Reinterpreting the famous train/embankment experiment of relativity This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2003 Eur. J. Phys. 24 379 (http://iopscience.iop.org/0143-0807/24/4/356) Download details: IP Address: 128.104.46.196 The article was downloaded on 03/09/2013 at 04:33 Please note that terms and conditions apply. View the table of contents for this issue, or go to the journal homepage for more Home Search Collections Journals About Contact us My IOPscience
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Reinterpreting the famous train/embankment experiment of relativity
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
2003 Eur. J. Phys. 24 379
(http://iopscience.iop.org/0143-0807/24/4/356)
Download details:
IP Address: 128.104.46.196
The article was downloaded on 03/09/2013 at 04:33
Please note that terms and conditions apply.
View the table of contents for this issue, or go to the journal homepage for more
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Received 18 February 2003Published 19 May 2003Online at stacks.iop.org/EJP/24/379
AbstractEinstein’s well-known train/embankment experiment, frequently used tointroduce the relativity of simultaneity, is revisited. Though present innumerous textbooks, the traditional analysis is shown to be incomplete andinadequate to demonstrate that simultaneity is relative. Completing the analysisproduces not the relativity of simultaneity in its usual form but rather a newcontradiction. It is shown that, under the conditions of the experiment, asimultaneity in one inertial frame does indeed hold in another inertial framebut is perceived as a non-simultaneity by an observer outside the given frame.The source of the paradox is traced to the experiment itself, which is posited inNewtonian space–time but under Einsteinian postulates. Though crafted as anintroduction to relativity, the experiment is seen, ironically, to require relativityto analyse it. The contradictions are resolved by a relativistic analysis.
1. Introduction
1.1. Author’s foreword
Let me begin with what this paper does not do. It does not question the relativity of simultaneitynor quarrel with any aspect of special relativity. (Given the subject, I want to make clear thatthis paper is not about some alleged ‘error’ in the theory.)
What this paper does do is investigate the capability of one thought experiment todemonstrate the relativity of simultaneity. The experiment is found to be inadequate to thetask, and this is shown several ways (sections 3–7, 9).
Arguably, one showing should suffice. But I have found that, for some, relinquishingwhat has been accepted analysis and pedagogical practice for six-sevenths of a century doesnot come easily. So I have presented the argument in multiple forms, hoping that at leastone will be persuasive. The conceptual foundation of all the arguments is summarized insection 10. A relativistic resolution, derived in the appendix, is presented in section 11.
One of Albert Einstein’s most famous thought experiments was on the relative natureof simultaneity. The train/embankment experiment has become a staple of introductorycontemplations on relativity. It has been reproduced numerous times in books and textbooksdevoted to or with chapters on the special theory of relativity (e.g., [1–10]).
For many, this is the very starting point of the theory. Indeed, in 1916 [11] when Einsteinhimself wrote the book Relativity [12] for popular consumption, his point of departure fromGalilean relativity was this thought experiment. Moreover, in the preface Einstein wrotethat he had ‘endeavour(ed) to present the main ideas . . . , on the whole, in the sequence andconnection in which they actually originated’ [13]. In fact, in his landmark 1905 paper, wherehe introduced the theory of special relativity to the world, Einstein began with derivations ofsimultaneity and synchrony using a model which could be considered an embryonic form ofthis experiment [14].
Because of its fame, its invention by Einstein, and its near ubiquity [15] in textbooks,the train/embankment experiment has an important pedagogical and historical pedigree. Thispaper examines the interpretation of that thought experiment.
2. Einstein’s presentation of the experiment
In the book Relativity, Albert Einstein begins chapter nine, ‘The Relativity of Simultaneity’,with the following [16]:
‘Up to now our considerations have been referred to a particular body of reference, whichwe have styled a ‘railway embankment’. We suppose a very long train travelling along therails with the constant velocity v and in the direction indicated in figure 1. People travellingin this train will with advantage use the train as a rigid reference-body (coordinate system);they regard all events in reference to the train. Then every event which takes place along theline also takes place at a particular point of the train. Also the definition of simultaneity canbe given relative to the train in exactly the same way as with respect to the embankment. As anatural consequence, however, the following question arises:
‘Are two events (e.g. the two strokes of lightning A and B) which are simultaneous withreference to the railway embankment also simultaneous relatively to the train?’ We shall showdirectly that the answer must be in the negative (italics Einstein’s).
‘When we say that the lightning strokes A and B are simultaneous with respect to theembankment, we mean: the rays of light emitted at the places A and B , where the lightningoccurs, meet each other at the mid-point M of the length A −→ B of the embankment. Butthe events A and B also correspond to positions A and B on the train. Let M ′ be the mid-pointof the distance A −→ B on the travelling train. Just when the flashes1 of lightning occur, thispoint M ′ naturally coincides with the point M , but it moves towards the right in the diagramwith the velocity v of the train. If an observer sitting in the position M ′ in the train did notpossess this velocity, then he would remain permanently at M , and the light rays emitted bythe flashes of lightning A and B would reach him simultaneously, i.e. they would meet justwhere he is situated. Now in reality (considered with reference to the railway embankment)he is hastening towards the beam of light coming from B , whilst he is riding on ahead of the
1 As judged from the embankment. (Footnote in text.)
Reinterpreting the famous train/embankment experiment of relativity 381
beam of light coming from A. Hence the observer will see the beam of light emitted fromB earlier than he will see that emitted from A. Observers who take the railway train as theirreference-body must therefore come to the conclusion that the lightning flash B took placeearlier than the lightning flash A.’
