# Relativity of Simultaneity

1. Feb 7, 2012

### mangaroosh

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]
This was effectively how I understood it, but in a discussion on here I was told that wasn't the case. As with the other concepts of relativity I'm trying to get a better understanding of it.

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. Feb 7, 2012

### Staff: Mentor

Relativity of simultaneity is a particular feature of the Lorentz transform (in units where c=1):
$t'=\gamma (t-vx)$
$x'=\gamma(x-vt)$

Here is a transform which has length contraction and time dilation, but not the relativity of simultaneity:
$t'=\gamma (t)$
$x'=\gamma(x-vt)$

Here is a transform which has the relativity of simultaneity, but not length contraction or time dilation:
$t'=t-vx$
$x'=x-vt$

3. Feb 7, 2012

### ghwellsjr

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. Feb 7, 2012

### mangaroosh

thanks Dalespam; I think you mentioned that before. I don't fully understand it from that, but all information is helpful

5. Feb 7, 2012

### mangaroosh

Thanks gh.

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. Feb 7, 2012

### ghwellsjr

It's meaningless to consider RoS for the universe as if it is something intrinsic to the universe that we have to or could learn about or discover. This is an issue of remote time. We can't talk about it until we define what we mean and since there are an infinite number of ways to define remote time, it's not going to be something that we get from nature, rather it's something we put into nature.

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. Feb 7, 2012

### mangaroosh

Please forgive the use of imprecise terminology; I used the inverted commas to try and demonstrate that I know that isn't necessarily how we would talk about them, but in the absence of proper terminology I thought they would convey the meaning. We can abandon any mention of "feature" of the universe and "experience" and replace them with whatever words make sense when talking about contractions and simultaneity.

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 events; if they are simultaneous then does that mean that absolute simultaneity prevails and not RoS? In saying that they can be simultaneous with each other but not with others, we are not limiting it to two, but to an undefined number of events. Of course, if they are simultaneous with each other but not with other [undefined] events, then there RoS prevails.

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 co-ordinate is different from all the rest [who have the same time co-ordinate]. 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. Feb 7, 2012

### ghwellsjr

The universe contains an infinite number of events: every different location at every different instant of time is a different event. All the events that occur at the same time are simultaneous with each other. But remember, the times are all defined according to our FoR. There is no issue of RoS within a single FoR.
This sounds like a repeat of what you just said, so ditto what I just said.
There's no absolute simultaneity in SR. At every instant in time, all the locations throughout the entire universe are simultaneous events, because they all have the same time coordinate but different spacial coordinates. At the next instant in time, there is a new set of events throughout the entire universe that are another set of simultaneous events. We keep repeating this forever. But if you pick one event from the first set and another event from a subsequent set, they are not simultaneous.
As I keep saying RoS is not a factor until you transform the coordinates for a pair of events in one frame into the coordinates for the same pair of events into another frame in motion with respect to the first frame. You can continue to transform any number of events to see which pairs remain simultaneous.

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. Feb 7, 2012

### Staff: Mentor

The question about RoS isn't whether two arbitrary events are simultaneous or not, but whether two events which are simultaneous in one frame are also simultaneous in other frames.

10. Feb 8, 2012

### mangaroosh

Sorry, I phrased that all wrong; I meant to talk about reference frames, not events, but I lost myself on that one.

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 co-ordinate is different from all the rest [who have the same time co-ordinate]. 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?

Last edited: Feb 8, 2012
11. Feb 8, 2012

### Staff: Mentor

This does not make sense. The number of reference frames is not a property of a universe. Universes don't "have" reference frames, they are mathematical devices for analyzing physics, not physical features themselves. You could use an infinite number of reference frames to describe even the simplest possible universe.

If you want to talk about something "having" reference frames, then it would be an analysis which has reference frames.

Yes to all the above.

No. Even if everything were at rest to each other you could still analyze it in different reference frames, and if the transformation of the time coordinate included a spatial term then there would be relativity of simultaneity.

Strictly speaking, I don't think that is mathematically possible since coordinate systems are required to be smooth, but essentially yes.

