Illustrating the relativity of simultaneity Bernhard Rothenstein 1), Stefan Popescu 2) and George J. Spix 3)

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1 Illustrating the relativity of simultaneity ernhard Rothenstein 1), Stefan Popesu ) and George J. Spix 3) 1) Politehnia University of Timisoara, Physis Department, Timisoara, Romania, bernhard_rothenstein@yahoo.om ) Siemens AG, Erlangen, Germany, stefan.popesu@siemens.om 3) SEE Illinois Institute of Tehnology, USA, gjspix@msn.om Abstrat. We present a relativisti spae-time diagram that displays in true magnitudes the readings (daytimes) of two inertial referene frames loks. One referene frame is the rest frame for one lok. This diagram shows that two events simultaneous in one referene frames are not ompulsory simultaneous in the other frame. This approah has a bi-dimensional harater. 1. Introdution While teahing speial relativity the physis professor is in the same position as a parent who helps his little hild solving a math problem. Knowing algebra the parent ould solve the problem in two lines, but he may have serious problems explaining it in simple but insightful terms that are omprehensible for the hild. Without knowing or applying the algebra the math professor should invent intuitive solutions for eah problem despite the fat that algebra would otherwise work omfortably in these ases. In physis the Lorentz-Einstein transformations (LET) are the algebra of speial relativity. Using them orretly we an solve all the problems enountered in relativisti kinematis and avoid paradoxes at same time. It is onsidered that 1 : The LET is the standard way to derive formulas for relativisti phenomena. Although the derivation of these formulas is straightforward, it is rather formal and not very transparent from the point of view of physis. Those who use the LET should be aware of the physis behind them. Presenting them as: x = γ ( x + t) (1) y = y () t = γ ( t + x ) (3) 1 1/ using the well established notations = V and γ = (1 ), physiists know that they relate the spae-time oordinates of events E ( x, y, t) and E ( x, y, t ) deteted from the inertial referene frames K(XOY) and respetively K (X O Y ). These equations hold exatly if additionally the following onditions are fulfilled: 1

2 The orresponding axes of the two frames are parallel to eah other and the OX(O X ) axes are ommon. K moves with onstant veloity V = relative to K in the positive diretion of the ommon axes. Events E and E take plae at the same point in spae, whose position is defined in K by the spae oordinates (x,y) respetively in K by the spae oordinates (x,y ). The loks of the two referene frames are synhronized following a proedure proposed by Einstein. The symbol in the LET represents the speed of the light signal propagating in vauum and performing the synhronization of the distant loks. The time oordinates t and t (date times) represent the readings of loks C ( x, y) and C ( x, y ) shortly loated in front of eah other at the point where the events take plae. When the loks of the two referene frames read a zero time (t=t =0) the origins O and O of the two frames are loated at the same point in spae. Homogeneity and isotropy of spae makes that eah point in spae is a onvenient host for the origins O and O and that eah orientation of the axes of the two referene frames is as good as all the others. Homogeneity of time makes that there is no preferred origin for time. Omission of anyone of the onditions imposed above ould lead to paradoxes and to fiere debates in order to eliminate or avoid them. This is the reason why we onsider that the investigation of the diffiulties enountered by students while learning the speial relativity should start with finding out if these students are aware of the onditions imposed above,3. Generating simultaneous events When we speak about a physial quantity it is advisable to mention the observer(s) or the devie(s) that detet or measure it, when and where the detetion takes plae and the measuring devies involved. In our paper we onentrate on the onept of simultaneity and on its relative harater. A thought experiment introdued by Einstein 4 and an alternative of it 5, subsequently borrowed by ountless text books and journal papers, gave rise to disussions onerning the foundational and pedagogial aspets of these experiments 6,7,8,9,10,11,1. Analysing the different approahes to the problem of teahing the relative harater of simultaneity we onlude that they either involve the somehow non-intuitive fundamental relativisti effets (time