3. Einstein’s presentation in perspective
But, given the experiment as presented, this is not what the observer (OT ) on the train at M ′ willsee, nor how he will analyse what happened. The narrative is, rather, how the embankmentobserver (OE ) describes the encounter between OT and the two light beams. Conclusionsdrawn in the text about observers (such as OT ), who use the train as a reference frame, arebased solely on what OE says OT should see. To be accepted, these conclusions must beverified by OT ’s own observations—a verification that, as it turns out, cannot be made.
OT could not proffer OE ’s description of events for the following reason: OT and histrain are at rest in an inertial frame of their own. In that frame, as in any inertial frame, thespeed of light equals c. At his location at M ′, OT can detect the arrival of light beams anddetermine the directions whence they came. But he could not detect his moving toward onelight beam and away from another. For if he could, it would indicate his having a differentvelocity with respect to one light beam versus the other. Or, equivalently, it would meanthat the two light beams have different relative velocities with respect to OT . But this wouldviolate the requirement that the speed of light be constant throughout the inertial frame and socannot happen. The conclusion must be, therefore, that OT could not witness the experienceOE ascribes to him. OT must give a different account—a different, dare we say, tale of twoelectricities.
There is also the problem that the experiment has only been partially analysed, since theresults in both frames have been presented from the perspective of only one frame—that of theembankment. Indeed, the first lesson of relativity is that one cannot rely on conclusions aboutevents in one frame from an observer in another (moving) frame. Inquiry into OT ’s experiencefrom OT ’s own point of view (i.e., from the perspective of the train frame) therefore becomesnecessary to fulfil the purpose of the experiment—which is to ascertain whether the frame ofreference affects the description of events.
In this case, the events are the times of occurrence of the lightning strokes at A and B—that is, the question of their simultaneity. To analyse the timing of the lightning strokes inboth frames solely from the perspective of the embankment frame is, thus, insufficient. It alsogives an unjustified preference to the embankment frame, when, according to the principle ofrelativity, the two frames are equivalent. Therefore, the view from the train frame must alsoand separately be investigated. The principle of relativity says the results should be the same.Otherwise, there must be a reconciliation.
4. View from the train frame
If OT is ‘unmoved’by OE ’s account of the experiment, how would OT describe the adventure?OT sits in the middle of the long train (at M ′) and, as the train is an inertial frame, senses nomotion at all.
By stipulation, the lightning strokes are simultaneous at A and B on the embankment.Then Einstein adds, ‘but the events A and B also correspond to positions A and B on thetrain’. These are the same A and B—as is confirmed by the diagram in figure 1. Note thatwhile there are M and M ′, there is no A′ or B ′, only A and B . We must conclude, therefore,that A(embankment) coincides with A(train) and B(embankment) coincides with B(train) atthe instant the lightning flashes occur. Put another way, A(embankment) and A(train) arespace–time coincident—i.e., they are one and the same point—when the strokes occur, so astroke that hits A(embankment) also hits A(train) at the same instant. Likewise, a stroke hittingB(embankment) hits B(train).
382 A Nelson
4.1. A digression
At this juncture it is important to emphasize, especially for those familiar with special relativity,that this is a pre-relativistic perspective—that is, the discussion takes place prior to anyknowledge of relativity. Indeed, the objective of the experiment is to introduce relativityby showing (qualitatively) that simultaneity is frame-dependent. For purposes of this analysis,then, there are no relativistic transformation effects for space and time. (Note: none are usedin the Einstein description.) Relativistic results such as length contraction and time dilationdo not yet exist.
So, length and time are unchanged going from one frame to another. That is, in thisexperiment length and time remain dimensionally absolute in the Newtonian sense; therefore,Galilean relativity applies. As regards length, this means there is correspondence in the twoframes for A and B , and the separation between A and B (i.e., the length of the train) is thesame in both frames. With respect to time, this means that clocks read the same and timepasses at the same rate in the two frames. Therefore, the lightning bolts must hit A and B inthe two frames at the same instant.
On the other hand, the experiment is already quasi-relativistic or Einsteinian because ofthe second postulate and the definition of simultaneity. This translates into the communicationor conveyance of time being done at the speed of light, not infinitely quickly.
The combination of conditions produces an experiment hypothesized under an unusual(and impossible) hybrid of Newtonian and Einsteinian conceptions of space and time. Butthese are the conditions given, so it is under these that the experiment must be analysed. Onecannot ‘cheat’ ahead and apply results one already ‘knows’ from relativity, or the experimentwould no longer be introductory. In fact, were we to use the result of the derivation to make thederivation, we would engage in a tautology and render the experiment meaningless. (Moreover,students uninformed about relativity, for whom this experiment is posed, would find such ananalysis incomprehensible.)
Given that the hybrid is physically impossible, however, one could argue that we shouldsimply acknowledge the contradiction and proceed down another track. But countless studentshave been (and continue to be) led through this experiment in its hybrid form—and to anunwarranted conclusion. So there is value in following a derivation which adheres strictlyto experimental constraints, especially since it produces a different outcome. The resultingparadox is instructive in its own right and motivates a relativistic analysis (section 11 andappendix). End of digression.
4.2. View from train frame (continued)
We return to OT in the train frame. OT is aware of no motion, and he is located midwaybetween A and B in a domain where the speed of light equals c. Therefore, light from thestroke at A will take the same time to reach him as light from the stroke at B . Since light fromthe strokes arrive at the mid-point M ′ at the same time (the definition of simultaneity), OT willconclude that the lightning strokes were simultaneous. OT will also observe that light fromthe stroke at A reaches OE before light from the stroke at B . Therefore, OT will declare thatthe strokes were not simultaneous for OE .
