Undergrad Confusion about Simultaneous Events in Relativity

[Thecla]

In Lee Smolins book Time Reborn he discusses simultaneous events and says events that
are far from each other we find that there is no absolute ordering that all observers can agree on. For some observers the two events may be simultaneous for other observers one event may be in the past of the other.

Also earlier in the book he says that observers do agree about what can be called the causal structure. Take 2 events x,y.There are 3 pssibilities
x could be the cause of y
y could be the cause of x
neither could be the cause of each other.

Does the second paragraph contradict the first?

I will describe a scene where two events are far from each other:
A rocket ship takes off from Earth and 4 months later it lands on Mars.
Are there some observers who see the take-off and landing as simultaneous?
Are there some observers who see the landing on Mars before the take-off from earth?

 

[andrewkirk]

In Lee Smolins book Time Reborn he discusses simultaneous events and says events that
are far from each other we find that there is no absolute ordering that all observers can agree on. For some observers the two events may be simultaneous for other observers one event may be in the past of the other.

Also earlier in the book he says that observers do agree about what can be called the causal structure. Take 2 events x,y.There are 3 pssibilities
x could be the cause of y
y could be the cause of x
neither could be the cause of each other.

Does the second paragraph contradict the first?

I will describe a scene where two events are far from each other:
A rocket ship takes off from Earth and 4 months later it lands on Mars.
Are there some observers who see the take-off and landing as simultaneous?
Are there some observers who see the landing on Mars before the take-off from earth?

It comes down to the concepts of timelike vs spacelike separation of events.
Two events are timelike-separated if it is theoretically possible for a particle with mass to travel from one to the other.
They are spacelike-separated if it is impossible for any particle to travel from one to another.
There is an in-between condition called lightlike separation, in which it is possible for light (massless particles) to travel from one to another, but not for particles with mass.

For timelike-separated events, we can say without ambiguity that one is earlier than the other. For spacelike-separated events, we cannot.
In the timelike case, the earlier event is the one which it is theoretically possible for a particle with mass to use as its starting point of its journey to the other event. Speaking causally, we can say the earlier of two timelike-separated events can be a 'cause' of the later event - although one has to bear in mind that the notion of cause is ill-defined and rubbery. We can speak precisely of causality (time-ordering) but not of causes, since any event can be thought of as having infinitely many causes.

In your rocket example, the takeoff and landing are timelike separated, because massive particles (those in the rocket) do travel from one event to the other. So the two events are unambiguously ordered in time and no observer can see the arrival as happening before, or simultaneous with, the departure.

Smolin's use of the phrase 'far from each other' is an attempt to refer in a non-technical way to the notion of spacelike separation. But it's not really about how large the distance separating them is. Rather it is - in a very loose sense - the ratio of the distance separation to the time separation.

 

[Ibix]

In Lee Smolins book Time Reborn he discusses simultaneous events and says events that
are far from each other we find that there is no absolute ordering that all observers can agree on. For some observers the two events may be simultaneous for other observers one event may be in the past of the other.

This paragraph is not correct. Either Smolin oversimplified or you misunderstood him.

In relativity, nothing can exceed the speed of light. So we end up with a natural distinction between events that are far enough apart that not even light can cross the gap in the time between them and those that are close enough. The former type are said to be spacelike separated and cannot possibly cause one another because nothing could get from one to the other. The latter type are said to be lightlike or null separated if only light is fast enough to make the crossing, and timelike separated otherwise. In both null and timelike separated cases, one event may be the cause of the other. All frames agree on the type of separation between any two events, because the speed of light is invariant. They do not necessarily agree on the time ordering of spacelike separated events. They do agree for timelike and null separated events.

In your rocket example the rocket travels slower than light. Thus light can (easily!) cross the gap between the departure and arrival events so they are timelike separated. All frames will agree the ordering.

 

[LURCH]

So we end up with a natural distinction between events that are far enough apart that not even light can cross the gap in the time between them and those that are close enough. The former type are said to be spacelike separated and cannot possibly cause one another because nothing could get from one to the other. The latter type are said to be lightlike or null separated if only light is fast enough to make the crossing, and timelike separated otherwise.

Perhaps this goes to the heart of the OP’s question. Would it be accurate to say that, if two events are simultaneous, then the time between them is zero? If so, then I think that we could say that for two simultaneous events, at any arbitrary distance from one another, light cannot get from one to the other in the time between them, because the time between them is zero.

In light of the fact that simultaneity is relative, we can also say that two events which are simultaneous in one reference frame are not simultaneous in another. This would appear to mean that one could change the separation of the two events from space-like to time-like or light-like, just by selecting the appropriate reference frame. Is that a correct statement? Also, is it helpful @Thecla?

