aa r X i v : . [ phy s i c s . c l a ss - ph ] D ec The four particles paradox in specialrelativity
J. Manuel Garc´ıa-Islas ∗ Departamento de F´ısica Matem´aticaInstituto de Investigaciones en Matem´aticas Aplicadas y en SistemasUniversidad Nacional Aut´onoma de M´exico, UNAMA. Postal 20-726, 01000, M´exico DF, M´exicoMay 30, 2018
Abstract
We present a novel paradox in special relativity together with its solu-tion. We call it the four particles paradox. The purpose of this paradox ispedagogical and therefore directed towards students and lecturers of physics.Even if most paradoxes in special relativity theory are very interrelated andsome are special cases of others, the paradox we present here is original andilluminates on the very nice subject and the literature of special relativity.
Ever since its appearance [1], Einstein’s special relativity theory has been filled withinteresting paradoxes. We couldn’t agree more with Bernard Schutz’s [2] when inhis opinion paradoxes do not exist, as these are only misunderstood problems. There may only be two reasons about the existence of many paradoxes of spe-cial relativity in the literature. These are only misunderstood problems from asuperficial knowledge of the subject, or they are posed by lecturers and researchersin depth knowledge of the subject who are interested in illustrating these problemsto students of physics, like in [3], [4], [5], [6].From this latter perspective, we can say that paradoxes in special relativity areinteresting problems which are at first confusing, wrongly pointing to inconsisten-cies with the theory, but that after a better understanding of the subject, they arefinally very good exercises for students to master the subject.In this work, we present a novel paradox along with its solution. We call it thefour particles paradox.The main purpose of this work is at the pedagogical level, and will be veryuseful and a very nice example for students as well as for lecturers in relativitytheory. Moreover, the paradox along with its solution requires elementary conceptsof special relativity only. ∗ e-mail: [email protected] See reference [2], pages 23-24 The paradox
We now present the paradox, and its solution. We invite the student to think aboutit before reading the solution.We will consider inertial frames which we denote S , S ′ and S ′′ . Mathemati-cally, let us consider that points at inertial frames are given coordinates ( x, y, z ),( x ′ , y ′ , z ′ ) and ( x ′′ , y ′′ , z ′′ ) respectively. We also suppose that they all move withrespect to each other along the x, x ′ , x ′′ direction and that all their axes are paral-lel. Let us pose the ’paradox’
The four particles paradox:
Two inertial frames S and S ′ move towards eachother with respect to an inertial frame S ′′ and with the same speed v as measuredby S ′′ . Eva (an observer) at rest in S places two classical particles in her frame, onelocated at A = ( x , y , z ) = (0 , ,
0) and the other at B = ( x , y , z ) = ( ℓ, , d ).Manuel (an observer) at rest in S ′ places two identical particles to Eva’s in his frame,one located at A ′ = ( x ′ , y ′ , z ′ ) = (0 ′ , ′ , ′ ) and the other at B ′ = ( x ′ , y ′ , z ′ ) =( ℓ ′ , ′ , d ′ ), such that | ℓ | = | ℓ ′ | and | d | = | d ′ | . (See Figure 1).The experiment consists of the following:According to Eva the identical particles B and B ′ will collide and vanish earlierthan the identical particles A and A ′ because of length contraction along x, x ′ . (SeeFigure 2). However, just after the collision of particle B and B ′ , she decides tocollect particle A before it collides with particle A ′ .Analogously, to Manuel the identical particles A and A ′ are the ones which willcollide and vanish earlier than the identical particles B and B ′ , because of lengthcontraction along x, x ′ . (See Figure 3). However, just after the collision of particle A and A ′ , he decides to collect particle B ′ before it collides with particle B .To an anonymous observer at S ′′ the four particles will collide and vanish si-multaneously and neither Eva nor Manuel will have their corresponding particlesin their hands.So, how is it possible that Eva has in her hand particle A if Manuel saw itvanished when it hit particle A ′ ? In the same way, how is it possible that Manuelhas in his hand particle B ′ if Eva saw it vanished when it hit particle B ? How isit possible that to the anonymous observer neither Eva nor Manuel have a particlein their hands.Who is right? In other words; Eva will claim she has the A particle in her handand that particle B and B ′ have vanished. Manuel will claim he has the B ′ particlein his hand and that particle A ′ and A have vanished. The anonymous observerwill claim the four particles have vanished. Let us now present the solution. We will use basic special relativity concepts only. It is important to mention that we only need two spatial dimensions to describe the problem.However, we stick to three spatial dimensions for aesthetic reasons. Just because physical objectssuch as trains, spaceships, cars, which are represented by inertial frames, are three dimensional. Throughout this article, particles will refer to classical particles, not to quantum ones. Andwhen we say that they vanish as they collide, it means that they will scatter and the observer willno longer see them. We insist one more time to the student to think of the solution before reading it. S and S ′ moving towards each other at speed v , as seenfrom an inertial frame S ′′ . Particles A and B are drawn as seen by observer at S and particles A ′ and B ′ are drawn as seen by observer at S ′ .Due to the addition of velocities in special relativity, Eva and Manuel are movingtowards each other at speed w = 2 v v (1)According to Eva, particles at Manuel’s frame are longitudinally separated a dis-tance L = ℓ ′ p − w (2)due to length contraction along the direction of motion. Moreover, they are ver-tically separated a distance | d | = | d ′ | , since there is no contraction along theperpendicular direction of motion.Therefore, according to Eva, the identical particles B and B ′ will collide andvanish earlier than the identical particles A and A ′ . Just after the collision ofparticle B and B ′ , she decides to collect particle A before it collides with particle A ′ . Analogously, Manuel will observe particles at Eva’s frame longitudinally sepa-rated a distance L ′ = ℓ p − w (3)due to length contraction along the direction of motion. Moreover, they are ver-tically separated a distance | d | = | d ′ | , since there is no contraction along theperpendicular direction of motion.Therefore, according to Manuel, the identical particles A and A ′ will collideand vanish earlier than the identical particles B and B ′ . Just after the collision ofparticle A and A ′ , he decides to collect particle B ′ before it collides with particle B . Let us now see that it is not possible that Eva collects particle A before itcollides with particle A ′ . Particles A and A ′ will collide and vanish before sheprevents them from colliding. And the same applies to Manuel, it is not possible3igure 2: Inertial system S ′ moving towards S at speed w . Particles A ′ and B ′ asseen by observer at S .that he collects particle B ′ before it collides with particle B . Particles B and B ′ will collide and vanish before he prevents them from colliding.If Eva were located just where her A particle is situated , then this is whathappens. Recall that in special relativity all signal information is transmitted, atmost, at the speed of light. Therefore, when particle B and B ′ collide and vanish,a clock situated at the point of collision will read t = 0. Then, Eva will haveknowledge of this collision when light coming from the point of collision gets toher.The point of collision of particles B and B ′ is separated from particle A adistance r = √ ℓ + d . Therefore, information about the collision of particles B and B ′ will reach Eva at proper time t = √ ℓ + d . It will be enough to considerthe longitudinally separation of the point of collision of particles B and B ′ andparticle A given by ℓ so that information of the collision of particles B and B ′ willreach Eva at proper time t = ℓ < √ ℓ + d .From Eva’s point of view, at the moment of collision of particles B and B ′ ,particles A and A ′ are longitudinally located a distance D apart given by D = | ℓ | − | L | = | ℓ | − | ℓ ′ | p − w = | ℓ | − | ℓ | p − w = | ℓ | [1 − p − w ](4)and therefore particles A and A ′ will collide and vanish at Eva’s proper time givenby t = Dw = ℓ [1 − √ − w ] w (5)It can easily be checked that t < t . Let us check this strict inequality Like sitting on top of it, so that she collects it as fast as possible. In units where where c = 1 B w x
Figure 3: Inertial system S moving towards S ′ at speed w . Particles A and B asseen by observer at S ′ . ℓ [1 − √ − w ] w < ℓ ⇒ [1 − p − w ] < w ⇒ − p − w < w − ⇒ − w > [1 − w ] ⇒ > w [ w −
1] (6)and this latter inequality is true, since w < A and A ′ will collide and vanish before Eva knows thatparticles B and B ′ have collided, and therefore, she cannot collect particle A beforeit collides with particle A ′ . By the time she knows that particle B and B ′ havecollided, particles A and A ′ will also be vanished.The same method applies to Manuel with the conclusion that he will not be ableto collect particle B ′ before it collides with particle B , since by the time he realisesabout the collision of particles A and A ′ , particles B and B ′ will be vanished.The paradox is solved. Neither Eva, nor Manuel will have collected a particle,thus agreeing with the anonymous observer.The paradox we presented here can be seen as a smart variation of the twocolliding inclined rods paradox presented in [7]. However the solution presentedhere deals with pure simple relativistic concepts. It does not involve the idea of’extended present’ as invoked to solve the paradox in [7]. In our opinion the term’extended present’ does not exist. The solution we presented here solves both, thefour particles paradox and the one presented in [7].It can easily be seen that in terms of the space-time geometry the observerat S concludes that the separation of the events corresponding to the collision ofparticles B and B ′ and the collision of particles A and A ′ is space-like, as wellas the observer at S ′ concludes that the separation of the events corresponding to Compare with [7]. A and A ′ and the collision of particles B and B ′ is alsospace-like.It is a trivial exercise (for students) to find the Lorentz transformation betweenthe inertial frame S and S ′ which sends the space-like separated events at S intothe space-like separated events at S ′ . Recall that Lorentz transformations sendspace-like vectors into space-like vectors. To sum up, the paradox has been solved using only basic concepts of specialrelativity, and it is suitable to be presented as a good exercise for students. Itilluminates on the subject of relativity and can be used at the pedagogical level byteachers in the area. It also sends time-like vectors into time-like vectors and null vectors into null vectors. eferences [1] Einstein, Albert, Zur Elektrodynamik bewegter K¨orper, Annalen der Physik (10): 891921, 1905.[2] Bernard Schutz, A First Course in General Relativity, Cambridge UniversityPress, Second Edition, 2009.[3] W Rindler, Length contraction paradox, Am. J. Phys. , 365, 1961[4] Øyvind Grøn, Steinar Johannesen, Computer simulation of Rindler’s lengthcontraction paradox, Eur. J. Phys. (5) 563-567, 2012.[7] Chandru lyer, G.M.Prabhu, Reversal in the time order of interactive events:the collision of inclined rods, Eur. J. Phys,27