc events. Only the physicallaws, and not the description 0( any givenphenomenon, mus~ remain invariant as we passfrom one frame of reference to another .

For particles of class 111 this description canbe so chosen that in any one frame only particlesof positive energy appear. Such reinterpretationis made possible by the fact that particles in thenegative-energy portion of the (E,p) hyperboloidappear to travel backward in time. These twofacts in effect restore positive definiteness ofenergy for all observers even though the hyper-boloid is single-sheeted so that all points on it canbe transformed into each other under properLorentz transformations. .

Let us elaborate on this point by examining aspecial case.

(1) Assume that the following events takeplace ina reference frame x. The source SI atXI ~o emits a particle with v>c at time tl =0 andthe sink S2atx2 absorbs it at a time t2(t2>tJ (seeFig. 2). Consider another frame x' in which thetime component of the interval becomes negativeas shown in Fig. 3. In this x' frame the energy isalso negative. Therefore as viewed from theframe x' , the particle moves ~ith negative energy

T= T 0(1- (vi C)2Jt (5)

must be real. In turn, this implies that for parti-cles with v>c the proper length Lo and propertime T o are imaginary. Any objection to thisconclusion is overruled on the grounds that Loand T o are not accessible to measurement byan observer, who by definition must belong toclass I.

Imaginary mass, or m02<O, of the class IIIparticles implies that the (E,p) surface describedby Eq. (1) is now a single-sheeted hyperboloid ofrevolution around the E axis, as shown in Fig.1 (c). If the framework of the special theory ofrelativity is preserved, then all points on thesheet can be transformed into each other underproper Lorentz tra:l1sformations. The featureof the single-sheeted hyperbo.loid of not beingbounded in either the + E and -E directionappears to introduce the possibility of havinginfinite energy sources, which would violate afundamental concept of physics, according towhich no such sources can exist. This question is

51

3 Max Jammer, Concepts of Mass in Classical and ModernPhysics (Harvard University Press, Cambridge, Massa.chusetts. 1962). n. 153.

s 2~ ~.

(XI'tl) (X2.t2)

tl < t2 for E > O

t2< tl for E < O

FIG. 2. Interchange of the roles of source and sink forclass III nart;"les.

722BILAN1UK. DESHPANDE, AND SUDARSHAN

(a)

u<c

(u=t>

(b)

u:c

FIG. 5. Simultaneous collision of two class III particleswith a class { particle is interpreted as fusion of four classIII particles with the class I particle in a frame in which thetwo class III particles seem to acquire negative energies inthe collision.

(c)

particles could indeed serve as an infinite sourceof energy, the notion of their existence wouldviolate the fundamental physical concept whichexcludes the existence of such sources. Theresolution of this apparent contradiction is againachieved by properly reinterpreting the process.In any frame ill which the energy of a class IIIparticle appears negative ( -E above) the par-ticle will also appear to be moving backwards intime. Thus the time sequence of the process willbe such that the observer will see a fusion of fourparticles with the class I particle, see Fig. 5. Theincrease of energy of the class I particle by 2E isaccounted for by the two fusing particles'lwhichbring in an energy E each from external sources.

u>C

(u= 2C)

w=_-9--"u

,, i I I

FiG. 6. Graphical representation of the assumption thatthe relativistic velocity addition v= (u+w)/(1+uw/c2)holds (a) for class I particles, (b) class II particles, (c)class III particles._If u is the velocity of a particle in ourframe of reference, the graphs indicate the velocity v of thesame particle as measured by an observer who moves withrespect to our frame with a velocity w.. Graphs are re-stricted to jw I<c since all observers are assumed to belongto class I.