 

[Dale]

This would appear to mean that one could change the separation of the two events from space-like to time-like or light-like, just by selecting the appropriate reference frame. Is that a correct statement?

No, this is not a correct statement. You can select any reference frame you like, but it will not change the separation from spacelike to timelike or vice versa. I recommend working through the math on this one.

 

[PeroK]

Perhaps this goes to the heart of the OP’s question. Would it be accurate to say that, if two events are simultaneous, then the time between them is zero? If so, then I think that we could say that for two simultaneous events, at any arbitrary distance from one another, light cannot get from one to the other in the time between them, because the time between them is zero.

In light of the fact that simultaneity is relative, we can also say that two events which are simultaneous in one reference frame are not simultaneous in another. This would appear to mean that one could change the separation of the two events from space-like to time-like or light-like, just by selecting the appropriate reference frame. Is that a correct statement? Also, is it helpful @Thecla?

It's not correct at all. The property of timelike, spacelike or null separated for a pair of events is frame independent. Events cannot be one thing in one frame and another thing in another frame.

Second, as simultaneity is relative, there is no such thing as simultaneous events, only events that are simultaneous in a given reference frame.

IF two events are simultaneous in a given frame, then by definition they are spacelike separated, hence spacelike separated in all frames.

 

[Dale]

IF two events are simultaneous in a given frame, then by definition they are spacelike separated, hence spacelike separated in all frames.

And conversely, if two events are spacelike separated then there exists a frame where they are simultaneous. In all other frames they are not simultaneous, but are still spacelike separated.

 

[Ibix]

Would it be accurate to say that, if two events are simultaneous, then the time between them is zero?

Yes. Note that frames won't generally agree on whether two events are simultaneous.

If so, then I think that we could say that for two simultaneous events, at any arbitrary distance from one another, light cannot get from one to the other in the time between them, because the time between them is zero.

As measured by the frame in which they are simultaneous, yes. Other frames will say that the time is non-zero but will always agree that the time is too short for light to cross the distance.

This would appear to mean that one could change the separation of the two events from space-like to time-like or light-like, just by selecting the appropriate reference frame.

No. If that were the case, frames would be able to disagree on whether events were causally connected or not. The character of an interval (timelike, null, or spacelike) is invariant.

This is easiest to see by writing down the interval between the two events, $\Delta s^2=c^2\Delta t^2-\Delta x^2$ in one frame. Convince yourself that this is positive for timelike separated events, zero for null separated events, and negative for spacelike separated events. Write down the interval in some other frame, $\Delta s'^2=c^2\Delta t'^2-\Delta x'^2$. Use the Lorentz transforms to write $\Delta x'$ and $\Delta t'$ in terms of $\Delta x$ and $\Delta t$ and grind through the algebra to show that $\Delta s^2=\Delta s'^2$ - and hence that the character is invariant.

 

[LURCH]

I see; so in the case where two events are timelike separated, there does not exist any frame in which they can be simultaneous? And Thecla’s “Rocket to Mars” illustration would be an example of this, yes?

 

[PeroK]

I see; so in the case where two events are timelike separated, there does not exist any frame in which they can be simultaneous? And Thecla’s “Rocket to Mars” illustration would be an example of this, yes?

Yes, in all reference frames, timelike-separated events follow the same sequence.

 

[LURCH]

Thank you all so much!

Well, I don’t know if we have helped the OP, but this has made a huge difference in my understanding. Must be 30 years I’ve been hearing people say “simultaneity is relative”, and this is the first time I’ve ever heard anyone include the caveat about events being inside or outside of each other’s light cone. Now, I’m trying to retrofit a whole lot of my old perceptions with this new information.

With the “rocket to Mars” illustration it’s pretty easy, but I’m struggling a bit with Einstein’s two ambassadors on a train (a favorite of mine because of the international intrigue, and subtle implication that the key to world peace is a correct understanding of Special Relatively). I thought that the whole point of that story was that the ambassadors think that they both pushed their buttons at the same time, but to the observers outside, the two events happened at notably different times. Is that not how that story goes? I’m off to see if I can find a copy in his original words (‘cept, y’know, English).

I doubt that I’m the only one with this misunderstanding. Maybe this is what had the OP confused, as well.

 

[m4r35n357]

The key to world peace is a correct understanding of Special Relatively).

Imagine if it that were true . . . on second thoughts we're doomed!

 

[Thecla]

Your discussion has helped me very much

 

[Herman Trivilino]

In Lee Smolins book Time Reborn he discusses simultaneous events and says events that
are far from each other we find that there is no absolute ordering that all observers can agree on.

If you can, in principle, be present at both events then the events have a definite order.