5. Collinear depiction
The argument becomes more compelling with the figure redrawn. Figure 1 is presentedschematically, with the train and embankment separated for clarity. But conceptually, train andembankment are collinear. This is depicted in figure 2—which also shows that A(embankment)and A(train) are a single point, as are B(embankment) and B(train). The dots represent thelightning strokes having hit A and B on the embankment simultaneously.
Time is the same throughout both frames. Figure 2 is thus a snapshot, or the situationfrozen in time, at the instant of the strikes. Since the speed of light equals c in both frames,
Reinterpreting the famous train/embankment experiment of relativity 383
(OT
)
M' Train
A M B Embankment
(OE)
v
Figure 2. Collinear representation of the train/embankment experiment.
OE (at M) will perceive that light signals from the dots will reach him at the same time. ButOT (at M ′) will likewise perceive that light signals from the dots will reach him at the sametime. That is, each observer will perceive the lightning strokes to have been simultaneous inhis own frame (and non-simultaneous in the other observer’s frame).
Described another way, from figure 2, AB(embankment) = AB(train) = L, so the trainspans the same length in each frame. This means the train has its ‘proper’ length in bothframes. If that strikes the relativistically informed reader as impossible, that reader has justconfronted the reason the pre-relativistic train/embankment experiment derails.
6. The traditional result applied in the train frame
Let us come at the problem from the reverse angle. Suppose, ignoring the findings of the lasttwo sections, we accept the (traditional) conclusion—that the lightning bolts are simultaneouson the embankment but not simultaneous in the train frame. What does this say, then, aboutevents in the train frame?
Figure 3(a) presents the initial conditions of the experiment—simultaneous events in theembankment frame. In the train frame, the embankment moves to the left with velocity v, andin the conventional analysis event 1 takes place first. For convenience, let event 1 occur whenB coincides with B ′. At that instant (and only then) the train stretches from A to B on theembankment (figure 3(b)).
Since in the train frame the events are not simultaneous, event 2 must take place atsome later time, and during the intervening period the embankment must move relative to
384 A Nelson
K' A' O' B' v
K A O B(a)
K' A' O' B'
v K A B(b)
K' A' O' B'
K A O B(c)
O
Figure 4. Experiment with conceptually equivalent frames. (a) K -perspective, (t = 0): K ′ moveswith velocity v to the right; (b) K ′-perspective, (t ′ = 0): K moves with velocity v to the left; (c)snapshot of either frame, (t = t ′ = 0).
the train (figure 3(c)). Under pre-relativistic conditions time passes at the same rate in bothframes, and AB = A′ B ′ = L.
But in figure 3(a) a lightning bolt (event 2) strikes A on the embankment, whereas infigure 3(b) there is no lightning bolt hitting A. In fact, by the time event 2 takes place(figure 3(c)), the embankment (with its points A, M , and B) has moved some distance to theleft, and the second bolt hits at D (not A) on the embankment. So, A is never struck at all. Alightning strike, however, is a verifiable event. Its absence at A and presence at D would benoted by the embankment observer and would thus be a contradiction of figure 3(a). Moreover,if we accept the analysis from the train frame, that the bolts hit the embankment at B and D,the lightning strokes would not, in general, be simultaneous in the embankment frame—acontradiction of the initial conditions of the experiment.
So, the event pattern stipulated in the embankment frame (figure 3(a)) is incompatible withthe event sequence in the train frame (figures 3(b) and (c))—even though the latter derives fromthe former following the conventional analysis. Since the two scenarios are contradictory andirreconcilable, something in that analysis must be inconsistent. We find, as before, therefore,that the traditional result is untenable [17].
7. The experiment analysed with generic frames
This commends a return to the result of sections 4 and 5. The symmetry and paradox ofthat derivation can be shown to advantage if we abstract the frame designations of trainand embankment (thereby equalizing the frames conceptually). Figure 4 presents the sameexperiment but with two generic frames K and K ′. From a K -perspective, K ′ moves withvelocity v to the right (figure 4(a)). From a K ′-perspective, K moves with velocity v to theleft (figure 4(b)) [18]. For both figures AB = A′ B ′ = L.
Now we take a snapshot of each frame at the instant the lightning strokes hit (t = t ′ = 0).Photographs freeze all motion, and, again, time is the same throughout both frames. So wehave a picture of figure 4(a) and a picture of figure 4(b), with the v-vectors absent. But then
Reinterpreting the famous train/embankment experiment of relativity 385
K' A' O' B' v (a)
K A O B
K' A' O' B' v (b)
K A O
K' A' O' B' v
K A O (c)
K' A' O' B' v
K A O(d)
B
B
B
Figure 5. K -perspective: (a) t = 0; (b) B-flash arrives at O ′, tB = L/2(c + v); (c) both flashesarrive at O ′, tAB = L/2c; (d) A-flash arrives at O ′, tA = L/2(c − v).
the two snapshots are identical. That is, when the lightning bolts strike, figures 4(a) and (b)are indistinguishable; looking just at one of the photographs one could not tell whether itwas a picture of figure 4(a) or (b). Put another way, figure 4(a) turned upside down becomesfigure 4(b) and vice versa. The isotropy of space then requires indistinguishability. The singleresulting image is shown in figure 4(c)—which is the same as figure 2.
Yet from figure 4(a) comes figure 5, from figure 4(b) comes figure 6. These are equivalentdepictions of the event sequence starting from exactly the same fact pattern. But in figure 5,observer O determines that the strokes were simultaneous, because the AA′-flash and theB B ′-flash arrive at the same time. O also concludes that O ′ will say the strokes were notsimultaneous, because, from O’s perspective, O ′ receives the B B ′-flash before the AA′-flash.