Yes, I showed an example above.

OK, let's look at the equations $t'=t-vx$ and $x'=x-vt$ in a little more detail.

Suppose we have three events with coordinates $(t_A,x_A)=(0,0)$, $(t_B,x_B)=(0,1)$, and $(t_C,x_C)=(1,0)$. A and B are simultaneous, since $t_A=t_B$, and the time between A and C is 1.

Now, transforming to the primed coordinates using the above formulas (v=0.5) gives $(t'_A,x'_A)=(0,0)$, $(t'_B,x'_B)=(-.5,1)$, and $(t'_C,x'_C)=(1,-.5)$. So we see that $t_A \ne t_B$ 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.

It doesn't have to do with motion, but with the transformation between different reference frames.

Yes to the above, although again mathematically I don't think that it is possible for only one event to be non-simultaneous.

12. Feb 8, 2012

### ghwellsjr

What happened to gamma?

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.

Last edited: Feb 8, 2012
13. Feb 8, 2012

### Staff: Mentor

I was describing this transformation:
Which has no gamma. I was showing a transformation (not the Lorentz transform) which had relativity of simultaneity, but not length contraction nor time dilation. The transformation above is not a useful transform for physics, just an example showing that the relativity of simultaneity is not the same thing as length contraction and time dilation.

14. Feb 9, 2012

### mangaroosh

Apologies, I am aware of that, but don't often use terminology that makes that clear.

That would be another thing that I don't understand, namely how, or why, the time co-ordinate would include a spatial term.

Thanks for going through the above; the part I don't understand is the initial equations; I read $t'=t-vx$ as meaning $t'$ equals t minus the velocity along the X-axis, but I don't understand why the velocity comes into it.

and $x'=x-vt$ I read as $x'$ 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 co-ordinates 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 (co-ordinates).

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. Feb 9, 2012

### harrylin

Just a primer: If you start with a reference clock in your hand, how would (indeed, how could!) you determine what time it is at a distance? That has little or nothing to do with clock rate.

16. Feb 9, 2012

### mangaroosh

You probably couldn't, but the time at a distance would either be the same or it wouldn't; if it is the same [for all clocks at a distance] then absolute simultaneity prevails; if any of the clocks is different, then RoS prevails. What I'm wondering is, what would cause any of the clocks not to tell the same time?

17. Feb 9, 2012

### harrylin

Sorry, I could not decipher your method to set a distant clock "on time", such that you assume (or pretend) that both clocks indicate the same time perfectly simultaneously, even if only shortly. How would you do that? How could you make distant clocks tell exactly the same time?

18. Feb 9, 2012

### mangaroosh

I mightn't be making the point very lucidly, but the intention isn't to set two distant clocks to the exact same time; the question is, how might it arise that they don't tell the exact same time?

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?

19. Feb 9, 2012

### ghwellsjr

We have no way of knowing if a clock remote from us has the same time on it as our local clock. That's the problem. Once you recognize that there is no test, no measurement, no way to detect, no way to determine, etc., etc., etc., the time on a remote clock, then you can follow Einstein's process. He said unless you define the time on the remote clock, it is impossible to deal with the problem. And you can define it arbitrarily in many different ways. So rather than suppose, like everyone else did, that there is an absolute universal time that nature is ticking away at, he postulated that the time on a remote clock is equal to the time on a local clock when a light signal takes the same amount of time to get from the local clock to the remote clock as it does for a light signal to get from the remote clock to the local clock. Under this defintion, RoS prevails. Under the previous assumption of an absolute universal time, RoS is not a factor.

20. Feb 9, 2012

### harrylin

We understood your intention, which appears to be based on an unfounded assumption. Clocks are man-made and when you put a battery in it you can set it at any time you want. Thus, in order to have two clocks tell the same time, you have to do that.

You seem to have already a difficulty with getting two distant clocks synchronized according to yourself, despite your suggestion that all clocks will be automatically synchronized with all other clocks according to everyone. Nevertheless it was only an introduction to the next question: how can you do that in such a way that everyone will agree?