Relativity of Simultaneity: A question about the train paradox

[Vikrant94]

Hi, I was reading the book "Spacetime Physics" by Taylor/Wheeler. In their discourse on the relativity of simultaneity, the example they have used is Einstein's Train Paradox. See and also the attachment for the Train Paradox.

My question is this: The analysis of what the train observer(TO) observes, is fine... but only in the station observer's(SO's) frame of reference. Suppose for a moment that the lightning strikes DO occur simultaneously in both frames. Then the SO will say that the light from first strike reaches the TO before the light from the second strike, which is correct. But the TO will observe light from both ends reaching him/her at the same time, since speed of light is invariant in both frames. They now disagree not on the simultaneity of the lightning strikes, but the simultaneity of the light flashes reaching the TO. What I'm saying is that I don't understand why they have to agree on when the light flashes reach the TO. Why is our supposition at the start of this question incorrect?

PS: I wasn't sure whether to post this here or in the homework section. It seemed to me more a general question about what I'd read, than a specific problem, so I posted it here. Help is much appreciated!

 

[pervect]

The point is that if two events happen at the same location, all observers agree on whether or not they are simultaneous - because two events that are simultaneous and at the same position are the same event.

If two events occur at different locations, observers do not in general agree about what event are simultaneous.

If you draw the set of simultaneous events on a space-time diagram, it might help clarify the point.

Are you familiar with Einstein's method of clock synchronziation, and space-time diagrams both? If so, you should be able to draw a space-time diagram of what simultaneity means for moving and stationary obserers. You can also use the "radar" method to draw "lines of simultaneity" on space-time diagram, said method being that the midpoint of the emission and reception event on the radar worldline is simultaneous with the event where the radar beam is reflected.

 

[Vikrant94]

I'm still not clear on this. Here's why:

If two events happen at the same place and the same time, aren't they still two different events, occurring at the same point in spacetime? For example if a particle collides with another particle, that's one event. It so happens that at the same time the particle collided (and so the same place in this frame of reference) it also disintegrated. Surely we cannot say these are the same events, even though they occur at the same time and place?

I understand that observers do not agree on time and place of events, and hence on simultaneity. However my question is that assuming indeed they agree on simultaneity in this case, I cannot find a necessary contradiction.

I tried making a spacetime diagram, I've uploaded the image. This is in the frame of reference of the SO, events A and B are the lightning strikes, C and D are receiving of left and right strikes respectively by the SO and C' and D' the same for the TO. It didn't help me get any further insight.

 

[Mentz114]

In these diagrams the observer in the middle of the train thinks the light beams were emitted simultaneously ( grey worldline) from the back and front of the train but in the maroon frame the beams are clearly not emitted simultaneously.

The diagrams are accurate ( barring user clumsiness) and are related by a Lorentz transformation. Time is on the vertical axis per convention.

 

[JesseM]

If two events happen at the same place and the same time, aren't they still two different events, occurring at the same point in spacetime?

Yes, in this case all frames agree that the events are simultaneous, it's only when the events occur at different positions in space (and there is a space-like separation between them) that different frames disagree about whether they are simultaneous. In the case of the strikes, all frames must agree whether the light from each strike reaches the observer at the center of the frame at the same moment or not, since if it did this would be a pair of events occurring at the same point in both space and time. But if the strikes were simultaneous in the track observer's frame when he was next to the observer at the center, then if the postulate about both beams moving at c applies in the track observer's frame, this implies the track observer predicts the two beams do not reach the observer at the center of the train at the same moment. So, this must be true in the frame of the observer at the center of the train as well, precisely because different frames can't disagree about whether events coincide at the same place and time. But if the observer at the center of the train receives the light from each strike at different moments, and yet is equidistant from the two ends of the train where the strikes occurred, the only way to reconcile this with the idea that both beams move at c in his frame is to say the strikes occurred at different moments in his frame.

 

[pervect]

I'm still not clear on this. Here's why:

If two events happen at the same place and the same time, aren't they still two different events, occurring at the same point in spacetime? For example if a particle collides with another particle, that's one event. It so happens that at the same time the particle collided (and so the same place in this frame of reference) it also disintegrated. Surely we cannot say these are the same events, even though they occur at the same time and place?

Let's call these two events C (collision), and D (disintegration). An observer in one frame, call it the stationary frame, finds the space-time coordinates of event C. Then he finds the space-time coordinates of event D. But he finds that the space-time coordinates of event C are essentally the same coordinates as the event D, because the events happen "at the same place" and "at the same time", or at least very very close together.