In contradiction, however, in figure 6, O ′ determines that those same lightning strokeswere actually simultaneous, because the AA′-flash and the B B ′-flash reach him at the sametime. Moreover, and conversely, O ′ concludes that O will find that the strokes were notsimultaneous, because, from O ′’s perspective, O receives the AA′-flash before the B B ′-flash.
O’s description of O ′’s experience is the mirror image of O ′’s description of O’sexperience. Each ascribes non-simultaneity to the other, but the order of arrival of the flashesis reversed, because the relative velocity of the two is reversed.
The paradox is, then, that O (OE ) thinks that two events he observes to be simultaneousshould be viewed by O ′ (OT ) as non-simultaneous. For his part, O ′ (OT ) observes those sameevents to be simultaneous but argues that O (OE ) should witness them as non-simultaneous.This would seem to show that simultaneity in one inertial frame does indeed hold in anotherinertial frame but does not appear that way to an observer outside the frame. This is verydifferent from (almost opposite to) what the thought experiment purports to demonstrate.
386 A Nelson
K' A' O' B' (a)
v K A
A
A
O
O
O
B
K' A' O' B' (b)
v K B
K' A' O' B' (c)
v K B
K' A' O' B' (d)
v K A O B
Figure 6. K ′-perspective: (a) t ′ = 0; (b) A′-flash arrives at O , t ′A = L/2(c + v); (c) both flashesarrive at O ′, t ′AB = L/2c; (d) B ′-flash arrives at O , t ′B = L/2(c − v).
The outcome also appears to violate the principle of relativity, which says that experimentalresults in a given frame should be the same regardless of the frame of observation. (Ultimately,these conclusions turn out to be incorrect, but only when we go to a relativistic analysis.)
Note, the paradoxical result (where the observers contradict each other) emerges because ofthe Newtonian–Einsteinian hybrid. In a purely Newtonian analysis both ‘observers’ would holdthe strokes to be simultaneous in both frames. Relativistically, as we shall see, both observersfind the strokes to be simultaneous in the K -frame (embankment) and non-simultaneous in theK ′-frame (train). The hybrid case, however, produces internal inconsistency.
The analysis in this and prior sections has revolved around one key point—that, underpre-relativistic conditions, the following statement holds in both frames: A coincides with A′when B coincides with B ′, and that is the instant when the lightning strokes hit. From suchinitial conditions we cannot obtain the traditional relativity of simultaneity.
(As the foreword forewarned, I have, perhaps, belaboured the point. But this is the cruxof the reinterpretation, and with an experiment as fundamental, famous, and familiar as this,no generosity is accorded a would-be reinterpreter.)
8. The experiment using non-light signals
The (traditional) interpretation given in the text would be correct for any signal mode otherthan light. For example, consider again figure 2 but now with sound waves travelling throughair. For convenience, let the air be still relative to the embankment. The signals still arrivesimultaneously for OE , but now both OE and OT agree that they are not received simultaneouslyby OT in the train frame.
Reinterpreting the famous train/embankment experiment of relativity 387
Table 1. Values of TD, T ′D by observer and frame. Each observer sees the lightning strokes as
simultaneous in his own frame but as non-simultaneous in the other frame.
Frame
Observer K , embankment K ′, train
O, OE 0Lv
c2 − v2
O ′, OT − Lv
c2 − v20
OE ’s observation is the same as before; he sees OT moving toward the B-signal and awayfrom the A-signal. But OT ’s analysis changes. He still regards himself (and his train) asmotionless, but now he detects a head-wind—that is, a wind blowing from the front of the traintoward the back. Because of this head-wind, in the train frame the sound wave from B travelstoward OT faster than the sound wave from A. That is, the velocity of sound is anisotropic inthe train frame. Hence, by his own analysis, OT receives the signal from B before that from A.
Einstein’s analysis dealt with light not sound, but it was done only from the perspectiveof the embankment frame, for which the results are the same for both signal modes. That thespeed of light also equals c throughout the train frame was posited, but it was never used. Forno aspect of the experiment was evaluated from the perspective of the train frame.
This left open the possibility that agreement with the embankment frame result wouldemerge when the signal velocity was anisotropic in the train frame—as proved to be the casewith sound. In fact, since constant (therefore, isotropic) velocity in all inertial frames is uniqueto light, all signal modes other than light will yield agreement with the Einstein derivation.
It is only when the analysis is extended to include the train frame perspective that we canapply the requirement that the speed of light equal c (isotropically) in that frame, too. Thisleads to the paradoxical results described in prior sections.
9. Pre-relativistic quantitative analysis
Before resolving the paradox and the conflicts, let us consider the problem quantitatively,as this invites comparison with the relativistic results. Einstein does not quantify the result(neither do other authors). One probable reason is that it sufficed to show that simultaneity isrelative to the frame of reference; for Einstein realized that if simultaneity is relative, so toomust be time itself. A second probable reason emerges from the relativistic results.
A quantitative analysis comes from figures 5 and 6. In K , we define TD as the timedifference between reception of the two signals from A and B for a given observer—i.e., thedeparture from simultaneity. So, TD = tA − tB , and TD = 0 means the signals were receivedsimultaneously. Similarly for T ′
D in K ′. The quantified results are presented in table 1.
9.1. Experimental results, invariants, and causality
The single-frame (traditional) analysis of the text produces only the first row of the table. Butit is the disparity of results along the columns that is problematic—confounding causality.Consider, for example, the K ′-frame results in the second column. T ′
D = 0 represents a space–time coincidence in that frame, and this should be an invariant—i.e., agreed to by all observers.We can test for this2 by having a lamp on the train at M ′, which is switched on only whenit receives the signals from AA′ and B B ′ simultaneously. But according to the results, O inthe K -frame concludes the lamp remains off; O ′ in the K ′-frame concludes the lamp goes on.The invariance is not there.
