EDUCATION Revista Mexicana de Fsica E 60 (2014) 107110 JULYDECEMBER 2014The four particles paradox in special relativityJ.M. Garca-IslasDepartamento de Fsica Matem atica, Instituto de Investigaciones en Matem aticas Aplicadas y en Sistemas,Universidad Nacional Aut onoma de M exico,Apartado Postal 20-726, 01000, M exico DF, M exico,e-mail: jmgislas@iimas.unam.mxReceived 22 May 2014; accepted 22 October 2014We present a novel paradox in special relativity together with its solution. We call it the four particles paradox. The purpose of this paradoxis pedagogical and therefore directed towards students and lecturers of physics. Even if most paradoxes in special relativity theory are veryinterrelated and some are special cases of others, the paradox we present here is original and illuminates on the very nice subject and theliterature of special relativity.Keywords: Special relativity; length contraction; paradox; general physics.PACS: 03.30.+p; 01.55.+b1. IntroductionEver since its appearance [1], Einsteins special relativity the-ory has been lled with interesting paradoxes. We couldntagreemorewithBernardSchutzs [2] wheninhis opin-ion paradoxes do not exist, as these are only misunderstoodproblemsi.Theremayonlybetworeasonsabouttheexistenceofmany paradoxes of special relativity in the literature. Theseare only misunderstood problems from a supercial knowl-edgeofthesubject, ortheyareposedbylecturersandre-searchers in depth knowledge of the subject who are inter-ested in illustrating these problems to students of physics, likein Refs. 3 to 6.From this latter perspective, we can say that paradoxes inspecial relativity are interesting problems which are at rstconfusing, wrongly pointing to inconsistencies with the the-ory, but that after a better understanding of the subject, theyare nally very good exercises for students to master the sub-ject.In this work, we present a novel paradox along with itssolution. We call it the four particles paradox.The main purpose of this work is at the pedagogical level,and will be very useful and a very nice example for studentsaswellasforlecturersinrelativitytheory. Moreover, theparadox along with its solution requires elementary conceptsof special relativity only.2. The paradoxWe now present the paradox, and its solution. We invite thestudent to think about it before reading the solution.We will consider inertial frames which we denoteS,S

andS

. Mathematically, letusconsiderthatpointsatin-ertial frames are given coordinates (x, y, z), (x

, y

, z

) and(x

, y

, z

) respectively. We also suppose that they all movewith respect to each other along thex, x

, x

direction andthat all their axes are parallelii.Let us pose the paradoxThe four particles paradox:Two inertial frames S andS

move towards each other with respect to an inertial frameS

and with the same speed v as measured by S

. Eva (an ob-server) at rest in S places two classical particles in her frame,one located atA=(x1, y1, z1)=(0, 0, 0) and the other atB=(x2, y2, z2)=(, 0, d). Manuel (an observer) at restin S

places two identical particles to Evas in his frame, onelocated at A

=(x

1, y

1, z

1) =(0

, 0

, 0

) and the other atB

=(x

2, y

2, z

2)=(

, 0

, d

), such that ||=|

| and| d |=| d

|. (See Fig. 1).The experiment consists of the following:According to Eva the identical particlesBandB

willcollide and vanishiiiearlier than the identical particles A andA

because of length contraction alongx, x

. (See Fig. 2).However, just after the collision of particleBandB

, shedecides to collect particleA before it collides with particleA

.Analogously, to Manuel the identical particles A and A

aretheoneswhichwillcollideandvanish earlierthantheidentical particlesBandB

,because of length contractionalong x, x

. (See Fig. 3). However, just after the collision ofparticle A and A

, he decides to collect particle B

before itcollides with particle B.To an anonymous observer at S

the four particleswill collide and vanish simultaneously and neither Eva norManuel will have their corresponding particles in their hands.108 J.M. GARCIA-ISLASFIGURE 1. Inertial systems S and S

moving towards each other at speed v, as seen from an inertial frame S

. Particles A and B are drawnas seen by observer at S and particles A

and B

are drawn as seen by observer at S

.So,how is it possible that Eva has in her hand particleA if Manuel saw it vanished when it hit particleA

?In thesame way, how is it possible that Manuel has in his hand par-ticle B

if Eva saw it vanished when it hit particle B?Howis it possible that to the anonymous observer neither Eva norManuel have a particle in their hands.Who is right? In other words; Eva will claim she has theA particle in her hand and that particle B and B

have van-ished. Manuel will claim he has theB

particle in his handand that particleA

andA have vanished. The anonymousobserver will claim the four particles have vanished.2.1. The solutionLet us now present the solutioniv. We will use basic specialrelativity concepts only.Due to the addition of velocities in special relativity, Evaand Manuel are moving towards each other at speedw =2v1 +v2(1)According to Eva, particles at Manuels frame are longitudi-nally separated a distanceL =

1 w2(2)due tolengthcontractionalongthe directionof motion.Moreover, they are vertically separated a distance | d |=| d

|,since there is no contraction along the perpendicular directionof motion.Therefore, according to Eva, the identical particles B andB

will collide and vanish earlier than the identical particlesA andA

. Just after the collision of particleB andB

, shedecides to collect particleA before it collides with particleA

.Analogously, Manuel will observe particles at Evasframe longitudinally separated a distanceL

