A common example used to demonstrate the Relativity of Simultaneity goes like this, more or less:
A train of length D is passing by a platform of length D. An observer P stands in the middle of the platform, and there's another observer T who sits in the middle of the train. P notices that the front end of the train coincides with the front edge of the platform AT THE SAME TIME as when the rear end of the train coincides with the rear edge of the platform. The two 'coincidence events' are thus simultaneous in P's frame of reference.The correct answer in Special Relativity is "No".
The question is, does T agree that both 'coincidence events' are simultaneous in T's frame of reference? (It is implicitly assumed that this question is posed in P's frame of reference and that both reference frames are inertial.)
Call the front coincidence event F and the rear coincidence event R. Then, according to T, F happens before R. The events are NOT simultaneous in T's frame of reference even though they are simultaneous in P's frame of reference.
The Bad Explanation
Often times, the bad explanation for this is an incomplete version of a correct explanation provided by Einstein himself [Ein1924]. The bad explanation goes as follows:
Because P is in the centre of the platform, the light from both 'coincidence events' reaches her at the same time. Hence, P considers both events as simultaneous. But, as T is moving towards F, light from F reaches T sooner than does light from R. Hence, T concludes that the event at F happened before the one at R.
This is Wrong on Many Levels
Whereas the explanation provided by Einstein himself [Ein1924] appears correct in all respects (and I'll talk about it below), by skipping some tiny, but important, details the bad explanation is totally out-of-whack.
Apparent Time vs. Real Time
The fact that light from both ends reaches the observer P at the same time is immaterial to the notion of simultaneity.
Would P's definition of simultaneous change if we had placed her at one edge, say the front edge, of the platform? Going by the explanation given above, P would then say that F happened before R because light from R takes some time (D/c) to reach her at F. However, the correct thing for P to do (whether in Relativity or in Newtonian physics) is to compute the real time at which R occurred based on her knowledge of
- The distance between R and her; and
- The time on her clock when the light from R reached her.
If P does this, she'll recognize that F and R did happen at the same time in her frame of reference regardless of her exact position.
In fact, all discussions use the real time of the event as measured in a frame of reference and NOT the apparent time of the event (i.e. when light from the event reaches the observer). This point, which is frequently glossed over, is true whether we're talking about Special Relativity or Newtonian physics.
Où vous êtes, Relativité Spéciale?
How many results from Special Relativity are used in the bad explanation above?
Absolutely NONE!
The bad explanation requires no knowledge of Special relativity at all. A Newtonian Physicist would as easily 'accept' the explanation as any layman would. A Newtonian physicist would additionally point out the need to compute the real time of both events in T's frame. She would then demonstrate that both the events were simultaneous in T's frame of reference too! (Prove this!) "Galilean Relativity is thus NOT violated; simultaneity is indeed absolute in all inertial frames. Phew!"
(Indeed, how can one hope to explain a Special Relativistic phenomenon using concepts from Newtonian mechanics? If that were possible, the Relativity of Simultaneity would have been appreciated decades before Einstein arrived on the scene.)
The Correct Explanation using Special Relativity
There are many ways to correctly explain the Relativity of Simultaneity in this particular example. Einstein himself provided one which employs the central postulate of Special Relativity (i.e. the speed of light in vacuum is the same in all inertial frames) [Ein1924]. In what follows, I employ the result of Length Contraction from Special Relativity.
We have to consider the events from T's perspective to understand what happens in her frame.
Let L denote the Lorentz factor between the two frames. Thus, if the train was moving with a velocity v as measured in P's frame, then
Since the (rest) length of the stationary platform is D in P's frame, the length of the platform in T's frame is D/L.
In other words, the train is longer than the platform by a factor of L^2 (> 1) as measured in T's frame. (The actual factor by which it is longer is completely immaterial in what follows. I've run through this just to reinforce the fact that in T's frame of reference the platform is shorter than the train.)
From T's perspective, when the front end of the train coincides with the front edge of the platform (event F), the rear end of the train is NOT coincident with the rear edge of the platform. See position 1 in the figure below.
Some time later, T coincides with P who is standing in the middle of the platform. At this point neither ends of the train are coincident what the edges of the platform. However, this is exactly when P perceives both events F and R. See position 2.
Some more time later, the rear end of the train coincides with the rear edge of the platform (event R). At this point, the front end of the train is obviously NOT coincident with the front edge of the platform. See position 3.
Thus, T's perspective on these events is completely different from P's perspective. The events are not simultaneous, and in fact the ordering of the events in T's frame of reference is as presented above.
This is basically Relativity of Simultaneity.
Einstein's Original Explanation
The basic statement of Einstein's original thought-experiment is somewhat different, but only in the details [Ein1924]. The resolution proceeds as follows:
Because P is in the middle of the platform, light from both 'coincidence events' reaches her at the same time. P knows she's equidistant from both edges of the platform so light from both edges should take the same time to reach her. Thus, she deduces that both F and R were simultaneous.
Now, as seen by P, T is moving towards F. So, in P's frame of reference, light from F reaches T before the light from R does. (Now Einstein implicitly assumes that this order of events must be true even in T's frame of reference. Why?) But, T knows she's in the centre of the train. So light from both ends of the train should take the same time to reach her as light travels with a constant speed in her frame too. (Einstein fails to clarify this specific point in his explication). But, as the light from F reached her first, she concludes that F happened before R.