2 My thanks to Professor Edwin F Taylor for suggestions leading to the development of this argument.
388 A Nelson
Looked at the other way, the order of reception of the flashes should also be an invariant.Here the test can be done by having the reception of the first flash turn the lamp on, while thereception of the second flash turns the lamp off. In this case, however, O determines that thelamp goes on (for a period of Lv
c2−v2 ); O ′ says the lamp stays off (more precisely, goes off theinstant it goes on). Again the results are not invariant. A similar analysis could be done in theK -frame.
These opposing observers’ conclusions cannot coexist. For there is a physical reality tolamplight. As Paul Revere can testify, either a lamp shines or it does not, and both observersmust either see its light or darkness. In other words, since invariants are observable in all frames,they represent physical realities. Results that repudiate invariants are flat out contradictions ofreality—and unacceptable.
10. Contradictions caused by pre-relativistic conditions
The cause of these conflicts is the impossibility of the pre-relativistic domain, the Newtonian–Einsteinian hybrid, wherein we have been compelled to analyse the experiment. In this pre-relativistic setting, the scales of space and time have been held invariant. But that has causedthe negation of both the invariance of space–time coincidences and the invariance of thereception order of signals—producing impossible (anti-physical) results. Consequently, thepre-relativistic conditions are not viable, and the thought experiment in this form is not valid.
Put more conceptually, the traditional (pre-relativistic) train/embankment presentationwould have us apply the Einsteinian definition of simultaneity and the constancy of the velocityof light without altering space and time. But that cannot be. The second postulate includesthe changes forced on space and time because of motion. Even the idea of consequence in thatregard is too weak. One is not a consequence of the other, so much as there is an inextricableconnection between the two. The constancy of the speed of light and the transformationproperties of space and time form an integrated, unified description of the world. We can nomore take one without the other than, in the relativistic domain, we can take length contractionwithout time dilation.
The notion of light moving with constant speed in all inertial frames has no rationalinterpretation without the associated (relativistic) spatial and temporal effects. It is precisely thedivorcing of those two concepts that undoes the conventional analysis of the train/embankmentproblem.
The relativity of simultaneity is not in dispute. It is just not demonstrable from a pre-relativistic analysis of the train/embankment experiment. What does emerge from such ananalysis is that Galilean relativity and Newtonian space–time are incompatible with the secondpostulate and the Einsteinian definition of simultaneity.
In short, the contradictory observations of table 1 and section 9 obtain because the effectsof special relativity have been ignored. A relativistic analysis is required. This is ironic,since the thought experiment is being used to introduce relativity, yet one needs the results ofrelativity to analyse it.
11. Contradictions resolved by a relativistic analysis
In going from a pre-relativistic to a relativistic setting for the train/embankment experiment,we, at first, encounter another paradox. The transition does not produce a unique set of initialconditions. To see this, we recall the pre-relativistic experimental environment, in which thereare no transformation effects, so the length of the train is the same in both frames. From theembankment perspective, let t = 0 when the mid-points M and M ′ coincide; then the trainstretches from A to B . At that instant, the lightning bolts strike.
Now there are two stipulatory scenarios: (1) the strokes hit A and B on the embankment,or (2) the strokes hit the ends of the train. Pre-relativistically, these are the same, as shown
Reinterpreting the famous train/embankment experiment of relativity 389
vTrain
(a) M'
A M B Embankment
vTrain
(b) M'
A M B Embankment
vTrain
(c) M'
A D M E B Embankment
Figure 7. Embankment perspective at t = 0: (a) pre-relativistic—no length contraction; (b)relativistic—bolts posited to hit A and B on the embankment; (c) relativistic—bolts posited to hitthe ends of the train.
in figure 7(a). Relativistically, however, the embankment observer sees a contracted train. Soin the first scenario, the bolts strike A and B on the embankment, but they hit in front of andbehind (not the ends of) the train (figure 7(b)). In the second scenario, the bolts hit the ends ofthe train, but they strike the embankment at points D and E , not A and B (figure 7(c)). Note:the frequently invoked char marks would make no difference. The disparity of the lightninghitting the train pre-relativistically and missing it relativistically is a contradiction wrought bypositing the (impossible) pre-relativistic conditions.
For purposes of a relativistic analysis, either figure 7(b) or (c) is a valid choice of initialconditions. Textbook examples make the assumption that, as viewed from the embankment,the lightning bolts simultaneously strike the two ends of the train, or the two ends of OT ’sobservation car on the train (e.g., [3, 4, 6, 8]). But the earlier narrative and comparison withthe non-relativistic case suggest taking the strokes rather to hit A and B on the embankment(cf figure A.1). This slightly more complicated calculation is presented in the appendix.
Actually, the relativity of simultaneity can be calculated starkly from one Lorentztransformation equation (equation (A.2), appendix), but a relativistic analysis, following themethodology that led to table 1, is valuable to show how perspectives and outcomes changewhen relativity is applied. The seminal difference relativistically is that for both observers Adoes not coincide with A′ when B coincides with B ′.
Indeed, the derivation shows that in the train frame the strokes hit neither at the same timenor at the same distances from the origin as in the embankment frame. The differential timeresults are presented in table 2. We now see the second probable reason that the experimentwas not quantified in its original form: the relativistic derivation gives a different result.