3 dilation and length ontration) or they perform the LET of the events involved in the experiment (from the referene frame where the experiment takes plae toward the referene frame relative to whih the experimental devie moves). In our approah we start with some very lear definitions for the simultaneity of two distant events in an inertial referene frame: Two distant events are simultaneous in a given inertial referene frame if the loks loated at the points where the events take plae display the same time. Two distant events are simultaneous in a given inertial referene frame if the light signals originating from the points where the events take plae arrive simultaneously at the middle point in-between these origin points. Two distant loks display the same running time after synhronization. All Einstein synhronized loks within an inertial referene frame display the same running time. If a point-like light soure loated at the origin of an inertial referene frame starts emitting light at t=0 in vauum then all the events it generates after the propagation time = t further ourring on a sphere of radius r = t are simultaneous. Two light signals emitted simultaneously by point-like soures loated at equal distanes from the origin are deteted simultaneously at the origin. Two light signals emitted at different times by two point-like soures loated at different points (events E 1( x1, y1, t1) and E ( x, y, t ) ) are reeived simultaneously at the origin O if: x1 + y1 t1 = (4) x + y t =. (5) The LET take a partiular form if we use them for transforming the spae-time oordinates of events generated by light signals when expressed in polar oordinates ( r, θ ) in K and respetively ( r, θ ) in K. We assoiate the event E 0 (0,0,0) with the emission of a light signal from the origin O at a time t =0 having the same spae-time oordinates in all inertial referene frames. The light signal propagates along a diretion θ and generates the event E ( x = t osθ, y = t sinθ, t ) after a propagation time t with t = r. When deteted from K the same event is haraterized by the spae-time oordinates E( x = t osθ, t sinθ, t) with t = r. If the two events represent the same event then the orresponding spae-time oordinates are related by: 3

4 x = γ r (osθ + ) (6) y = r sinθ (7) r = x + y = γ r (1 + osθ ) (8) r t = = γ t ( osθ ) (9) osθ + os θ = osθ (10) We obtain the inverse transformations by exhanging the orresponding physial quantities in K and K and by inverting the sign of V. In partiular after these operations (10) reads: osθ osθ =. osθ (11) Expressing the right side of (8) and (9) as a funtion of θ via (11) they beome: 1 γ r = r osθ (1) and respetively: 1 γ t = t osθ (13) The important onlusion is that we transform the position vetor lengths ( r, r ) and the time oordinates via the same transformation fator. If the irle r = (14) represents the geometri lous of simultaneous events in K, assoiated with the reeption at t = of the light signals emitted at t = 0 from the origin O then (1) and (13) tell us that when deteted from K these events are no longer simultaneous. Eah event E (, θ, ) in K loated on the irle r = has its orrespondent event E(r,θ, t) in K loated on the ellipse (1). 3. Illustrating the relativity of simultaneity and the thought experiments assoiated with it Equations (1), (13) and (14) leads to a relativisti spae-time diagram that enables us to find out the loation of an event in K and the loation of its Einstein-Lorentz transformed orrespondent in K. Figure 1 shows the diagram that displays the irle (14) and the ellipses (1) and (13) for =

5 Y Y =0.6 R o,r o O O θ θ r,t X, X k -1 R o kr o k = 1 Figure 1. The relativisti spae-time diagram that enables us to establish the loations of the same events 1 and 1 deteted from the referene frames K and K. The invariane of distanes measured perpendiular to the diretion of relative motion tell us that the event 1 in K is the orrespondent of event 1 in K. One fous of the ellipse oinides with the entre of the irle. Let 1 ( R,, t 0 θ 1 ) and ( R,, t 0 θ ) be two simultaneous events in K. The orresponding events as deteted from K are 1( r1, θ 1, t1 ) and ( r, θ, t) separated by a time interval t = t t1 that depends on the loation of events 1 and in K as shown in Figure. Y Y =0.6 R o,r o -1 r,t 1 1 R o,r o -1 O O θ θ r 1,t 1 X, X k = 1 k -1 R o kr o Figure. The events 1 and are simultaneous in K. The orresponding events 1 and as deteted from K are no longer simultaneous. 5

6 Consider one of the thought experiments used in the teahing the speial relativity 5 skethed in Figure 3 at rest in K. A R o Figure 3. A thought experiment used in order to illustrate the relative harater of simultaneity. A light soure is plaed midway between reeivers A, A respetively. The distane between the two observers is R 0. The light soure emits a flash (events O,O ) towards the observers A and A. Arriving there they generate the events A'((,0, ) and (,0, ) whih are simultaneous in K. When deteted from K the event A( x A,0, t A ) is the orrespondent of A whereas the event ( x,0, t ) is the orrespondent of event. With ( θ = 0 ) we have: t = (14) and with ( θ = π ) we have respetively: t A =. (15) As intuition tells us and the theory onfirms, the event E A takes plae before the event E ( t > t A) and the two events are separated by a time interval: t = t t A = γ (16) whih is proportional with the distane separating the two events in the rest frame of the experimental devie and with the relative veloityv =. Figure 4 shows a thought experiment proposed by Einstein 4. It is performed in two steps. We made some minor hanges in the senario, whih do not alter the physis behind the experiment. Y' S O R o X 6

