# Distance role in simultaneity measurement

1. Jan 30, 2014

### analyst5

My new question is quite connected to the famous Andromeda paradox. So it's stated that the Andromeda is far away and moving towards the galaxy will cause the observer to have the distant future of Andromeda as his present, in his reference frame (of course future compared to what the stationary observer considers present). So my question is, if we consider a third observer, which is also heading towards Andromeda with the same velocity as the previously mentioned one, but is much more close to Andromeda than the other one, will he consider the same event that the previously one mentioned as his present or will it be also the future one, but one that happened before the event that the observer that is more distant considers present. To sum up, what role does distance to the event play in defining what happens relative to an observer, if they are both having the same velocity, but one is much closer to the event?

This is a sub-question of the first one, but is very connected with it. I know Lorentz Ether Theory is not relevant anymore, but I want to know the postulates to understand SR better. So LET basically says that every observer that is at rest with aether is in a position of privilegy, and that his 'view' is correct, his plane of simultaneity is the universal one. But how is this compatible with the first topic of my post? If two observers are mutually at rest, and of course at rest with aether, they still may have different distances to the objects moving, for example, towards them, therefore they won't consider the same events as their present. Of course, I mean this, if distance plays a role in differences what observers with the same velocity that isn't zero relative to an object consider to be their present. So what is then considered to be the priviliged frame or criteria for absolute simultaneity in LET? In a hypotethical scenario, since it's basically abandoned.

2. Jan 30, 2014

### Staff: Mentor

Observers at rest wrt each other agree on simultaneity. More precisely, simultaneity is unchanged by either spatial or temporal translations of an inertial coordinate chart.

3. Jan 30, 2014

### analyst5

So no matter what the distance they have from Andromeda in this case, if they are inertially moving towards it, they will agree what event is 'the present' (i hate to use this word in this context) for them?

4. Jan 30, 2014

### analyst5

On the original interpretation of the Andromeda Paradox, it is stated that distance plays a big role because the Andromeda is very far away from the Earth. So if we put an observer closer, but still in a position of motion towars the Andromeda, how come that doesn't change anything?

5. Jan 30, 2014

### Staff: Mentor

Distance magnifies the effect of the velocity difference. This will be apparent if you look at the Lorentz transform for time: $t'={\gamma}(t-vx)$. For small values of $v$ $\gamma$ is very close to one so the difference between $t$ and $t'$ is small unless $x$ is enormous; for $v=0$ $t'$ is always equal to $t$ and there's no time difference regardless of the distance.

6. Jan 30, 2014

### analyst5

@Nugatory, could you please do a comparision using Lorentz transformations, on the concrete example of Andromeda, the comparision between an observer that is in space (much closer to Andromeda than the Earth) but is heading towards Andromeda with the same velocity as the Earth observer (which is moving) and the stationary Earth observer so I can notice how the temporal difference remains the same as in the case of the Earth observer that is at rest with Andromeda and the Earth observer which is moving towards Andromeda? Or anybody else please?

7. Jan 30, 2014

### WannabeNewton

Your original question involved two inertial observers at rest with respect to one another. In this case if they Einstein synchronize their clocks initially they will remain synchronized forever and the two will always agree on simultaneity. Now you've changed it to any and all inertial observers who may or may not be at rest with respect to one another in which case it's obviously not true because an initial Einstein clock synchronization of two relatively moving inertial clocks will fail to keep them synchronized forever (time dilation).

8. Jan 30, 2014

### analyst5

Two observers at rest with each other (that are moving towards Andromeda) and one Earth observer that is at rest with Andromeda. That's the scenario. Now I need comparance between frames, the one that is closer to Andromeda and travelling towards it, and the earth one which is at rest with Andromeda.

9. Jan 30, 2014

### Staff: Mentor

If the two observers are moving relative to one another, they will not get the same time values for distant events, and the difference will be magnified by distance.

If they are not moving relative to one another, they will see the same time value for all events, regardless of distance.

Note that the velocity relative to Andromeda is irrelevant, except to the extent that if one of them is moving relative to Andromeda and the other is not, then of course they must be moving relative to one another.

10. Jan 30, 2014

### analyst5

But in my scenario, the distance from Andromeda to the observer in space which is moving towards to it is much smaller than the distance from Andromeda to the Earth observer. How would the Lorentz Transformations look for those two observers? P.S. I've understood the truth behind the equations for observers which are mutually at rest.

11. Jan 30, 2014

### Staff: Mentor

If two inertial charts are at rest wrt each other then they will agree on simultaneity.

If two inertial charts are moving wrt each other then they will disagree on simultaneity. The degree of disagreement will depend on the distance between the two events being considered, not on the distance between the origins of the inertial charts.

12. Jan 31, 2014

### ghwellsjr

You're welcome.

I gave you a couple of diagrams with some explanation like you asked me for. Did they resolve all your issues? Please don't just leave the thread hanging.

Here are a couple of diagrams, again, like you asked for, showing an example of your scenario. Earth is the thick blue line and Andromeda is the thick red line. I show a star exploding at 2.5 million years in the mutual Earth/Andromeda rest frame. Two observers are moving toward Andromeda at 22% of the speed of light shown in black and green. The dots show the Proper Times for each object/observer. The bottom dots are time zero and the top dots are time five million years. You have to count between them to see the times for the other dots.

