# Relative simultaneity

1. Oct 23, 2009

### Grimble

Ref:http://www.bartleby.com/173/9.html" [Broken]

I am having a little bother with this and hope that someone will be able to explain it for me

If lightening strikes A & B simultaneously then, as those strikes are space-time 'events' they have no motion, only a time and a place. OK?

Then, as they also occur adjacent to points A' & B' on the train, and, if the light were reflected by mirrors attatched to those points, it would travel at 'c' relative to the train, in which co-ordinate system the observer at M' is not moving!:(

Therefore the two lightening strikes at A & B, A' & B', are also simultaneous to the observer on the train, as perceived by that observer

Einstein wrote that is the observer on the train
Then was he not saying that the lightening strikes were not simultaneous with respect to the train, as perceived from the embankment?

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2. Oct 23, 2009

### ZikZak

No, you're equivocating a specific "time and place" with "spacetime event." A spacetime event is a specific geometrical location in spacetime, which to any given observer will be at a specific time and place. But different observers will *disagree* on the particular time and place of the event.

In other words, yes, spacetime events are specific locations in spacetime, but the actual x and t coordinates assigned to them by different observers will differ.

The example is meant to illustrate that the strikes cannot be simultaneous in both frames, because if the light flashes reach M simultaneously (a single event, since it happens at a specific time and place), then they cannot have reached M' simultaneously (since M' is not at M at the event when both signals arrive).

There is only one event where the light flashes meet, and in the example, this is given to be at observer M. Since observer M' is not at that event, the light flashes cannot have reached him simultaneously.

Since the light flashes did not reach M' simultaneously, even though he is equidistant from A' and B', where the lightning struck, he can only conclude that the flash at B' (=B) happened first. Thus in his reference frame the lightning flashes are not simultaneous. i.e., even though observer M assigns the same value of the time coordinate t to the two lightning flashes, observer M' does not.

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3. Oct 23, 2009

### HallsofIvy

Staff Emeritus
I might add that you go off the track when you start with "If lightening strikes A & B simultaneously". There is no such thing without stating a frame of reference in which the two events are simultaneous.

4. Oct 24, 2009

### Rasalhague

In Einstein's example, the lightning strikes are simultaneous in the rest frame of the embankment, and therefore not simultaneous in the rest frame of the train:

"Are two events (e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative."

In fact they can't be simultaneous in the rest frame of the train if they're simultaneous in the rest frame of the embankment. Likewise, if another pair of lightning bolts struck A and B simultaneously in the rest frame of the train, then this second pair of lightning strikes would not be simultaneous in the rest frame of the embankment.

If the train stopped relative to the embankment, then the rest frames of train and embankment would coincide and be the same frame; only then could lightning strike A and B simultaneously in the rest frame of the train and the embankment. But the lightning strikes could only be simultaneous in all frames moving at some nonzero velocity relative to each other if the two bolts of lightning struck the same place as each other and at the same time as each other. This is what "relativity of simultaneity" means; in general, whether two events are simultaneous depends on which frame of reference they're described with respect to.

Here are three animations illustrating the relativity of simultaneity.

(Animation begins at 08:00 minutes.)

(Animation begins 01:16.)

I found it very confusing when I first saw these. I remember watching the animation in The Mechanical Universe again and again as I tried to get some intuition for it, but I think they're well worth a study, even if it's not obvious at first. Also worth scrutinising spacetime diagrams.

A personal way I have of imagining it that I have is to think of a line (or plane) of simultaneity in a Minkowski spacetime diagram as being like a speedboat that tilts upwards in the direction that the speedboat is going (and down at the back). But that's just my personal way of remembering which way round it goes. So if that makes no sense, just ignore this paragraph!

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5. Oct 24, 2009

### Rasalhague

Yes, this is the source of a lot of apparent paradoxes. For the statement "A and B are simultaneous" to be meaningful, you have to say in which frame of reference they're simultaneous - because if they're simultaneous in one frame of reference, they won't be in others.

6. Oct 24, 2009

### Grimble

OK, sorry, take out
No, let's start again.

Einstein wrote in http://www.bartleby.com/173/9.html" [Broken]

But then he writes,
that is the observer on the train
Was he not saying that the lightening strikes were not simultaneous with respect to the train, as perceived from the embankment?