Under relativistic conditions, i.e., under the influence of the second postulate, thedimensions of space and time are no longer invariant. With this concession, space–timecoincidences and the order of reception of signals regain their rightful prominence as invariants.This is manifested by the consistent columnar results in table 2. The experiment stipulatesthat the lightning strokes are simultaneous in the embankment frame, and, with a relativisticanalysis, the two observers agree on that simultaneity. They also conclude (independently)that the strokes are not simultaneous in the train frame, and they agree on the magnitude ofthat non-simultaneity. The principle of relativity rests once more in the comfort of affirmation.
390 A Nelson
Table 2. Relativistic values of TD, T ′D by observer and frame. The two observers agree on the time
separation of the strokes in both frames.
Frame
Observer K , embankment K ′, train
O, OE 0 γvL
c2
O ′, OT 0 γvL
c2
The relativistic transformation effects of space and time, presumptively ignored in thepre-relativistic analysis, produce the relativity of simultaneity.
12. Conclusion
The well-known train/embankment experiment has been and continues to be widely used tointroduce the relativity of simultaneity. The objective is to make a qualitative inquiry employingonly Einstein’s definition of simultaneity, the two postulates of relativity, and experimentalobservations.
Pre-relativistic conditions apply; that is, no relativistic effects involving space or time areused, usually because they have not yet been derived. Since these effects follow directly fromthe postulates, however, applying the postulates while ignoring the changes that they force ontime and space is an invitation to difficulty.
The analysis is traditionally performed by stipulating simultaneity in one frame and then,from the perspective of that frame, examining the results in a moving frame. This gets the‘right’ answer. But the single-frame approach precludes any application of the constancy of thespeed of light in the moving frame. The postulate is stated but never used. This inappropriatelyignores the second postulate, which was supposed to be an important aspect of the experiment.
It also opens up the possibility that when the investigation is expanded to include theperspective from the ‘moving’ frame, compatible results will emerge for signals whichpropagate anisotropically in that frame. This was, indeed, found to be the case. Pre-relativistically, observers in the two frames agree about simultaneity and non-simultaneityonly when such signals are employed, i.e., signals other than light.
When light is used, each observer concludes that the flashes were simultaneous in his ownframe and that they were not simultaneous in the other frame. This would seem to suggest thatsimultaneity holds across inertial frames but does not appear that way to an observer outsidethe particular frame. Such a result does not attain (nearly contradicts) the objective of thethought experiment—which was to show that simultaneity is relative.
The conundrum, ultimately, is that the experiment is posited to take place in Newtonianspace–time but under Einsteinian postulates. The experiment is Newtonian in that time andspace are dimensionally absolute, but Einsteinian because of the second postulate and thedefinition of simultaneity—which require that time be conveyed at the speed of light (not atinfinite speed). Such a schizophrenic architecture spawns an experimental oxymoron. This iswhy we get a ‘correct’ answer from a partial analysis that ignores the second postulate, whilewe inherit a new paradox from a methodology that is complete and adheres to the stipulations.
All these contradictions and difficulties are resolved by analysing the problemrelativistically. This yields both the desired relativity of simultaneity and quantitativeagreement between observers in different frames.
Since we end up with God being in His heaven and all being right with the relativisticworld, one may ask ‘Why have you bothered us?’ The attempt here has been to make moreof a pedagogical than a theoretical point. This thought experiment introduces students tothe concept of relativity, hence, to the importance of perspective in evaluating the results
Reinterpreting the famous train/embankment experiment of relativity 391
of an experiment. What makes relativity novel and difficult for the new student is not themathematics but rather the counter-intuitive nature of the subject and the fact that the universepresents differently depending on the frame of reference. The early lesson, then, is to bevery aware as to wherefrom one views a sequence of events. For all its age and use, thetrain/embankment experiment appears to have been traditionally analysed with a method thatcould be described as incomplete—such that it nearly falls prey to the very error that it seeksto instruct against.
When investigation is done in both frames, a contradiction of a different sort emergeswhich compels a relativistic perspective with, arguably, even greater force. Completing theanalysis thus presents the physics better, and provides a keener instruction for the student inthe need for fastidiousness regarding frames of reference and postulates applied thereto. Thehope is to provide greater power for the educator, so that this venerable thought experimentbecomes an even more illuminated gateway through which students may pass to explore theremarkable and beautiful theory of relativity.
Acknowledgments
Special appreciation to Professors R Victor Jones (Harvard) and Edwin F Taylor (MIT) forguidance and encouragement. Thanks also to Dr Herbert Shulman and Alex Maloney (Harvard)for suggestions and challenges, and to Sheri and Nick Siraco for invaluable clerical assistancewithout challenges.
Appendix. Relativistic quantitative analysis of the train/embankmentexperiment
With a relativistic perspective, figure 1 must be redrawn. Viewed from the embankment frame,figure A.1 shows the relativistic juxtaposition of train and embankment when M coincides withM ′. We take this as the origin, so, with more conventional notation, xM = x ′
M = 0. The lengthof the train is contracted such that
x ′Ax ′
B = L ′ = L(1 − v2/c2)1/2.
By convention, L ′ = Lγ
, where γ = (1 − v2/c2)−1/2.At t = t ′ = 0, the strokes hit x A and xB on the embankment (cf figure 7(b)). For purposes
of OE ’s analysis of what OT should observe, there is initially no effectual difference betweenfigure 1 and figure A.1. So the calculation proceeds to give TD = Lv
c2−v2 as in figure 5. Butnow the relativistic effect of time dilation must be included. OE makes his calculation usinghis own clocks, i.e., clocks at rest on the embankment, and for each case he needs two clocks.So OE ’s derivation is done in improper time. Proper time is kept by the single clock OTcarries with him on the train. Since it is OT ’s experience that we seek, improper time must beconverted to proper time:
T ′D = TD
γ= Lv
γ (c2 − v2)= γ
Lv
c2(OE ’s conclusion).