=

1 w2(3)due tolengthcontractionalongthe directionof motion.Moreover, they are vertically separated a distance | d |=| d

|,since there is no contraction along the perpendicular directionof motion.FIGURE 2. Inertial system S

moving towards S at speed w. Particles A

and B

as seen by observer at S.Rev. Mex. Fis. E 60 (2014) 107110THE FOUR PARTICLES PARADOX IN SPECIAL RELATIVITY 109FIGURE 3. Inertial system S moving towards S

at speed w. Particles A and B as seen by observer at S

.Therefore, according to Manuel, the identical particles Aand A

will collide and vanish earlier than the identical parti-cles B and B

. Just after the collision of particle Aand A

, hedecides to collect particle B

before it collides with particleB.Let us now see that it is not possible that Eva collects par-ticle A before it collides with particle A

. Particles A and A

will collide and vanish before she prevents them from col-liding. And the same applies to Manuel, it is not possiblethat he collects particle B

before it collides with particle B.Particles B and B

will collide and vanish before he preventsthem from colliding.If Eva were located just where her A particle is situatedv,thenthisiswhathappens. Recallthatinspecialrelativityall signal information is transmitted, at most, at the speed oflight. Therefore, when particle B and B

collide and vanish,a clock situated at the point of collision will readt0=0.Then, Eva will have knowledge of this collision when lightcoming from the point of collision gets to her.The point of collision of particles B and B

is separatedfrom particle A a distance r=

2+d2. Therefore, infor-mation about the collision of particlesBandB

will reachEvaviiat proper timet1=

2+d2. It will be enough toconsider the longitudinally separation of the point of colli-sion of particles B and B

and particle A given byso thatinformation of the collision of particles B and B

will reachEva at proper time t1 = [1 w]2 0 > 2w[w 1] (6)and this latter inequality is true, since w < 1.Therefore, particles A and A

will collide and vanish be-fore Eva knows that particlesBandB

have collided,andtherefore, she cannot collect particle Abefore it collides withparticleA

. By the time she knows that particleBandB

have collided, particles A and A

will also be vanished.The same method applies to Manuel with the conclusionthat he will not be able to collect particleB

before it col-lides with particle B, since by the time he realises about thecollision of particlesA andA

,particlesBandB

will bevanished.The paradox is solved. Neither Eva, nor Manuel will havecollected a particle, thus agreeing with the anonymous ob-server.The paradox we presented here can be seen as a smartvariation of the two colliding inclined rods paradoxviipre-sented in Ref 7. However the solution presented here dealswith pure simple relativistic concepts. It does not involve theidea of extended present as invoked to solve the paradox inRef 7. In our opinion the term extended present does notexist. The solution we presented here solves both, the fourparticles paradox and the one presented in Ref 7.It can easily be seen that in terms of the space-time ge-ometry the observer at S concludes that the separation of theevents corresponding to the collision of particlesBandB

and the collision of particles Aand A

is space-like, as well asthe observer at S

concludes that the separation of the eventscorresponding to the collision of particles A and A

and thecollision of particles B and B

is also space-like.Rev. Mex. Fis. E 60 (2014) 107110110 J.M. GARCIA-ISLASIt is a trivial exercise (for students) to nd the LorentztransformationbetweentheinertialframeSandS

whichsends the space-like separated events at S into the space-likeseparated events atS

. Recall that Lorentz transformationssend space-like vectors into space-like vectorsviii.To sum up, the paradox has been solved using only ba-sic concepts of special relativity, and it is suitable to be pre-sented as a good exercise for students. It illuminates on thesubject of relativity and can be used at the pedagogical levelby teachers in the area.i. See Ref. 2, pages 23-24.ii. Itisimportanttomentionthatweonlyneedtwospatialdi-mensions to describe the problem. However, we stick to threespatial dimensions for aesthetic reasons. Just because physicalobjects such as trains, spaceships, cars, which are representedby inertial frames, are three dimensional.iii. Throughout this article, particles will refer to classical particles,not to quantum ones. And when we say that they vanish as theycollide, it means that they will scatter and the observer will nolonger see them.iv. We insist one more time to the student to think of the solutionbefore reading it.v. Like sitting on top of it, so that she collects it as fast as possible.vi. In units where c = 1.vii. Compare with [7].viii. It also sends time-like vectors into time-like vectors and nullvectors into null vectors.1. A. Einstein, Annalen der Physik 17 (1905) 891921.2. B. Schutz, AFirst CourseinGeneral Relativity(CambridgeUniversity Press, Second Edition, 2009).3. W. Rindler, Am. J. Phys. 29 (1961) 365.4. yvind Grn, S. Johannesen, Eur. J. Phys. 14 (1993) 97-100.5. yvind Grn, Phys. Scr. 87 (2013) 035004.6. R. Cacioppo, A. Gangopadhyaya, Physics Education 47 (2012)563-567.7. Ch. lyer, G.M. Prabhu, Eur. J. Phys 27 (2006) 819-824.Rev. Mex. Fis. E 60 (2014) 107110