Now, in a different frame, call it the moving frame, a different observer finds the space-time coordinates of event C, and she gets different coordinates than her brother in the stationary observer did. She then goes on to find the coordinates of event D. Because events C and D are very close together , she'll also finds that event D has the same coordinates as event C, even though she assigns different values to said coordinates than her brother did.

So, the conclusion is - everyone agrees that C and D happen "close together" - they can't be "close together" for one observer and "far apart" for another.Now, onto my diagrams...

There are two diagrams, which show how clocks are synchronized using the Einstein midpoint method, one for a stationary observer and one for a moving observer.

We have three worldlines in the diagram - the left observer, the midpoint observer, and the stationary observer. All three wordlines are the same, they are worldlines in a stationary frame.

At some event O, the midpoint observer emits a pulse of light. It's received at events a and b. In the stationary frame, a and b are then regarded as happening "at the same time"

In the second diagram, we having the case for a moving fame. Again, we have the worldlines of three observers, a left observer, a midpoint observer, and a right observer. Again, the midpoint observer emits a pulse of light at time O. It's received by the left and right observers at events a' and b', which are regarded by the moving observer as simultaneous.

Inspection of the picture, though, will show that while a-b is a horizontal line, the line a'b' is not horizontal, but slopes. This demonstrates that notions of simultaneity are different for moving and stationary observers.

If you put one diagram on top of the other, super-imposing them, you can do this so that event a and event a' are close ("at the same time and at the same place"). Because event b is on a horizontal line, and event b' is on a sloped, if events a and a' are made to occur at the same space-time coordinates, events b and b' can NOT be made to occur at the same space-time coordinates.

In other words, the set of events simultaneous to a, in the stationary frame, is a different set of events than the set of events simultaneous to a', in the moving frame.

I hope this helps.

 

[Vikrant94]

In these diagrams the observer in the middle of the train thinks the light beams were emitted simultaneously ( grey worldline) from the back and front of the train but in the maroon frame the beams are clearly not emitted simultaneously.

The diagrams are accurate ( barring user clumsiness) and are related by a Lorentz transformation. Time is on the vertical axis per convention.

I appreciate your help, but I do find spacetime diagrams to be a little artificial, in that mostly you can only draw them correctly once you have already figured out how things are supposed to be, so the only insight you gain is visual. A little bit like saying x²+y²=1 when plotted on a graph is a circle, so every point must be equidistant from the centre (origin), which is something we could have analysed without the diagram. Of course this could be entirely because of my own naivety :)

So, the conclusion is - everyone agrees that C and D happen "close together" - they can't be "close together" for one observer and "far apart" for another.

This helps, so the spacetime interval being invariant, if the light flashes reach any observer at the same time in one frame, the also reach him at the same place, hence the interval between the flashes reaching him is zero, so it must also be zero in any other frame. That is, the train observer must agree with the station observer on whether or not the light flashes reached her at the same time. That's exactly what I wanted to know, thanks a lot for your help! :)

 

[ghwellsjr]

I think the "paradox" that is presented in the Word document is stating that the two flashes occur at the same time without stating which frame that is true in but then it goes on to illustrate how the explosion takes place in one frame and not in the other. The problem is to resolve the "paradox". The correct explanation is to point out that the problem as stated cannot exist because it assumes that there is an absolute meaning to the sentence, "Now, we fire the two flashes at the same time."

In the continuing description of the two situations, two different things happen at the end (explosion vs no explosion) but that is because two different things happen at the beginning (two flashes occur at the same time but in two different frames). So it's not like we are analyzing the same situation from two different frames and getting two different results, it's that we are analyzing two different situations and, of course, getting two different results.

Your statement in your first post, "Suppose for a moment that the lightning strikes DO occur simultaneously in both frames" is an impossibility and that is what needs to be pointed out to resolve this "paradox".

 

[Mentz114]

I appreciate your help, but I do find spacetime diagrams to be a little artificial, in that mostly you can only draw them correctly once you have already figured out how things are supposed to be, so the only insight you gain is visual. A little bit like saying x²+y²=1 when plotted on a graph is a circle, so every point must be equidistant from the centre (origin), which is something we could have analysed without the diagram.

This is not true. All the special relativistic relations can be worked out from space-time diagrams without reference to anything else. SR is a geometric theory. The diagrams show immediately that simultaneity is relative.

Of course this could be entirely because of my own naivety :)

Yes.

 

[Vikrant94]

This is not true. All the special relativistic relations can be worked out from space-time diagrams without reference to anything else.

Really? Isn't the fundamental principal of SR that the laws of physics hold in any inertial frame, or that it is impossible to distinguish one inertial frame from another, physically? From here, we arrive at the invariance of the spacetime interval, and from there, the Lorentz transformations. This is independent of all our attempts to draw diagrams, the physics holds with or without the diagrams. It seems to me that a diagram is superficial, supplementing our understanding only because we understand things visually faster.

