# Events can be simultaneous in one time frame but not so in others

1. Jul 13, 2010

### stevmg

A recent forum thread went into a long disussion of simultaneity. The term "ansible line" was used (later referred to as a "simultaneity line.")

Einstein demonstrated with a thought experiment with regards to lightning and a moving train that events can be simultaneous in one time frame but not so in others.

1) Can two events which are simultaneous in one frame of reference (FOR) be simultaneous in another - that is - THE SAME TWO EVENTS, or is the simultaneity limited to only ONE FOR.

2) Can one always find an FOR in which any two event pairs are simultaneous?

3) It would seem that an event A can be simultaneous with an event B in one FOR and also be simultaneous with event C in a second FOR. This does not mean that B and C are simultaneous but it should be possible to find an FOR in which they are. Again, this would NOT mean that $$\exists$$ an FOR in which A, B and C are simultaneous mutually, or does it?

2. Jul 13, 2010

### espen180

Re: Simultaneity

1) If you take a look at a Minowski diagram, you will notice that the slope of the simultaneity line is linearly proportional to the velocity. Therefore, the answer is no.

2) No. Mark the two events in a spacetime diagram and draw a straight line from one to the other. If the slope of this line is greater than or equal to c, there is no intertial frame in which they are simultaneous.

3. Jul 13, 2010

### yossell

Re: Simultaneity

I *think* the answer to (a) is yes. Imagine a frame F travelling in the x direction compared with frame G, and think about events that lie on the y axis - that is, events that lie on a line at right angles to the direction of motion. I think both frames will agree on the times of such events, and two events simultaneous in G will be simultaneous in F.

4. Jul 13, 2010

### stevmg

Re: Simultaneity

Thanks - have to get ahold of a Minkowsky diagram somewhere. Don't have one on hand.

Still need the answer to part 3...

stevmg

5. Jul 13, 2010

### stevmg

Re: Simultaneity

Thanks!

A special case - OK

Still need an answer to part 3

stevmg

6. Jul 13, 2010

### yossell

Re: Simultaneity

Ah 3 - I'm even less sure of my answer here - but

I think if A and B are spacelike related and A and C are space-like related, then B and C are space-like related. If there's a frame where A and B are simultaneous, then they're space-like related; ditto for A and C. So B and C are space-like related, so there's a frame where they're simultaneous.

But this certainly doesn't mean you can find a frame where all events are simultaneous.

But I would wait until a professional confirms or disproves what I say here!

7. Jul 13, 2010

### starthaus

Re: Simultaneity

The correct answer is "yes".

$$x'=\gamma(x-vt)$$
$$t'=\gamma(t-\frac{vx}{c^2})$$

so:

$$dx'=\gamma(dx-vdt)$$
$$dt'=\gamma(dt-\frac{vdx}{c^2})$$

If there is a frame where $$dx=0$$ and $$dt=0$$ then, in all frames $$dt'=0$$ (and $$dx'=0$$)

Last edited: Jul 13, 2010
8. Jul 13, 2010

### starthaus

Re: Simultaneity

You can start answering your own questions using only the prototype at post 7. All you need is the differential form of the Lorentz transforms.

9. Jul 13, 2010

### espen180

Re: Simultaneity

I am unable to make sense of this statement. How can an event (a point) lie at right angles to anything?

Here's a small Minowski diagram applet i threw together. (http://sites.google.com/site/espen180files/Minowski.html?attredirects=0&d=1"

You can vary the speed of frame S'. The movable line parallel to the x' axis is the S'-frame's simultaneity planes. Please convince yourself that two events can only be simultaneous in one frame.

Last edited by a moderator: Apr 25, 2017
10. Jul 13, 2010

### starthaus

Re: Simultaneity

Yes, you are correct, no doubt about it.

11. Jul 13, 2010

### starthaus

Re: Simultaneity

Look at post 7, it gives a rigorous explanation.

12. Jul 13, 2010

### espen180

Re: Simultaneity

I think you're missing the point.

Assume two events which are simultaneous in S. Their spacetime coordinates are
E1=(x1,t1)
and
E2=(x2,t2)
with t1=t2

Frame S' is travelling relative to S with velocity v. So we have
x'1=γ(x1-vt1)
x'2=γ(x2-vt2)=γ(x2-vt1)

t'1=γ(t1-vx1/c2)
t'2=γ(t2-vx2/c2)=γ(t1-vx2/c2)

There is only two ways the events E1 and E2 can be simultaneous in S':
1) x1=x2
2) v=0

In the case of 1), there is only one event, and simultaneity is not a useful concept here.
In the case of 2), S=S', there is only one frame.

This proves the statement that a pair of events can only be simultaneous in one frame.

13. Jul 13, 2010

### starthaus

Re: Simultaneity

I don't think so.

Err, no. You forgot the fact that you have two other coordinates , y and z , that can separate the events. Event means E(x,y,z,t).yossell already explained that to you in post 6. You need to pay attention.

Last edited: Jul 13, 2010
14. Jul 13, 2010

### DrGreg

Re: Simultaneity

The events lie on a line that is at right angles to the direction of motion

You are thinking in two dimensions only (1 space + 1 time). You need to think in three dimensions (2 space + 1 time). That's why yossell referred to the y axis, at right angle to both the t and x axes.

Last edited by a moderator: Apr 25, 2017
15. Jul 13, 2010

### my_wan

Re: Simultaneity

Without me providing the math, I suggest you remain skeptical of my answer. I haven't actually done the math myself to prove it.

What I see here in 1) is that if the two events have a spacelike separation, then it's possible to define their positions and timing in such a way that more than one frame will define them as simultaneous. This wouldn't work if the two events are in the same place. Also note that the two observers may note agree on where the events occurred.

2) Again, my guess is yes.

3) This is much more difficult, but I can imagine special cases might exist where it should be possible in principle.

The way I picture this, to justify my answers, is to consider two observers with some arbitrarily defined time horizon encircling each of them, such that any event on their respective time horizons are simultaneous. If those two horizons overlap, like overlapping circles (note that what is round for one observer is not for the other), then the two points where the time horizons overlap meets is the event time and locations that make it possible. Such time and locations would change for each moment in time, if we are talking solely in terms of Special Relativity.

16. Jul 13, 2010

### yossell

Re: Simultaneity

I'm pretty sure that espen180's `no' answer to this question is correct. There's no frame where two time like events are simultaneous.

17. Jul 13, 2010

### espen180

Re: Simultaneity

Ah, I see! My bad.

18. Jul 13, 2010

### my_wan

Re: Simultaneity

It looks to me like espen180's numbers are almost valid in post #12, but the OP conditions were misstated.

x1 does not have to equal x2, but t1 must equal t2. This does not mean t1 will equal t'2. Simultaineiety only depends on the time difference between events, not the proper time at which the events occured.

No, there are two event and two observers in 1). Note the original wording:
Again, it's the exact same physical conditions as provided in 1), only with the question generalized for certain specific two observers defined in relation to ANY two events. If there's only one event in the first question then there's ony one in the second, same for frames. Thus the conclussion is invalid.

19. Jul 13, 2010

### espen180

Re: Simultaneity

For simultaneity in S', you need t'1=t'2. t1=t2 is assumed. In 2D spacetime, 1) and 2) are the only way to acheive this in two frames simultaneously.

Even if two events occur at the same time in the same location, you consider it two separate events?

If v=0 between two frames with a common origin and parallel axes, you still consider them two separate frames?

20. Jul 13, 2010

### stevmg

Re: Simultaneity

Getting sleepy...

Later...

stevmg