The numbers can be crunched to see whether the observations in P's frame actually transform correctly to those in T's frame using Einstein's interpretation. As one would expect, they do. Details are in the accompanying figure.
In Closing
I believe that the curious way in which the problem is formulated leaves significant scope for confusion. Einstein, ostensibly, chose the basic structure of the problem in this manner (i.e. with both observers in the 'middle') to simplify the overall explanation and provide an intuitive feeling for Special Relativity. However, this leads to the situation where the order in which light from the two events reaches the observers is identical to the order in which their respective reference frames place the events. It is very easy for lay students of Special Relativity to lose sight of the fact that the order in which light reaches an observer is inconsequential in defining simultaneity (or general ordering between events). So, one might leave with the impression that light reaching one's eyes is the only thing that matters.
The example used in Wikipedia to demonstrate the Relativity of Simultaneity is definitely better in my opinion.
References
[Ein1924] Albert Einstein (Tr. Robert W. Lawson), "Relativity - The Special and General Theory", Methuen & Co., Ltd., 1924.
In fact, all discussions use the real time of the event as measured in a frame of reference and NOT the apparent time of the event (i.e. when light from the event reaches the observer). This point, which is frequently glossed over, is true whether we're talking about Special Relativity or Newtonian physics.
Où vous êtes, Relativité Spéciale?
How many results from Special Relativity are used in the bad explanation above?
Absolutely NONE!
The bad explanation requires no knowledge of Special relativity at all. A Newtonian Physicist would as easily 'accept' the explanation as any layman would. A Newtonian physicist would additionally point out the need to compute the real time of both events in T's frame. She would then demonstrate that both the events were simultaneous in T's frame of reference too! (Prove this!) "Galilean Relativity is thus NOT violated; simultaneity is indeed absolute in all inertial frames. Phew!"
(Indeed, how can one hope to explain a Special Relativistic phenomenon using concepts from Newtonian mechanics? If that were possible, the Relativity of Simultaneity would have been appreciated decades before Einstein arrived on the scene.)
The Correct Explanation using Special Relativity
There are many ways to correctly explain the Relativity of Simultaneity in this particular example. Einstein himself provided one which employs the central postulate of Special Relativity (i.e. the speed of light in vacuum is the same in all inertial frames) [Ein1924]. In what follows, I employ the result of Length Contraction from Special Relativity.
We have to consider the events from T's perspective to understand what happens in her frame.
Let L denote the Lorentz factor between the two frames. Thus, if the train was moving with a velocity v as measured in P's frame, then
L = (1 - v^2/c^2)^(-1/2) > 1Since the length of the moving train is D in P's frame, the (rest) length of the train in T's frame is LD.
Since the (rest) length of the stationary platform is D in P's frame, the length of the platform in T's frame is D/L.
In other words, the train is longer than the platform by a factor of L^2 (> 1) as measured in T's frame. (The actual factor by which it is longer is completely immaterial in what follows. I've run through this just to reinforce the fact that in T's frame of reference the platform is shorter than the train.)
From T's perspective, when the front end of the train coincides with the front edge of the platform (event F), the rear end of the train is NOT coincident with the rear edge of the platform. See position 1 in the figure below.
Some time later, T coincides with P who is standing in the middle of the platform. At this point neither ends of the train are coincident what the edges of the platform. However, this is exactly when P perceives both events F and R. See position 2.
Some more time later, the rear end of the train coincides with the rear edge of the platform (event R). At this point, the front end of the train is obviously NOT coincident with the front edge of the platform. See position 3.
Thus, T's perspective on these events is completely different from P's perspective. The events are not simultaneous, and in fact the ordering of the events in T's frame of reference is as presented above.
This is basically Relativity of Simultaneity.
Einstein's Original Explanation
The basic statement of Einstein's original thought-experiment is somewhat different, but only in the details [Ein1924]. The resolution proceeds as follows:
Because P is in the middle of the platform, light from both 'coincidence events' reaches her at the same time. P knows she's equidistant from both edges of the platform so light from both edges should take the same time to reach her. Thus, she deduces that both F and R were simultaneous.
Now, as seen by P, T is moving towards F. So, in P's frame of reference, light from F reaches T before the light from R does. (Now Einstein implicitly assumes that this order of events must be true even in T's frame of reference. Why?) But, T knows she's in the centre of the train. So light from both ends of the train should take the same time to reach her as light travels with a constant speed in her frame too. (Einstein fails to clarify this specific point in his explication). But, as the light from F reached her first, she concludes that F happened before R.
The numbers can be crunched to see whether the observations in P's frame actually transform correctly to those in T's frame using Einstein's interpretation. As one would expect, they do. Details are in the accompanying figure.
In Closing
I believe that the curious way in which the problem is formulated leaves significant scope for confusion. Einstein, ostensibly, chose the basic structure of the problem in this manner (i.e. with both observers in the 'middle') to simplify the overall explanation and provide an intuitive feeling for Special Relativity. However, this leads to the situation where the order in which light from the two events reaches the observers is identical to the order in which their respective reference frames place the events. It is very easy for lay students of Special Relativity to lose sight of the fact that the order in which light reaches an observer is inconsequential in defining simultaneity (or general ordering between events). So, one might leave with the impression that light reaching one's eyes is the only thing that matters.
The example used in Wikipedia to demonstrate the Relativity of Simultaneity is definitely better in my opinion.
References
[Ein1924] Albert Einstein (Tr. Robert W. Lawson), "Relativity - The Special and General Theory", Methuen & Co., Ltd., 1924.