Parts (a) and (b) of Fig. 6 represent the famil-iar situations as encountered with class land

Iclass II particles. Part (c) of Fig. 6 brings intofocus some of the striking properties of class IIIparticles. First of all, it should be noted that therole which c plays as the limiting velocity for

particles of class I is still with us in class Ill,except that c is the lower limit for the velocity

here. This result reflects the fact that the energy-

momentum hyperboloid of Fig. 1 (c) does not

comprise any points with p2<mo2c2. There is no

VELOCITY ADDITION

Further light can be shed on the properties ofthe hypothetical "meta" particles by consideringthe question of velocity addition. Let uand v bethe velocities of a particle as measured by twoobservers 01 and O2, respectively, whose relativevelocity is w. Our assumption that class IIIparticles obey the invariance of Eq. (1) impliesthat they comply with the rela1;ivistic law ofvelocity addition

v = (u+w)/[1 + (uw/c2) J. (10)

The consequences of this generalization aregraphically represented in Fig. 6, where v isplotted as a function of w, the relative velocity ofthe observers, for the three special cases ofu < c, u = c and U > c. Since all observers belong to

class I. the range of w is restricted to tw1 <c.

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723 IVITY'ME A RELA

be greater than the limiting angle of the latter .The question of the radiation output, however 1is not so straightforward. The frequency cut-off,which in the case of class I particles leads to afinite value of energy loss per unit length, c;;annotbe used here and only a detailed re-examinationof the formalism can lead to a prediction of theintensity of the Cerenkov radiation resultingfrom energy loss of meta particles.

Qualitative considerations seem to indicatethat a meta particle losing energy in a mediumwould actually undergo an acceleration.2 Thiscan also be seen from Figs. l(a) and 6(c), whichshow that for class III particles loss of energyimplies increase of velocity. Whereas an ordinaryclass I particle upon loss of momentum stops withzero velocity and finite rest energy, a meta-particle upon loss of energy disappears withinfinite velocity but finite mqmentum. As theenergy of the meta-particle decreases, theCerenkov angle would go to 90°. Has anyone ofyou gentlemen discarded a set of data on suchaccount? It may have been caused not by faultyelectronics, as you assumed, but by a shower ofmeta particles !

Lorentz frame in which the meta particle wouldtravel with a velocity equal to or smaller than c.

As the velocity of the observer O2 relative to01 approaches w= -C2/U, the velocity of themeta particle, according to O2, tends to infinity.Such a result in itself would suffice to disqualifythe hypothesis, because it appears to violate thepostulate that no energy propagation can takeplace with infinite velocity. Yet when the energyof a particle is evaluated in the system in whichthe particle velocity v tends to infinity it is seenthat the energy E=moC2/[1- (v/c)2Ji tends tozero, so that the above principle, too, staysinviolate. It is interesting to note that the situ-ation of w= -c2/u for class III particles cor-responds to the state of rest of class I particles.The latter have zero momentum and minimumenergy, the former have zero energy and min-imum momentum, p2=mov/[1- (v/c)2Ji=E2/C2-mo2c2= -mo2c2(>O). In terms of the energy-momentum space, the w= ~C2/U situation cor-responds to a Lorentz transformation into apoint lying on the E = O girth of the hyperboloidin Fig:. 1 (c).

DETECTION

'The only sure way to ascertain the physicalcontent of the hypothesis is to detect a metaparticle. Assuming the hypothetical class IIIparticles to carry electric charge, a possibleavenue for their discovery may lie in the Ceren-kov effect. Simple geometric arguments indicatethat the coherence condition4 which determinesthe unique angle of emission of Cerenkov radi-ation remains in force for class III particles. Thissuggests that class III particles could be clearlydistinguished .from the class I particles by theCerenkov angle which for the former must always

CONCLUSION

At least in one respect, the speculations above

have proved very successful. When introduced

by the way of problems or illustrations in an

introductory special relativity course, they have

invariably led to lively and penetrating debates

amonQ' the students.

ACKNOWLEDGMENTS

The authors wish to acknowledge numerousstimulating discussions with many of theiracademic friends whom they have tried to con~vince, not always with success, of the consistencyof the above hvpothesis.

, J. v. J elley , (jerenkov Radiation (Pergamon Press,

London, 1958), p. 5, and references therein.