SR is a geometric theory. The diagrams show immediately that simultaneity is relative.

What do you mean by a geometric theory? And of course they will show us that simultaneity is relative... immediately, because visual understanding is faster, even the equations will tell us 'immediately'. And it tells us that only because the equations say it must!

I don't mean to question the validity of what you're saying, I know it must be valid, and not only because physicists smarter than me have gone through this simple stuff for ages. I just want to understand better myself :)

 

[Mentz114]

Really? Isn't the fundamental principal of SR that the laws of physics hold in any inertial frame, or that it is impossible to distinguish one inertial frame from another, physically? From here, we arrive at the invariance of the spacetime interval, and from there, the Lorentz transformations. This is independent of all our attempts to draw diagrams, the physics holds with or without the diagrams. It seems to me that a diagram is superficial, supplementing our understanding only because we understand things visually faster.

I see it the other way round. Starting with Minkowski spacetime and the Lorentz transformation allows us to calculate the values of time dilation, length contraction, doppler shift and all the commonly used equations of SR.

What do you mean by a geometric theory? And of course they will show us that simultaneity is relative... immediately, because visual understanding is faster, even the equations will tell us 'immediately'. And it tells us that only because the equations say it must!

The equations come from the geometry, not vice-versa.

I don't mean to question the validity of what you're saying, I know it must be valid, and not only because physicists smarter than me have gone through this simple stuff for ages. I just want to understand better myself :)

I don't think we are disagreeing about anything. But spacetime diagrams are not merely illustrations.

I've attached a pdf doc showing how straightforward geometry can be used to calculate SR equations.

 

[Vikrant94]

I see it the other way round. Starting with Minkowski spacetime and the Lorentz transformation allows us to calculate the values of time dilation, length contraction, doppler shift and all the commonly used equations of SR.
.
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I don't think we are disagreeing about anything. But spacetime diagrams are not merely illustrations.

I see how you used Minkowski geometry to formulate the SR equations. But my point is that it is much more convincing to start at the premise that Physics does not change, and deduce from there that the geometry of spacetime is Minkowski, than to start by assuming that it follows a particular geometry of our choosing, and arrive at the same laws. In that sense I disagree with you that SR is a geometric theory; although it is appealing to use geometry to explain the equations, it cannot be the premise.

Since you say the diagrams are not merely illustrations, would you tell me how they could be used to find new knowledge that could not have been possible with the equations?

 

[GrayGhost]

I see how you used Minkowski geometry to formulate the SR equations. But my point is that it is much more convincing to start at the premise that Physics does not change, and deduce from there that the geometry of spacetime is Minkowski, than to start by assuming that it follows a particular geometry of our choosing, and arrive at the same laws. In that sense I disagree with you that SR is a geometric theory; although it is appealing to use geometry to explain the equations, it cannot be the premise.

The Minkowski model was built from the OEMB math, and the recognition that the LTs are consitent with graphical Euler rotations ... so the graphical model obeys the OEMB math model. Therefore, the Minkowski model is built upon the 2 postulates as foundation, because the OEMB was. That includes the requirement that the same mechanics hold equally in any/all inertial frames.

Since you say the diagrams are not merely illustrations, would you tell me how they could be used to find new knowledge that could not have been possible with the equations?

Well, the Minkowski illustrations brought some things to light that Einstein had not himself imagined. Even he said so. It was not until the Minkowski model that space and time were considered a single fused entity (coined spacetime), each meaningless w/o the other, and each dependent upon one another. The additive notion came from the realization that an inertial observer's own passage thru time must be a traversal thru 4 dimensional space.
The Minkowski model also suggests that all entities travel at speed c, relative velocity being the result of non-parallel speed c vectors thru 4-space. Add, since the LTs were consitent with euler rotations, Minkowski's model showed that relative velocity (and thus relativistic effects) is the result of relative frame orientation differentials within a 4d spacetime continuum. I'm not certain if Einstein envisioned it prior, but this would have helped Einstein envision gravitation as a warped spacetime, which would aid in conception of his general theory.

GrayGhost

 

[ghwellsjr]

Well, the Minkowski illustrations brought some things to light that Einstein had not himself imagined. Even he said so.

Can you please provide a reference?

 

[GrayGhost]

Can you please provide a reference?

Sorry GhwellsJr. I generally do not save such hyperlinks into archive for future reference. I should probably start doing so :) I'd figure some of that is floating about in Wiki somewhere.

GrayGhost

 

[JesseM]

Can you please provide a reference?