To determine relativistically what OT observes for himself, we invoke the Lorentztransformation in interval form [19]:
�x ′ = γ (�x − v �t), (A.1)
�t ′ = γ (�t − v �x/c2). (A.2)
From figure A.1,
�x = x A − xB = −L,
�t = tA − tB = 0.
392 A Nelson
x'A
x'M
x'B
v Train (K')
xA
xM
xB
Embankment (K)
Figure A.1. Train and embankment at t = 0 (OE ’s view).
Substituting in equation (A.2), and noting that x ′M = 0 remains at the mid-point of the flashes,
�t ′ = t ′A − t ′
B = γvL
c2= T ′
D (OT ’s conclusion).
At last we have agreement between the observers, for this is the same conclusion that OEdrew. But from OT ’s point of view, the result has nothing to do with moving away from theA′-flash and toward the B ′-flash, as OE averred. OT detected no motion and simply observedthat the flashes were emitted at different times and so arrived at his observation point at x ′
Mat different times. For his part, OE can attest that the flashes were emitted simultaneously,and he maintains that OT ’s motion caused OT to receive the B-flash before the A-flash.These are two different (but equivalent) descriptions of the same events, and they come to thesame conclusion. The disparity in the descriptions is a reflection of the different perspectivesor frames of reference of the two observers and illustrates why the analysis must be donerelativistically and in both frames.
The investigation can be carried further to complete a relativistic revision of table 1. Wehave looked at events in both frames from the perspective of the embankment frame and atevents in the train frame from the vantage point of the train frame. What remains is thedescription of events in the embankment frame from the perspective of the train frame.
In order to obtain this, we must first determine where and when the lightning strokesoccur in the train frame. These can be ascertained from the coordinate form of the Lorentztransformation [19]:
x ′ = γ (x − vt), t ′ = γ (t − vx/c2).
From figure A.1,
x A = − L
2, xB = L
2,
tA = 0, tB = 0.
To differentiate from the train coordinates, we call the stroke coordinates x ′S A and x ′
SB ,respectively. Then substitution gives
x ′S A = −γ L
2, x ′
SB = γ L
2, (A.3)
and the times for the strokes are
t ′A = γ vL
2c2, t ′
B = −γ vL
2c2. (A.4)
Because γ > 1, in the train frame the strokes hit in front of and behind the train—a descriptionwith which OE would concur, since he sees a train of contracted length (cf figure A.1).
Now with events in the train frame as the starting point, we can calculate what happensin the embankment frame. According to OT in the train frame, he is motionless, and theembankment is moving by him with velocity −v. The embankment, then, becomes the primedframe, the train the unprimed frame. Figure A.2 shows the situation at xM = x ′
M = 0, whichoccurs at t = t ′ = 0.
Figure A.2 actually depicts the juxtaposition of embankment and train between the twolightning strokes—since the B-flash (which hits to the right of xB at xSB) occurs before t = 0,and the A-flash (which hits to the left of x A at xS A) occurs after t = 0.
Reinterpreting the famous train/embankment experiment of relativity 393
Train (K)
xA
xM
xB
Lxx BA =
– v x'A
x'M
x'B
Embankment (K')
γL
Lxx BA =′=′′
Figure A.2. Train and embankment at t = 0 (OT ’s view).
xA
xM
xB
xSB
Train (K)
– v x'A
0 x'MB
x'B
Embankment (K')
Figure A.3. Train and embankment at the instant of the B-stroke (OT ’s view).
The course of events in the embankment frame can be determined by applying the intervalform of the Lorentz transformation, as before, except that now −v replaces v.
Thus,
�x ′ = γ (�x + v �t), (A.5)
�t ′ = γ (�t + v �x/c2). (A.6)
Equations (A.3) and (A.4) now represent unprimed coordinates and give
�x = xS A − xSB = −γ L, (A.7)
�t = tA − tB = γvL
c2. (A.8)
Note: since the strokes hit at the same time in the embankment frame, this is tantamount toa measurement in that frame of the distance between the strokes in the train frame. Therefore,one should expect to see the relativistic length contraction, and this is, indeed, the case, since
�x ′ = −L = −γ L
γ= �x
γ.
To calculate what OT says OE should see, we consider, first, the situation at the time ofthe B-stroke, tB = − γ vL
2c2 . OE at x ′M will not yet have reached the origin. Since OE moves
with velocity −v, he will be located to the right of the origin by (v)(γ vL
2c2
). Call this location
x ′M B . So,
x ′M B = γ v2 L
2c2.
At the same instant x ′B will be located at
x ′B = γ v2 L
2c2+
L
2γ= γ L
2= xSB (cf equation (A.3)).
So, OT will say that x ′B and xSB are space–time coincident when the B-stroke hits. The
configuration is shown in figure A.3.It may be noted in passing that figure A.3 is drawn for v < c/
√2, since the configuration
satisfies the inequality γ v2 L2c2 < L
2γ. For v = c/
√2, x ′
A is at the origin. For v > c/√
2, thecontraction is greater still, and x ′
A is to the right of the origin.
394 A Nelson
– cTB
xSB
x'MB
– vTB
2
2
22 c
LvL γγ −
Figure A.4. To calculate TB .
xSA
xA
xM
xB
Train (K)
– v x'A
x'MA
0 x'B
Embankment (K')
Figure A.5. Train and embankment at the instant of the A-stroke (OT ’s view).
The time (TB) for the signal from the B-stroke to reach OE can be calculated fromfigure A.4:
γ L
2− γ v2 L
2c2− cTB = −vTB ,
whence
TB = L
2γ (c − v).