7 Y' a A R o O R o X Y' b A R o O R o X Figure 4. Einstein s famous experiment largely used in order to illustrate the relative harater of simultaneity. Figure 4a shows the first part of the experiment performed in the rest frame of the experimental devie K. Two mirrors A and tilt at 45 0 relative to the O X axis are loated at equal distanes R 0 from O. Two rays of a plane eletromagneti wave, propagating in the negative diretion of the O Y axis, arrive simultaneously at the loation of the two mirrors generating the events A (,0, ) and (,0, ) respetively. Refleted by the mirrors the two rays return simultaneously to the origin O generating the event O '(0,0,0) that has the same spae-time oordinates in all inertial referene frames (Figure 4b). Deteted from K the orresponding events are O (0,0,0), A ( x A,0, t A ) and ( x,0, t ). Figure 5 shows the relativisti spae-time diagram that works in the new situation. 7

8 r 1,t 1 Y Y 1 1 =0.6 R o,r o -1 A A θ θ r,t O O X, X R o,r o -1 kr o k -1 R o k = 1 Figure 5. The relativisti spae-time diagram in the ase of onverging light signals. The diagram displays the geometri lous of the simultaneous events in K (the irle r = ) and the geometri lous of the orresponding events 1 γ deteted from K (the ellipse t = osθ (17) ). The invariane of distanes measured perpendiular to the diretion of relative motion makes that event 1 orresponds to event 1, orresponds to, A orresponds to A and orresponds to. We have: t A = (18) t =. (19) The time interval separating the events O and A is: t AO = (0) The time intervals separating the events O and and the events and A are respetively: t O = (1) t A = γ () The diagram illustrates the way in whih the Lorentz-Einstein transformations hange in different ways the time oordinates of 8

9 simultaneous events. The event assoiated with the return of the two rays to point O has the same time oordinates in all inertial referene frames. 4. Conlusions Nature has unfortunately denied me the gift of being able to ommuniate, so that what I write is orret, but also thoroughly indigestible says Albert Einstein about himself 13. However his transformation equations sueeded to beome a universal tool that redues ommuniation to zero and avoids paradoxes at same time. The relativisti diagram that we propose here illustrates quantitatively the way in whih the transformation equations differently hange the time oordinates of simultaneous events in a given inertial referene frame. Referenes 1 Asher Peres, Relativisti telemetry Am.J.Phys. 55, (1987) R. Sherr, P. Shaffer and S.Vokos, Student understanding of time in speial relativity, Am.J.Phys. 69, S4 (001) 3 R. Sherr, P.Shaffer and S.Vokos, The hallenge of hanging deeply held student beliefs about the relativity of simultaneity, Am.J.Phys 30, 138 (00) 4 A. Einstein, Relativity: The Speial and General Theory, a Popular Exposition, (New York; Crown 1961) Ch.4 5 L. Sartori, Understanding Relativity, (erkeley: University of California Press, 1966) p.54 6 A. Nelson, Reinterpreting the famous train/embankment experiment of relativity, Eur.J.Phys. 5, 645 (004) 7 D. Rowland, Comment on Reinterpreting the famous train/embankment experiment of relativity, Eur.J.Phys. 5, L45 (004) 8 J. Malinkrodt, On reinterpreting the famos train/embankment experiment of relativity, Eur,J.Phys. 5, L45 (004) 9 Garry E. owman, Gedanken experiments and the relativity of simultaneity, Eur.J.Phys. 6, (005) 10 Allen I. Janis, Simultaneity and speial relativisti kinematis, Am.J.Phys. 51, (1983) 11 Elisha R. Huggins, On teahing the lak of simultaneity, Am.J.Phys. 43, (1975) 1 G. Robinson and J.V. Denholm, Einstein s train on the embankment and the relativity of simultaneity, Am.J.Phys. 54, (1986) 13 A. Pais, Subtle is the Lord, (Oxford: Oxford University Press) pp