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Relativity of Simultaneity 
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#1
Feb712, 06:20 PM

P: 359

Hey guys,
this might seem like yet another basic question, but I was wondering about RoS. The impression that I got from reading about relativity was that relativity of simultaneity was a consequence of Lorentz contractions, primarily time dilation. Someone else made the point [emphasis is theirs, not mine] To try and illustrate my own understanding of it: if everything in the universe was at rest relative to each other, then there would be absolute simultaneity, but I thought that if an observer started moving relative to that previous rest frame then they would encounter time dilation and relativity of simultaneity would occur. It thought that RoS was a result of the time dilation. Just wondering what I'm missing, and if there are any online resources that clearly explain the distinction between RoS and Lorentz contractions, and how they are different from each other? 


#2
Feb712, 06:34 PM

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Relativity of simultaneity is a particular feature of the Lorentz transform (in units where c=1):
[itex]t'=\gamma (tvx)[/itex] [itex]x'=\gamma(xvt)[/itex] Here is a transform which has length contraction and time dilation, but not the relativity of simultaneity: [itex]t'=\gamma (t)[/itex] [itex]x'=\gamma(xvt)[/itex] Here is a transform which has the relativity of simultaneity, but not length contraction or time dilation: [itex]t'=tvx[/itex] [itex]x'=xvt[/itex] 


#3
Feb712, 06:38 PM

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P: 4,684

You have to understand the concept of a Frame of Reference in order to understand Relativity of Simultaneity. In Einstein's Special Relativity, a scheme to create a coordinate system is defined in which you have three coordinates for specifying a location (x,y,z) and one coordinate for specifying time (t). Just like we have three coordinates for specifying a point in space, these four coordinates specify an event in the Frame of Reference. If you pick any two events and they have the same time coordinate, then they are simultaneous. If you then pick another Frame of Reference moving with respect to the first one, you can transform the coordinates for those two events using the Lorentz Transformation which will give you a new set of coordinates for the same two events. If the two time coordinates in the new Frame of Reference are equal to each other, then the events are simultaneous in that FoR. In general, two events that are simultaneous in one FoR will not be simultaneous in another FoR, but not necessarily.
So it has nothing to do with what is at rest or what is moving but simply the time coordinates of a pair of events in one Frame of Reference compared to another FoR. 


#4
Feb712, 06:43 PM

P: 359

Relativity of Simultaneity



#5
Feb712, 07:00 PM

P: 359

I think I have a decent enough understanding of what a reference frame is ["I think" being the operative words]. I suppose, when thinking about simultaneity I consider it in the sense of simultaneity in the universe, as opposed to simultaneity between a limited number of events; because absolute simultaneity would be a universal phenomenon, as well as applying to a limited number of events. As per Dalespams example, I understand that two or more events can "experience" contractions but still be "absolutely simultaneous"; presumably it would be theoretically possible that all events could "experience" contractions and still be "absolutely simultaneous"; that, however, would mean that Absolute simultaneity, not relativity of simultaneity was a "feature" of the universe. Is it possible for RoS to be a "feature" of the universe without time dilation? 


#6
Feb712, 08:20 PM

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P: 4,684

Events do not "experience" anything, let alone contraction. They are numbers, three for space, one for time. If those numbers for the time coordinate are identical according to the synchronization established for that FoR, then the events are simultaneous. The reason that I limited it to two is because if you have more than two, some of them can be simultaneous with each other but not with some others. 


#7
Feb712, 09:07 PM

P: 359

Could we build on this, saying that three events are simultaneous with each other but not with [undefined] others; in that case RoS prevails again, and not absolute simultaneity. I presume we could do this exponentially until we arrive at a scenario where all events are simultaneous with each other  in this case absolute simultaneity prevail, wouldn't it. In order for RoS to prevail, I presume there would only need to be one single event where the time coordinate is different from all the rest [who have the same time coordinate]. Is this possible without there being "time" dilation? I see Dalespam's example seems to suggest that there might, but I'm not sure how. 


#8
Feb712, 10:05 PM

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P: 4,684

This has nothing to do with time dilation. Any clock that is moving in a Frame of Reference will be running at a slower rate than the coordinate clocks defining the Frame of Reference. You could have two clocks traveling at different speeds and in different directions and talk about the simultaneous events of where they both were at a particular time which has nothing to do with the times displayed on their two clocks. But when you consider a different Frame of Reference, all the coordinates of all the events take on a new set of values and events that used to be simultaneous in the first frame are no longer simultaneous in the second frame. Let me emphasize once more: unless you consider two different Frames of Reference, you don't have any issue with relativity of simultaneity. 