Last edited by a moderator: May 4, 2017
7. Oct 24, 2009

### Grimble

And again, if we were to consider the observer in the train, what does he perceive?

Again I quote

And
And it is surely not unreasonable to suppose that at night, for instance, the traveller is completely unaware of the embankment. No all he perceives is himself and his reference-body, the train.

Now when the lightning strikes he will see the light from A and the light from B, each travelling at c relative to the train reach him at the same time. For him, only his own reference-body exists and to him, this reference-body is not moving.

I do understand what Einstein was saying, the relativity of simultaneity; but it is, for one observer, between his own reference-body and his perception of another reference body; where he will of course also perceive length contraction and time dilation, not that they are relevant to this discussion.

8. Oct 24, 2009

### Rasalhague

In the first example, lightning strikes A and B simultaneously with respect to a coordinate system in which the embankment is still and the train moving, but these strikes are not simultaneous with respect to a coordinate system in which the train is still and the embankment moving. An observer sitting on the embankment at M will see the flashes at the same time, whereas an observer sitting on the train at M' will see the flash from B before the flash from A. This is no illusion. In the latter observer's rest frame (the rest frame of the train), the lightning will really have struck B before it struck A, even though in the rest frame of the observer at M (the rest frame of the embankment), both strikes happened at the same time as each other.

It's completely cointerintuitive. I'm sure everyone finds this a confusing idea when they first encounter it.

In the second example, Einstein imagines a different pair of lightning strikes which happen to be simultaneous with respect to the train's rest frame, in which case they're not simultaneous in the rest frame of the embankment. In this example, it's the train passenger at M' who sees the flashes at the same time, while the person sitting on the embankment at M will see the lightning strike A before it strikes B. In this case, in the rest frame of the embankment, the lightning really does strike A first, even though the strikes were simultaneous in the train's rest frame.

This is a good idea to think about.

This is not what will happen if the lightning strikes are simultaneous in the rest frame of the (invisible) embankment. Even at night and even if there is no embankment there at all, so long as the lightning strikes A and B simultaneously according to a frame in which the train is defined as moving, these events will not be simultaneous in a reference frame defined such that the train is at rest in it.

The choice of reference frame is arbitrary, a matter of convenience. We can define a frame in terms of some object being at rest in that frame, but we don't have to. We could just as well define a frame moving at some velocity relative to the train even if we didn't know of any object which was at rest in such a frame, or of any literal observer at rest with respect to it. Or if there was only an embankment and no train, we could just as easily define a frame moving at some velocity relative to the embankment and calculate time, distances and the order of events in such a frame, regardless of whether there is any physical object at rest with respect to that frame.

Last edited: Oct 24, 2009
9. Oct 24, 2009

### Grimble

Yes but...

Let me simplify my understanding to the very basics as I see them:

We have two entities, the embankment and the train which are moving relative to one another.

Each is stationary wth respect to its own reference-body and each is moving with respect to the other's reference body.

Lightning strikes points A & B which exist in both reference bodies.

Observers in each reference-body will surely see the same effects with respect to their own reference-body; which, at the expense of repeating myself, is stationary with respect to that observer.

The light will travel from A & B at the speed of light c relative to the observer's reference-body and so will meet at either M or M' depending on which reference-body the observer is in.

It is all a completely self contained, symmetrical, set of circumstances.

From the above, please tell me how one can determine that one case is simultaneous and the other is not

As far as I can see each would say that the lightning strikes were simultaneous for him but not for the other. What is it I am missing here

10. Oct 24, 2009

### Staff: Mentor

Think of A and B as fixed points on the embankment. Both reference frames agree that the lightning strikes A and B, but the train frame will measure the positions differently. Think of a position A' fixed in the train that coincides with A at the instant the lightning strikes A; similarly, think of train location B' coinciding with B at the instant the lightning strikes B.

Not sure what that means.

No. M' coincides with M at the moment the lightning strikes according to the embankment frame. Light traveling from A & B will meet at M, but not at M'. By the time that the light reaches M, M' has moved along.

Not symmetric at all.

The embankment frame sees the light go from A and B and meet in the middle at M. So they surely think that the lightning strikes were simultaneous. The train, on the other hand, sees the lightning striking at point A' and B' (equidistant from M'), but the light does not meet in the middle of the train at M'. So they must conclude that the lightning strikes did not occur simultaneously.