The A-stroke occurs at time tA = γ vL/2c2, which is γ vL/c2 after the B-stroke. Duringthat time interval, x ′
M and x ′A move a distance (−v)(γ vL/c2), putting OE at x ′
M A, where
x ′M A = γ v2 L
2c2− γ v2 L
c2= −γ v2 L
2c2,
and
x ′A = −
(L
2γ− γ v2 L
2c2
)− γ v2 L
c2= −γ L
2= xS A (cf equation (A.3)).
So, OT will say that x ′A and xS A are space–time coincident when the A-stroke hits. The
configuration is shown in figure A.5.Then, we can determine the transit time (TAM) for the signal from the A-stroke to reach
OE from figure A.6.So,
−γ L
2+
γ v2 L
2c2+ cTAM = −vTAM ,
whence,
TAM = L
2γ (c + v).
The total time elapsed from the B-stroke until the signal from the A-stroke reaches OE at x ′M
is then
TA = γ Lv
c2+ TAM = γ Lv
c2+
L
2γ (c + v).
Reinterpreting the famous train/embankment experiment of relativity 395
Figure A.6. To calculate TAM .
The departure from simultaneity is TD = TA − TB . But the prior calculations are madeby OT using his own clocks, i.e. clocks at rest in the train frame. These keep improper timeand must be converted to proper time to determine what OE sees.
T ′D = TD
γ= TA − TB
γ= 1
γ
(γ Lv
c2+
L
2γ (c + v)− L
2γ (c − v)
)= 0.
(Note that this time conversion is formally correct but technically unnecessary, because theparenthetical expression is zero.)
So OT now confirms that OE will view the strokes as simultaneous. But OT will say thatit is because the strokes at x A and xB were staggered in time, and OE had exactly the rightamount of speed to cancel out the time differential and arrive at the point x ′
M just as the twobeams intersected there. For his part, OE will wonder why OT is making this so complicated.OE simply saw two simultaneous lightning strokes.
Incorporating relativity, then, table 1 can be revised to present the data withoutcontradictions (table 2).
References
[1] Scherr R, Schaffer P and Vokos S 2002 The challenge of changing deeply held student beliefs about the relativityof simultaneity Am. J. Phys. 70 1238–48
[2] Scherr R, Schaffer P and Vokos S 2001 Student understanding of time in special relativity: simultaneity andreference frames Am. J. Phys. Suppl. 1 69 S24–35In this study volcanic eruptions replace lightning strokes, but the experiment is the same.
[3] Serway R, Beichner R and Jewett J Jr 2000 Physics for Scientists and Engineers (Pacific Grove, CA: Brooks/Cole)pp 1255–6
[4] Serway R and Faughn J 1999 College Physics 5th edn (Pacific Grove, CA: Brooks/Cole) pp 862–3[5] Gianconi D 1998 Physics 5th edn (Upper Saddle River, NJ: Prentice-Hall) pp 801–3[6] Young H 1992 University Physics 8th edn (Boston, MA: Addison-Wesley) pp 1075–6[7] Ohanian H 1989 Physics vol 2 (New York: Norton) pp 992–4 (expanded)[8] Sears F, Zemansky M and Young H 1987 University Physics 7th edn (Boston, MA: Addison-Wesley) p 958[9] Bergmann P 1976 Introduction to the Theory of Relativity (New York: Dover) pp 30–2 (with a foreword by
Albert Einstein)[10] Gardner M 1976 The Relativity Explosion (New York: Vintage Books) pp 43–5[11] Einstein A 1961 Relativity (New York: Three Rivers Press) p vi
(authorized translation by Robert W Lawson)This is the earliest reference to the train/embankment experiment I could find—after looking, inter alia, inStachel J et al (ed) 1987 The Collected Papers of Albert Einstein (Princeton, NJ: Princeton University Press)and after private conversations with J Stachel of Boston University. The German edition of [12] was published
in 1917. The first English edition came out in 1920. Einstein’s preface is dated ‘December 1916’. If anyoneis aware of an earlier disclosure of the experiment, I would appreciate learning of it.
[12] Einstein A 1961 Relativity (New York: Three Rivers Press)(authorized translation by Robert W Lawson)
[13] Einstein A 1961 Relativity (New York: Three Rivers Press) p v
(authorized translation by Robert W Lawson)
396 A Nelson
[14] Einstein A 1998 On the electrodynamics of moving bodies Einstein’s Miraculous Year ed J Stachel (Princeton,NJ: Princeton University Press) pp 127–30
Originally published inEinstein A 1905 Ann. Phys., Lpz. 17 891–921
[15] Nelson A 2002 Reinterpreting a familiar, textbook thought experiment of relativity (unpublished)This paper discusses an inversion of the train/embankment experiment, also, though less widely, used to introduce
the relativity of simultaneity.[16] Einstein A 1961 Relativity (New York: Three Rivers Press) pp 29–30
(authorized translation by Robert W Lawson)[17] Scherr R, Schaffer P and Vokos S 2002 The challenge of changing deeply held student beliefs about the relativity
of simultaneity Am. J. Phys. 70 1240–44This proffers a different analysis which this paper questions.
[18] Resnick R 1968 Introduction to Special Relativity (New York: Wiley) pp 53–5Resnick R and Halliday D 1992 Basic Concepts in Relativity and Early Quantum Theory 2nd edn (New York:
Macmillan) pp 41–3But the methodology is used for a different objective, incompatible with this paper.
[19] See, for example,Resnick R, Halliday D and Krane K 1992 Physics vol 1, 4th edn (New York: Wiley) p 475
