Motion Paradox from Einstein s Relativity of Simultaneity Espen Gaarder Haug Norwegian University of Life Sienes November 5, 7 bstrat We are desribing a new and potentially important paradox related to Einstein s theories of speial relativity and relativity of simultaneity. We fully agree on all of the mathematial derivations in Einstein s speial relativity theory and his result of relativity of simultaneity when using Einstein-Poinaré synhronized loks. The paradox introdued shows that Einstein s speial relativity theory leads to a motion paradox, where a train moving relative to the ground (and the ground moving relative to the train) must stand still and be moving at the same time. We will see that one referene frame will laim that the train is moving and that the other referene frame must laim that the train is standing still in the time window between two distant events. This goes against ommon sense and logi. However, looking bak at the history of relativity theory, even time dilation was going against ommon sense and a series of aademis attempted to refute it. Still, based on this new paradox we have to ask ourselves if the world really an be that bizarre, or if Einstein s speial relativity ould be inomplete in some way? We are not going to give an answer to the seond uestion in this paper, but we will simply present the new paradox. Key words: Speial relativity theory, relativity of simultaneity, paradox, motion. The Motion Paradox ornerstone predition in Einstein s theory of speial relativity is the relativity of simultaneity. This basially predits that two distant events that happen simultaneously in one referene frame annot happen simultaneously as observed from another referene frame. In other words, under speial relativity theory there is no absolute simultaneity. We will see here that relativity of simultaneity leads to a strange paradox that we have not seen mentioned before. ssume that two events L distane apart happen simultaneously in referene frame one. The distane between the two events is L, as measured from this frame. Stating that the events happen simultaneously basially means there an be no time di erene between the two events as measured in that frame, as they indeed happened at preisely the same time. In other words, the time window is zero. However, under the onept of the relativity of simultaneity, Einsteins speial relativity maintains that the two events happening simultaneously in referene frame one will, as observed from referene frame two, have happened with the following time di erene apart: t () where v is the speed of frame one relative to frame two as measured with Einstein synhronized loks. 3 In speial relativity this speed is reiproal. Both frames will observe the other frame moving We have not seen this paradox disussed before. Still, after having studied speial relativity for many years and having aess to one of the worlds largest libraries on speial relativity literature, we do not laim to have read every publiation on the topi. By 9 already, there were supposedly more than 3,4 papers written about relativity, aording to Maurie LeCat in Bibliographie de la Relativité, Bruxelles 94. Today, the onept that time is a eted by motion has been onfirmed by a series of experiments, see for example Bailey and et al. (977) and Haefele and Keating (97b,a). Time dilation was introdued by Larmor (9) who ombined it with length ontration to get a mathematial theory onsistent with the Mihelson and Morley experiment. 3 Some prefer to all them Einstein-Poinaré synhronized loks, as Henry Poinaré suggested a very similar way of synhronizing the loks, yet with a slightly di erent interpretation. Poinare e assumed that the true one-way speed of light was afuntionofitsveloityagainsttheether,butthatthisouldnotbedetetedandsoforloksynhronizationpurposes,one ould assume the one-way time of light was half of the round trip time of light. Einstein on the other hand simply abandoned the ether and assumed that the one-way speed of light was the same as the round-trip speed of light. Einstein s theory is the simplest as long as it truly is impossible to measure veloity against a preferred frame, suh as the ether frame.
at speed v relative to their own frame as measured with Einstein synhronized loks. Euation an be derived diretly from the Lorentz transformation and is well known from a series of soures in the speial relativity theory literature, see for example Comstok (9), Carmihael (93), Bohem (965), Shadowitz (969), and Krane (), see also ppendix. Sine the two distant events happened simultaneously in frame one, then no time an have gone by between the events. This also means that from the perspetive of frame one, the two referene frames annot have moved relative to eah other in the instant that the two events happened. Still, from frame two the time between the two events (happening simultaneously in frame one) must, in Einstein s speial relativity theory, happen t apart. This means from the perspetive of frame two the two referene frames must have moved the following distanes relative to eah other during the time di erene between the two events tv v L () But again from frame one s perspetive the two distant events happened simultaneously, that is to say at the same point in time and therefore the two frames annot have moved relative to eah other during the two events. One ould try to argue that no event takes zero time. However, we ould make the event itself as short as we would like. It is the time di erene between the two events that is important here, this time di erene is zero in one frame and t in the other frame. The paradox gets even more interesting when one ould argue that sine t must have gone by between the two events as observed from frame two, then the following time must have gone by between the two events as measured from frame one (simply taking into aount time dilation): ˆt where t is the time gone by between the two events as observed from frame two and onverted into frame one by using time dilation. This leads to the observation that the two referene frames must have moved the following distane relative to eah other as measured from frame one: t ˆtv L p v ˆt (3) v This is atually the same formula as given by Comstok (9), and is onsistent with formula when taking into aount length ontration when observed from the other frame. Based on this, both frames will agree: the two frames moved relative to eah other in the time between the two distant events, but then again from frame one s perspetive both events happened simultaneously and no time an have gone by between the two events. We ould have set up the entire problem the other way around and the paradox and the alulations would be reiproal. Two events happening simultaneously in frame two as measured from frame one annot happen simultaneously as measured from frame one. However, even if the paradox is reiproal between the two frames this does not solve the paradox desribed above, this just means the paradox is also reiproal. The paradox is naturally that both frames annot be right, at least not if we follow ommon sense. The two frames either have moved relative to eah other or they are standing still relative to eah other. The motion between the two frames ould naturally be heked. For example, if frame one was a train and frame two was the embankment, then the train ould have aelerated to a given speed v relative to the embankment. This speed ould be heked both from the embankment frame and the train frame using loks that were Einstein synhronized in eah frame respetively. Both frames would then agree that the frames were moving relative to eah other with speed v. The paradox first omes when we look at two events that are happening simultaneously as observed in one frame and not simultaneously as observed from another frame. The Minkowski perspetive Could the paradox be solved by simply looking at it from the Minkowski spae-time perspetive? Minkowski (98) showed that the spae-time interval was invariant:
3 ds t dx dy dz (4) ssume the two distant simultaneous events happen L meters apart on board the train as observed from the train. From this standpoint, the Minkowski spae-time interval is given by ds L L (5) while from the standpoint of the train platform the Minkowski spae-time interval is given by ds @ ds ds L L ( ) L @ L v ds L (6) Both the train observer and the train platform observer agree on the spae-time interval that has gone by between the two events. We would have obtained the same result if we used the Lorentz transformation for the dt term alternatively in our derivation as shown in ppendix B, this naturally gives the same result. We are not uestioning that Minkowski is onsistent with speial relativity theory and Einstein synhronized loks. The fat that the train observer and the embankment observer both agree on the spae-time interval does not remove or solve the paradox: that from the viewpoint of the train observer, the train was standing still during the time between the two distant events and from the viewpoint of the embankment observer, the train was moving relative to the embankment. They do not agree on whether or not the train has moved relative to the train platform in between the two events. It is well known that two referene frames not will agree on the time interval and the spae interval, but with regard to the spae-time interval, this is expeted, as we naturally aept that there is time dilation and length ontration and that Minkowski is onsistent with SR. It seems, however, when pushed to the limit of two events that happen simultaneously in one of the referene frames that we must also aept that the train did not at all move relative to the embankment, while from the embankment it must be moving. This is not new from a pure mathematial perspetive; it is the deeper philosophial and logial part we are uestioning here. One solution to the paradox is simply to aept that relativity of simultaneity leads to the following onlusion: we must agree that a train is both moving relative to the embankment and not moving in between two distant events, that is if the two events happen simultaneously as observed from one of the referene frames with Einstein synhronized loks. lternatively, we ould laim that there must something deeper here that is not fully overed by Einstein s speial relativity theory. We look forward to a disussion around this paradox. 4. Conlusion Einstein s relativity of simultaneity says that two events that are observed to happen simultaneously in one frame not are happening simultaneously as observed from another frame. We fully agree that this is what is predited and what one will observe when using Einstein synhronized loks. Still, this leads to the paradox that when we have two frames moving relative to eah other, for example a train and the embankment, then if two distant events are happening simultaneously on the train, then no time an have gone by in between the two distant events as observed from the train and the train annot have moved relative to the embankment during a zero time interval. On the other hand, although there is no time di erene between the two events that are happening simultaneously on the train, they are observed to happen with a time di erene from the embankment, and the train must have moved relative to the embankment during this time period. Either one must aept that Einstein s speial relativity leads to results that are far from ommon sense (and we would say rather bizarre) or one should onsider that there is possibly a deeper reality not fully unovered by Einstein s speial relativity theory. 4 This is not the first time someone has raised a disussion on the relativity of simultaneity, see for example the reent disussion by Bolós, Liern, and Olivert (), Brogaard and Marlow (3) and Manson (4).
4 ppendix The Einstein relativity of simultaneity euation an be derived diretly from the Lorentz transformation. ssume two loks and B with a distane of L apart. light signal is sent out simultaneously from points and B as observed from the frame these loks are at rest in, let s all it frame one. This means that the time it takes for the light signal going from or from B to the midpoint in frame one as measured from frame one must be t, t,b L From frame two, whih is moving at a speed of v relative to frame one, the time between the two events is given by the Lorentz time transformation. ording to frame two, the two distant light signals are reahing the midpoint in frame one with a time di erene of t, t,b L + L (7) In other words, the two distant light signals that have happened simultaneously in frame one must have happened with a time di erene of aording to frame two. Under Einstein s speial relativity p v theory this argument is also reiproal. Two distant light signals that have happened simultaneously in frame two must have happened with a time di erene of aording to frame one. p v Under speial relativity what is happening simultaneously in one frame is not happening simultaneously in the other frame. There is no uestion about this result as long as we use Einstein-Poinarè synhronized loks, one of the fundamental assumptions in Einstein s speial relativity theory. However, as shown in this paper this leads to a rather bizarre paradox. ppendix B ssume the two distant simultaneous events happen L meters apart on board the train as observed from the train. Sine the events happen simultaneously we have t. Thetwoeventshappenx L distane apart in the train frame. The Minkowski spae-time interval as observed from the train frame is then given by ds L L (8) while from the train platform the Minkowski spae-time interval is given by the following euation, this time solved with the Lorentz transformation rather than the relativity of simultaneity euation: ds @ t ds @ ds ds L L ( ) @ L tv L @ L v ds L (9) whih is the same as the result we derived diretly from the relativity of simultaneity euation. This is as expeted sine the relativity of simultaneity time interval is found from the Lorentz transformation. gain this only shows that Minkowski is learly onsistent with Einstein s speial relativity theory, it does not solve the paradox stated in this paper.
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