11. Oct 24, 2009

### Bob_for_short

If in one reference frame you have a solution Acos(ωt) (a long horizontal rod oscillating up and down as a whole), in a moving RF (with velocity V) you will see it as a wave: A'cos(ω't'-kx') with k being dependent on V.

12. Oct 24, 2009

### Staff: Mentor

Well, that's true. But I'm not sure how helpful it will be in sorting out the specific example being discussed in this thread.

13. Oct 24, 2009

### ZikZak

But not simultaneously in both. The example demonstrates how assuming that the strikes occur simultaneously in both reference frames leads to an inescapable contradiction.

The light does indeed travel at c relative to either reference body. Therefore, the fact that the flash from B' reaches M' before the flash from A' does requires that M' conclude that the strikes were not simultaneous.

No it isn't. You are forgetting the givens. The given is that the flashes reach M simultaneously. There is no such given for M'. The assumption of complete symmetry you are trying to make leads to a logical contradiction. That is the point of the exercise.

In the example, it is given that the flashes are received simultaneously by M. There is one event at which the world-lines of both flashes and M himself meet. Because all the worldlines meet at one event, all observers must agree that this event occurred. This is the basic assumption of reality in science: that observers might disagree on when and where an event happened, but they should always agree that it did.

Therefore, M' also observes that the light rays meet with observer M at a single event. If an external reality exists, this is it. There is no escaping this. Things that happen, happen. M' may label the "both light rays hit M" event with different coordinate values than M does, but it is one single event with one single set of coordinates in any particular reference frame.

Working in the reference frame of M, M observes the flash from B hit M' at a different time (different event) from the flash from A. Two separate events. Changing reference frames cannot possibly combine two different events into one. In other words, M observes that when flash A reaches M', flash B is not there at that event. When flash B reaches M', flash A is not there. When the two flashes meet each other, M' is not there. Merely changing reference frames cannot magically stitch all these different events together. M' therefore observes the flashes at two different times (two different events). He must conclude that the bolts were not simultaneous.

Summary: assuming that the bolts were simultaneous in M *forbids* them from being simultaneous in M'. Both observers agree on the events that occur: light flashes A and B reach M at a single event, and M' at two different events. All observers in the universe must agree that both light flashes and M meet at some event, while M' meets the light flashes separately at two different events.

You could do the same argument starting from the assumption that the flashes reach M' simultaneously, which would forbid M from seeing them simultaneously, but that would be a different universe. The two situations would be two completely different sets of events. Only one of them can occur, not both.

EDIT: And I should add, that since the given is that the light rays reach M simultaneously, and not that the bolts were simultaneous in all frames, that the correct calculation to make in the M' frame would be to ask when the bolts must occur in order for their flashes to meet at M, NOT at M'. Since M is traveling backwards, bolt B must occur before bolt A in order for the flashes to meet at M as given.

Last edited: Oct 24, 2009
14. Oct 24, 2009

### Rasalhague

Yes.

I'm more familiar with the synonyms "reference frame" and "coordinate system", but this is what the book says "reference body" means, a coordinate system. Bear in mind that although the names train and embankment can be given to these two coordinate systems, the coordinate systems themselves are abstract entities. There doesn't have to be a physical embankment present for us to define a coordinate system moving relative to the train.

Yes.

They won't see the same effect if lightning strikes A and B simultaneously in one frame. If this was the case, there would be no need for a theory of relativity. It's because the effects referred to one coordinate system differ from the effects referred to another, travelling at some velocity relative to it, that we need to take care to specify which coordinate system events are simultaneous in. If they're simultaneous in one frame, they can't be in the other (except in the trivial cases where A = B or the frames are the same).

The light travels at c in both frames. If the strikes were simultaneous in the rest frame of the embankment, the light from each strike will reach M at the same time. According to this frame, the light from B will reach M' first. In this frame, the light reaches B first because M' is moving to meet the light from B and away from the light from A, so the light from A has further to go to get to M' and so takes longer to reach M'. Likewise in the rest frame of the train, the light from B will reach M' first. This couldn't happen if the strikes were simultaneous in this frame too unless the light from B was approaching M' at a different speed than the light from A. But we know the light always travels at speed c, so that can't be the reason. Instead we conclude that B must have been struck first, since the light from this strike arrived first, and the only way it could arrive first is if it set off first.