Well, this page says that Einstein originally thought the "spacetime" concept was useless but later changed his mind:

Einstein did not quite finish the job, however. Contrary to popular belief, he did not draw the conclusion that space and time could be seen as components of a single four-dimensional spacetime fabric. That insight came from Hermann Minkowski (1864-1909), who announced it in a 1908 colloquium with the dramatic words: "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality".

...

Einstein initially dismissed Minkowski's four-dimensional interpretation of his theory as "superfluous learnedness" (Abraham Pais, Subtle is the Lord..., 1982). To his credit, however, he changed his mind quickly. The language of spacetime (known technically as tensor mathematics) proved to be essential in deriving his theory of general relativity.

Although I'm not sure how accurate that is, a post by "Old Fool" on http://www.thescienceforum.com/viewtopic.php?p=279694 suggests that the "superfluous learnedness" comment was more about using tensor mathematics to describe relativity, rather than the conceptual notion of spacetime:

This is a quote from the detailed biography of Einstein by Abraham Pais (Subtle is the Lord - The Science and Life of Albert Einstein):

Initially, Einstein was not impressed and regarded the transcriptions of his theory into tensor form as "uberflussige Gelehramkeit", (superfluous learnedness). However, in 1912 he adopted tensor methods and in 1916 acknowledged his indeptedness to Minkowski for having greatly facilitated the transition from special to general relativity.

Apparently his comment about "superfluous learnedness" was made to V. Bargmann who passed it on to Abraham Pais.

 

[Shark 774]

Just to clarify, in the youtube video about the train paradox: Both will agree that the light from the front reaches the woman before the light from the back does and both will agree that the two light beams will reach the man on the platform at the same time, however they will disagree about the simultaneity of the actual strikes. In other words they agree on the simultaneity/non-simultaneity of the strikes reaching either person but disagree on the simultaneity/non-simultaneity of the actual strikes. Is that correct?

 

[JesseM]

Just to clarify, in the youtube video about the train paradox: Both will agree that the light from the front reaches the woman before the light from the back does and both will agree that the two light beams will reach the man on the platform at the same time, however they will disagree about the simultaneity of the actual strikes. In other words they agree on the simultaneity/non-simultaneity of the strikes reaching either person but disagree on the simultaneity/non-simultaneity of the actual strikes. Is that correct?

Right, different frames always agree on local events like what time a particular observer's clock reads when some light ray reaches him, but here they disagree about whether the strikes that emitted the light were simultaneous or not.

 

[Grimble]

Let me first of all appologise to you JesseM, but are you not guilty of creating a preferred frame of reference here?

In the case of the strikes, all frames must agree whether the light from each strike reaches the observer at the center of the frame at the same moment or not,

No. All frames must agree whether the light from each strike reaches the event at the spatial midpoint between the two lightning strike events at the same moment.

since if it did this would be a pair of events occurring at the same point in both space and time.

Yes, the same point in both space and time, not the same point in one preferred frame of reference.
That point in space and time only matches that observer's postion in that frame. In any other frame it would be a different postion.

But if the strikes were simultaneous in the track observer's frame when he was next to the observer at the center, then if the postulate about both beams moving at c applies in the track observer's frame, this implies the track observer predicts the two beams do not reach the observer at the center of the train at the same moment.

As measured by the track observer.

So, this must be true in the frame of the observer at the center of the train as well, precisely because different frames can't disagree about whether events coincide at the same place and time.

No, not the same place and time but at the same space-time event. I am sorry but you are using a PREFERRED frame again.

That same space-time event will occur in different places in different frames.

Consider; any reference frame may be considered stationary in space-time; and therefore may beused to map space-time. Events will merely have different coordinates depending on the frame used.

For any reference frame then, the event where the lights meet will be stationary and the lights will therefore meet at the midpoint between the two strikes in that frame.

If we consider only space-time, and not any particular reference frame, we see that the light from the two strikes meeting is a single event that is the same for any frame. Also the two strikes having equal spatial separation from that single event and, as the speed of light is constant, they must have happened simultaneously in space-time for their light to meet at that single event.

The paradox only exists if one gives that frame preference and maintains that everyone agrees that the central station observer is the one who sees the strikes simultaneously in all frames.

He doesn't.

It boils down to where is that single light-meeting event in each frame; and it has to be at the mid-point between the strikes for each and every frame.

So the station observer sees them as simultaneous only for him; the train observer sees them as simultaeous only for him and the man on the bicycle cycling past at that moment sees them as simultaneous only for him.

The important thing to realize is that the station observer is where that single event occurs in his frame, the train observer is where that event occurs in her frame and the cyclist is where the event occurs in his frame; and that location is different in each frame because each frame is mapping space-time with a different set of coordinates