The thin blue line represents a radar signal going from the Earth and the black observer at their time zero. It also represents a radar signal going from the green observer at his time of one million years. The return echo from the exploding star is shown as the thin red line. It is received by the green observer at his time of three million years, by the black observer at his time of four million years and by the Earth at its time of five million years:

Although the above diagram shows that the star exploded at 2.5 million years, Earth has no awareness of this until its time reaches 5 million years. It can use the radar signals as I describe in your other thread to establish that Andromeda was 2.5 million light-years away at the time of the exploding star and that it happened at Earth's Proper Time of 2.5 million years.

But the black and green observers establish the time and distance to the exploding star differently because they also assume that the radar signals take equal times to get to the star as the echoes take to get back. For example, black sends the radar at his time of zero and receives the echo at his time of 4 million years so he establishes that the star exploded at his time of 2 million years at a distance of 2 million light-years. Green sends his radar signal at his time of 1 million years and receives the echo at his time of 3 million years so he establishes that the star exploded at his time of 2 million years (same as black) at a distance of 1 million light-years.

Now I transform to the mutual rest frame of the black and green observers:

As you can see, all the same radar signals are sent and the echoes received identically according to each object's/observer's clock as in the first diagram but this one is what the black and green observers establish based on their mutual rest frame.

I think maybe your point of confusion is that you didn't realize that we assume that the clocks for the green observer are synchronized with the black observer so that in their mutual rest frame, their Proper Times match the Coordinate Times and that is why they consider the times of distant events to be identical. If we don't do that, we can't meaningfully talk about simultaneity issues.

The only difference between LET and SR is that LET, as you say, assumes that light propagates at c in only one Inertial Reference Frame (IRF). If you decide that the Earth/Andromeda mutual rest frame is that privileged IRF, then you would consider the Coordinate Times and Distances of the first diagram to be the absolute times and distances of the universe and the coordinate times and distances of the second diagram to be considered artificial times and distances that are only conveniences and don't represent reality (or whatever, I don't really know what a LET adherent would call the second diagram). But if the LET adherent decided that the mutual rest frame of the two observers was the privileged IRF, then they would switch the interpretations of the times and distances between the two diagrams.

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13. Feb 1, 2014

### phyti

analyst5;
Here is a comparative illustration of the different perceptions of observers E earth, S1 & S2 ships moving toward A Alpha Centauri at .4c and .8c. ( a case that would be possible within a lifetime).
In the E frame, each sends a light signal at the origin, that reflects from an object at A. The coordinates (x,t) for the event are E(4, 4), S1(2.62, 2.62), S2(1.33, 1.33).
The magenta lines are the geometric equivalent of t/gamma (t = round trip time) for each observer. The coordinates are calculated using the SR convention of c*t/2. As speed increases, the observers frame is scaled by 1/gamma.
In the S1 and S2 frames, the observer perceives the universe to be contracted, since A arrives earlier than expected according to their local clock.
The same phenomena apply regardless of distance.

View attachment 66205

Last edited: Mar 9, 2014
14. Feb 1, 2014

### FactChecker

The travelers at the same rate all agree on what happened, where, and when. They also agree that Andromeda-reference-frame clocks are set wrong and that their synchronization errors grow with distance from Andromeda. So something happening at time t1 for one traveler, also happened at time t1 for all other travelers in the same reference frame. But they see that the Andromeda clocks at the different positions do not agree on those times. They report that the Andromeda reference frame is wrong and thinks that the same events happened earlier (by Andromeda clocks) the farther they are from Andromeda (toward the farther travelers to Andromeda) .

15. Feb 1, 2014

### ghwellsjr

Only if they agree to agree, that is, if they have previously agreed on a coordinate system and if they have synchronized their clocks. It doesn't happen automatically. In my previous diagrams in post #12, I picked up the scenario in the Earth/Andromeda rest frame when the black traveler left Earth and I defined that event as the origin of both diagrams. I also assumed that the Proper Times on all three of those clocks was set to the Coordinate Time and I pointed out that the Proper Time on the green traveler was not set to the Coordinate Time of that IRF but rather to the Coordinate Time of the mutual rest frame of both travelers. However, I didn't say how that was done and as a matter of fact, as is often done in scenarios like this, we don't concern ourselves with how clocks get synchronized but there is a required process that must be performed to actually make it happen.

Here is another spacetime diagram that starts prior to the green traveler leaving Earth:

At the beginning of this diagram, it's trivial for the Earth and the two future travelers to synchronize their clocks because they are all inertial, colocated and at rest with each other. But as soon as the green traveler leaves Earth at 22%c his clock immediately goes out of sync with the Earth and the future black traveler. Later on, when the black traveler leaves Earth at 22%c, both travelers are at rest in a mutual Inertial Rest Frame (IRF), but their clocks are not synchronized, even though they are now ticking at the same rate. Here is a spacetime diagram depicting that mutual IRF:

If they wanted to get their clocks synchronized, they would either have to spend a few million years doing it (essentially engaging in radar measurements) or they would have had a prior agreement on how the green traveler would modify his clock to anticipate the black traveler's launch from Earth, in other words, calculate ahead of time how far off the green traveler's clock would be in the coordinate system of the black traveler when he left Earth.