The symmetry becomes apparent when you compare what happens in the case of a pair of lightning strikes at A and B which are simultaneous in one frame with the case of a different pair of lightning strikes, this latter pair being simultaneous in the other frame. The amount of time that M' will have to wait in the first case after seeing the light from B till the light from A arrives will be the same as the amount of time that M will have to wait in the second case after seeing the light from A till the light from B arrives.

Did you look at the videos I linked to? They might not make it immediately clear. They certainly didn't for me. But I think that poring over them did eventually help get me a little closer to understanding. But it was really when I learnt about spacetime diagrams that the concept of relative simultaneity began to sink in.

EDIT: The book talks about "two strokes [sic] of lightning A and B", which we could think of as points in spacetime (events). Alternatively, we can think of a space coordinate A (constant in the embankment's rest frame) and distinguish it from a space coordinate A' (constant in the train's rest frame), so A makes a line in spacetime, as does A', such that their intersection is the point in spacetime where and when the lightning strikes on the left. And we can define B and B' likewise.

Last edited: Oct 24, 2009
15. Oct 24, 2009

### Grimble

What that means is that as each observer is stationary within his own reference-body and the points A&B are stationary with respect to each body of reference they will experience similar phenomena. Light from A&B will meet at the mid-point between A&B in each reference frame. i.e. at M or M' depending on which frame the observer is in.

But not according to the train the train's frame where it is permanently midway between A&B

Look, I am not saying that M and M' remain adjacent, They are in different reference-bodies.
Light is travelling at c within each reference body, relative to each reference body, so it will meet at two separate points in two seperate reference bodies. Surely that is fundamental to Relativity.

But why not? It is travelling at c from A' and from B' to M', MIDWAY, and permanently midway, between A' & B', from the train's reference body.

Last edited by a moderator: Oct 24, 2009
16. Oct 24, 2009

### Staff: Mentor

Note that A & B are stationary with respect to the embankment; while A' & B' are stationary with respect to the train. The light meets at M, not at M'!

Wrong: M is a fixed point in the embankment that is permanently midway between A & B. M' is a fixed point in the train that is permanently midway between A' and B'. M is not fixed from the train's viewpoint, just like M' (the middle of the train) is not fixed from the embankment's view.

The light meets at point M--not M'. Everyone agrees on that. M is midway between the A and B (as it always is). M is not midway between A' and B' at the moment the light reaches M.

No. Light does not travel from A' and B' and meet at M'--the light meets at M. Which is the entire point. Since everyone agrees that the light meets at M, and since M is much closer to the rear of the train when the light reaches M, the train observers conclude that the lightning strikes could not have been simultaneous. (If the lightning strikes were simultaneous according to train observers, then the light would meet in the middle of the train--at M', not M--since the lightning struck at points A' and B' which were equidistant from the middle of the train. But that doesn't happen.)

17. Oct 24, 2009

### ZikZak

No, a universe which would permit real paradoxes (M is not equal to M' and the light beams meet at M and the light beams meet at M') is anathema to Relativity, and all of science.

18. Oct 25, 2009

### Rasalhague

If there's a spacelike separation between two events (i.e. the sum of the squares of each space components of the separation vector is greater than the square of the time component (when space and time are measured in the same units)), then--given that nothing can go faster than c--it's never possible for a single particle to be present at both events, and so neither event can have an effect on the other. The lightning strikes are two such events. And when there is a spacelike separation between two events, there's no absolute way of ordering them in time; we can only say which happens first and how far apart they are in time with respect to a particular coordinate system, knowing that there are infinitely many other coordinate systems we could chose to describe them, in some of which the other event will happen first. Although bizarre and counterintuitive to us humans, these rules are consistent with causality because events which have no absolute order can't influence each other, so the contradiction of cause preceding effect never arises.