#9
Feb712, 11:43 PM

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#10
Feb812, 12:48 AM

P: 359

Limiting it to two is fine, but if we limit it to two then we speak about a universe in which there are only two reference frames; if all events are simultaneous across those reference frames then absolute simultaneity prevails and not RoS; would that be correct? In saying that an event can be simultaneous in two reference frames but not with others, we are not limiting it to two, but to an undefined number of reference frames. Of course, if events are simultaneous across two refrence frames but not with other [undefined] reference frames, then RoS prevails.I presume we could build on this, saying that all events are simultaneous across three reference frames but not with [undefined] others; in that case RoS prevails again, and not absolute simultaneity. I presume we could then extrapolate this exponentially [at least theoretically] until we arrive at a scenario where all events are simultaneous across all reference frames; in which case absolute simultaneity would prevail, wouldn't it? Would this only be possible if everything were at absolute rest, or perhaps at rest relative to each other? In order for RoS to prevail, I presume there would only need to be one single event that isn't simultaneous across all reference frames; namely, where the time coordinate is different from all the rest [who have the same time coordinate]. Is this possible without "time" dilation? I see Dalespam's example seems to suggest that there might, but I don't really understand the maths representing the logic. If the two scenarios, mentioned above, are the only possibilities where absolute simultaneity could prevail, then presumably there would have to be relative motion in order for RoS to prevail; or am I way off on that? An issue might be with the assumption I'm working from, namely, that if all events are simultaneous across all reference frames, then that is absolute simultaneity; if even one event is not simultaneous, that is RoS. EDIT: I think it is meangingful to contrast absolute simultaneity with RoS because without one there would be the other; is that accurate? 


#11
Feb812, 08:50 AM

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P: 16,969

If you want to talk about something "having" reference frames, then it would be an analysis which has reference frames. Suppose we have three events with coordinates [itex](t_A,x_A)=(0,0)[/itex], [itex](t_B,x_B)=(0,1)[/itex], and [itex](t_C,x_C)=(1,0)[/itex]. A and B are simultaneous, since [itex]t_A=t_B[/itex], and the time between A and C is 1. Now, transforming to the primed coordinates using the above formulas (v=0.5) gives [itex](t'_A,x'_A)=(0,0)[/itex], [itex](t'_B,x'_B)=(.5,1)[/itex], and [itex](t'_C,x'_C)=(1,.5)[/itex]. So we see that [itex]t_A \ne t_B[/itex] meaning that simultaneity is relative, and the time between A and C is still 1 meaning that time does not dilate. Therefore, the relativity of simultaneity is possible without time dilation. 


#12
Feb812, 11:57 AM

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P: 4,684

The way I calculate the three transformed events, I get: A' = (0,0) B' = (0.577,1.1547) C' = (1.1547,0.577) So A and C do not have the same time coordinates so they are not simultaneous. EDIT: I see that wasn't your point. I should have said, the time between A and C is not the same as before, it's longer in the primed frame. But I wouldn't call that time dilation, it's just different coordinates for a pair of events. 


#13
Feb812, 09:50 PM

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#14
Feb912, 02:03 AM

P: 359

and [itex]x'=xvt[/itex] I read as [itex]x'[/itex] equals x minus the velocity multiplied by the time  which makes a bit more sense to me. My interpretation of it would be that, if the clocks which give the time coordinates all ran at the same rate, then absolute simultaneity should prevail; and in order for RoS to prevail clocks would have to give different times (coordinates). I suppose, essentially, where I have trouble is how we can go from the scenario where an event (or all events) are absolutely simultaneous across all reference frames, to a scenario where there is RoS. Presumably the initial scenario of absolute simultaneity would involve a transform (to affirm absolute simultaneity); I don't understand where a different transform could result in [the conclusion of] RoS if the initial transform leads to the conclusion of absolute simultaneity. Hopefully that makes some bit of sense. 


#15
Feb912, 02:38 AM

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#16
Feb912, 08:43 AM

P: 359




#17
Feb912, 08:49 AM

P: 3,183




#18
Feb912, 09:04 AM

P: 359

There's only two possible scenarios: either the clocks do tell the same time, or they don't. If they do then absolute simultaneity prevails; if they don't RoS prevails; what would cause them not to tell the same time? 


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