So if we look at either of the above two diagrams and assume both travelers use the radar method to determine the time of the exploding star, the black traveler will establish that it happened at his time of 2 million years at a distance of 2 million light-years, but the green traveler will establish that it happened at his time of 2.1 million years (the average of 1.1 when the radar signal was sent and 3.1 when the radar echo was received) at a distance of 1 million light-years. So they won't agree on where and when the event happened unless they have previously agreed to establish a mutual coordinate system.

By the way, they could have agreed to use their original coordinate system where they synchronized their clocks with Earth and then they would agree that the event of the exploding star happened at the coordinates of 2.5 million years and 2.5 million light-years, just like the first diagram shows.

As I just said, they could have agreed to use their original Earth/Andromeda IRF and then they would not say that any Andromeda clocks are set wrong or have any synchronization errors at all.

Look, I just pointed out that the two travelers have a synchronization difference in their own mutual reference frame unless they proactively do something about it. Synchronization only has meaning when two clocks are inertial and at rest with each other so your assessment doesn't even apply.

This is a confusing statement. If you're talking about Coordinate Times, then something that happened at t1 in a particular coordinate system happened at t1 everywhere else in that coordinate system where the Coordinate Time was equal to t1, whether or not any traveler was present at any of those other places. But if you're talking about Proper Times, then it is not even true unless all the travelers happened to have synchronized their clocks together (assuming, of course, that "in the same reference frame" means "at rest in the same reference frame").

What Andromeda clocks? Aren't we considering Andromeda to have just one clock colocated with Andromeda, whatever that position might be?

Anyone who believes or promotes the idea that one reference frame is wrong just because it has different coordinates for the same event than another reference frame doesn't understand that there is no privileged (or better or unique) reference frame.

Furthermore, it sounds like you are saying that even in the mutual rest frame of the two travelers, the Coordinate Times for an event at Andromeda are different for the two travelers and that simply isn't true, is it?

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16. Feb 1, 2014

### FactChecker

If we are studying "simultanious" events at a distance, the first thing we should do is Einstein-synchronize all clocks in our inertial reference frame. We can hypothesize synchronized clocks in each reference frame and study the problem.

Andromeda is in an inertial reference frame and can Einstein-synchronize all the clocks in that reference frame. That is what I am talking about.

You are right, there is no privileged inertial reference frame. But clocks that are Einstein-synchronized in one inertial reference frame will only agree with Einstein-synchronized clocks in another reference frame at one point along a line of their relative motion. So people in one inertial reference frame will not agree that the Einstein-synchronized clocks in the other reference frame are properly synchronized. If they understand SR, this will not surprise them and they can easily calculate how much "error" there will be in the clocks at each position in the other reference frame.

17. Feb 1, 2014

### ghwellsjr

The issue of simultaneous events at a distance according to an Inertial Reference Frame (IRF) has nothing to do with synchronized clocks. When you transform from one IRF to another, do you think someone has to synchronize or resynchronize any clocks? The whole point of Coordinate Time is that it is not associated with any clocks.

18. Feb 1, 2014

### WannabeNewton

Of course it does. An inertial observer deems a distant event $q$ to be simultaneous with an event $p$ in his vicinity if and only if a clock in the vicinity of $q$ synchronized with the observer's clock reads the same time as the observer's clock at $p$. Using this prescription the inertial observer can build a global time coordinate and from it a global inertial frame.

The whole point of having a lattice of synchronized clocks is to build a global coordinate time relative to an IRF in the first place. Global coordinate time relative to an IRF has no meaning without a lattice of synchronized clocks to begin with.

19. Feb 2, 2014

### ghwellsjr

If there is a clock at the distant event and if it is inertial and if it is at rest with respect to the observer and if it has been previously synchronized with the observer's clock (by radar methods), then, of course, two events at those two clocks must read the same time on those two clocks in order for those two events to be considered simultaneous according to Einstein's convention.

But there doesn't need to be a clock at the distant event in order for the observer (who, of course, must have a clock) to establish such simultaneity. If that were true, then there is no point in discussing simultaneity at large distances where no clock has ever been such as at Andromeda and especially not one that is at rest with the two observers moving toward Andromeda.

Why did you ignore my question:
Taylor and Wheeler in Spacetime Physics (second edition) introduce the lattice of synchronized clocks on page 37 but on page 44 they point out that the lattice is imaginary and conceptual (as if they needed to point that out). The lattice is like training wheels, after you learn to ride, you get rid of them. That's an important step. We don't want to associate Coordinate Time (the basis of simultaneity) with a bunch of clocks pervading the universe.

20. Feb 2, 2014

### ghwellsjr

My statements don't disagree with the lattice of clocks so there's no red flag in my case. The red flag is if someone claims that there must be a lattice of real clocks or even a single clock connected by a rigid rod between an observer and a distant event in order to establish simultaneity.