But if there's a timelike separation between two events (i.e. the square of the time component of the separation vector is greater than the sum of the squares of the space components), or a lightlike separation (i.e. the square of the time component equals the sum of the squares of the space components), then it is possible for a particle to be present at both events. There is a causal connection between the two events: one can have an effect on the other. Events which a single particle is present at are said to lie on the particle's worldline. Worldline means a trajectory through spacetime. Consider the light from the lightning strike at B = B' reaching M' as one event, and the light from the lightning strike at A = A' reaching M' as another event. Both events lie on the worldline of M' because M' is present at both events. It doesn't matter which coordinate system we describe this pair of events in, this pair of events will always happen in the same order, otherwise there would be a genuine contradiction: no way of ordering cause and effect. If there wasn't an absolute order to events on a worldline, then we'd lose the causal structure of Minkowski spacetime. So if this pair of events (the light from each strike reaching M') happens in one order in the rest frame of the embankment, it must happen in the same order in all frames (coordinate systems), including the rest frame of the train.

Likewise the simultaneous arrival of the light from A and B at M. When things happen simultaneously at the same location in one frame, they must happen simultaneously in all frames otherwise there'd be a contradiction about real physical events.

19. Oct 25, 2009

### Grimble

OK OK, let us examine what you are saying here:

Everyone agrees (which somehow gives it authority?) that the lightning strikes at A & B are simultaneous as observed from the embankment. . . . . . (1)

At the instant that the lightning strikes hit, points A' & B' are adjacent to points A & B, so the lightning strikes them too. . . . . . (2)

In order to establish whether this appears to be simultaneous as observed from the train, we need to establish when the light from those lightning strikes reaches an observer at M', midway between A' and B'. . . . . . (3)

If the distance between the points A' & B' is 2L the light has to travel the distance L to reach M'. . . . . . (4)

Then the time it will take from A' to M' $t = \frac{L}{c}$ where c is the speed of light in the train's reference-body. . . . . . (5)

And the time it will take from B' to M' is also $t = \frac{L}{c}$.. . . . . (6)

So the transit times for the light to travel from A' to M' and from B' to M' are equal - (5),(6) above. . . . . . (7)

And as we saw in (2) above the lightning struck A' and B' at the same time as it struck A & B which strikes we have established was simultaneously - (1)

So if the light started from A' and B' at the same instant and the travel times were equal - (7)

The light must meet at point M' in the Train's reference-body! - in the same way that it meets at M in the embankment's reference-body.

That surely is relativity - What happens is relative to where it is viewed from.

Or is there something wrong with my maths???

It has to be straightforward and logical, there is nothing mysterious about relativity, it is not some sort of dark magic only known to a few initiates

Please note that my last comment is intended to be humerous and is in no way intended to be sarcastic.

Grimble

20. Oct 25, 2009

### Rasalhague

Here's where the problem is. It's not an algebraic mistake, but a logical one. A' and B' are indeed equidistant from M', but when you use (4) in your argument for simultaneity, you assume the conclusion you expect, namely that the lightning strikes are simultaneous in the rest frame of M' (the rest frame of the train). But if this was the case, we'd have a contradiction because, according to M (in the rest frame of the embankment), the light from the right reaches M' before the light from the left, not at the same time. In the rest frame of the embankment this is because M' has moved closer to B (while B' has moved further to the right beyond B). But M' hasn't moved in the train's rest frame. The light from B can't both arrive first at M' and not arrive first at M', nor (according to observation and Einstein's second postulate) can its speed be different. The only remaining alternative is that in the train's rest frame, the light struck B' first, before it struck A', and that's why it arrives first at M'.

This conclusion offends our intuition, and seems superficially like a contradiction too. But the fact that information can't travel faster than c ensures that causality is respected, i.e. that cause always precedes effect. The only events whose order isn't frame invariant are events which can't be affected by each other because there isn't time in any frame for information to travel from one to the other at speed c or less.

Similarly, if the light from both strikes does reach an observer simultaneously in one frame, it must reach that observer simultaneously in all frames. When two things happen at the same time and place, we call them a "spacetime coincidence". Anything that's a spacetime coincidence in one frame is a spacetime coincidence in all frames, and would have to be for causality to be preserved.

It is logical, but I think it's only natural that most people find it far from obvious at first (I certainly did, and there's lots of concepts I still struggle with). That's because it's so counterintuitive. In our everyday experience, at the human scales we're used to, simultaneity is not relative, and there's nothing directly comparable to this effect. But of course what one person can learn, another can, and with determination you'll get there in the end!

Last edited: Oct 25, 2009