Is Relative Simultaneity Real?

[Dale]

My geometric "proof" had no concept of time and had infinitely long lines moving

As soon as you have something moving you have a concept of time. Your proof didn't get away from any notion of time, it simply used a Galilean notion, which is perfectly fine geometrically, but disagrees with experiment.

The laterally moving line or a segment in stationary system could be seen as means to synchronise clocks instantaneously without violation of the speed of light limit.

Certainly. Although Einstein's synchronization convention uses light pulses, there are a few other equivalent synchronization mechanisms. Slow clock transport is the most famous, but I agree that this is another.

All I can say about your refutation is that my geometric problem cannot be resolved within
Euclidean Geometry alone because you need to invoke special relativity.

I agree that the problem cannot be resolved within Euclidean geometry alone, in fact, the problem cannot even be stated within Euclidean geometry alone. You need time to have moving lines, so you must have a geometry that includes time. Euclidean geometry does not, so you can pick Galilean or Minkowski geometry.

Perhaps using 3D Classic Geometry in the context of relativity is simply inappropriate and that point could be agreed upon.

I agree. However, to be clear, at any single instant in any frame the 3D space is Euclidean. All of Euclid's axioms and geometry apply at a single instant in any frame.

This can be seen directly from the metric: $ds^2=-dt^2+dx^2+dy^2+dz^2$. In a single instant we have $dt=0$ which clearly leaves the metric for Euclidean geometry.

So it seems quite naively to me that even without possibility of speed greater than light you can still achieve instantaneous synchronistion accomplished by to ends of a line segment

Certainly. But it is only "instantaneous" in one reference frame. This procedure is equivalent to Einstein's method.

I have already pointed out that Lorentz transformation which in my examples require two steps:
a) LX matrix/vector multiplication which preserves parallell lines which is a fundamental theorem in linear transformation theory.

Yes, the Lorentz transform can be represented as a linear transform, and the standard rules of linear algebra apply, and parallel lines do remain parallel in a linear transform. If you take any pair of worldlines which are parallel in one frame and perform a Lorentz transform then they will remain parallel in every other frame.

However, that is not relevant to the discussion here, because in the space where that linear transform exists (Minkowski spacetime) the X-X' axis is a plane, not a line, and that plane is not parallel to the T-T' plane. Please refer to my geometric description in post 32 above.

I was interested in step a) after which the line is still parallel, and the meaning of such fact.

It means that you are still using t, and t is coordinate time in the frame where they are parallel.

Then suddenly this becomes controversial and inappropriate issue in this thread.

There remains plenty to discuss, if you wish. The idea of the line as an equivalent synchronization procedure is interesting, possibly novel. As long as you don't repeat the disproven assertion that it is parallel in other reference frames, then there is nothing otherwise inappropriate with it.

 

[andromeda]

There remains plenty to discuss, if you wish. The idea of the line as an equivalent synchronization procedure is interesting, possibly novel. As long as you don't repeat the disproven assertion that it is parallel in other reference frames, then there is nothing otherwise inappropriate with it.

The idea is not novel although I have come up with it myself.
My later research found publication of Jackson and Pargetter [1] which of course attracted criticism in [2],[3],[4], however I remain unconvinced that the last word was said on this issue. But until I find sufficient presentable argument otherwise, I will shut up.

The good thing in this thread is that you get down from your ivory tower and face the real world and see the points of view different than your own, and if you think your point of view is sound and matters, you have to find the way to convince even the harshest critics.


[1] Jackson, F. and Pargetter, R. Relative Simultaneity in Special Relativity. Phil. Sci. 1977, Vol. 44, 3.
[2] Giannoni, C. Comment on "Relative Simultaneity in the Special Theory of Relativity". Phil. Sci. 1979, Vol. 46, 2.
[3] Torretti, R. Jackson and Pargetter's Criterion of Distant Simultaneity. Phil. Sci. 1979, Vol. 46, 2.
[4] Øhrstrøm, P. Conventionality of Distant Simultaneity. Found. Phys. 1980, Vol. 10, 3/4.
12.

 

[Dale]

The idea is not novel although I have come up with it myself.

Ahh, I know how frustrating that can be!

My later research found publication of Jackson and Pargetter [1] which of course attracted criticism in [2],[3],[4], however I remain unconvinced that the last word was said on this issue.

The fact that there was criticism and the expectation that there will be more discussion can be taken as a given since you are looking in the philosophy literature.

 

[andromeda]

Taking a break

As promised twice already I do not intend to add more to this thread unless there is a significant comment or some new really relevant issue comes to my attention.
The defence and the prosecution have rest their case and the jury is out.

However, if readers wish to comment in any way they can use the forum private messaging mechanism that remains open (I hope). I find every bit of critique or support (if any) very valuable,
and that is why I put this post despite my previous promise.

 

[andromeda]

Need some opinion

While digesting my apparent failure of the parallel line arguments I came to some issue that is related.

I would like an opinion about the following:

Given a long (say one light second long) straight rigid rod and an observer near one end of the rod.

If the observer moves away from the end of the rod in the immediate vicinity, does he instantaneously moves away from the other end or the change of distance is delayed due to speed of light.
More precisely, can the observer moving away from one end of the rod can assume he moves away instantaneously from the other end?
Is the problem formulated properly? Is there a valid explanation in agreement with STR where nothing can happen faster than at the speed of light?

 

[Nugatory]

Is there a valid explanation in agreement with STR where nothing can happen faster than at the speed of light?

"Happen faster than the speed of light" makes no more sense than "happens faster than I can walk" - "happens" isn't a speed so it can't be faster or slower than the speed of light or the speed of anything else.

You mean the observer is standing next to one end of the rod, and at time zero starts moving away from the rod with speed $v$? This is no different than if I take a step to the left and everything to my right is immediately one meter further away, even the remote star that went from being fifty light-years away to being fifty light-years plus one meter away. There's nothing faster than light going on here, our observer just started moving while the rod sits there.

 

[Fredrik]

All you have described is a person moving away from an object. There's an event where that person begins to move away, but to discuss simultaneity, you would have to specify at least two events. And if you're only going to move the person and not the rod, why does the rod need to be rigid?

I suspect that what you were trying to ask is this: Suppose that that guy pushes himself away from the rod by kicking it, so that he moves in one direction, and every part of the rod is simultaneously nudged in the other direction. Will an observer comoving with the rod describe the two events "my end of the rod starts to move" and "that guy starts to move" as simultaneous?

The problem with this scenario is that a perfectly rigid rod contradicts relativity. What would actually happen when you push the rod is that you would only be moving the first layer of atoms, which interacts electromagnetically with the second layer, which interacts with the third, and so on. This creates a longitudinal wave in the rod. Longitudinal waves in a solid are, by definition, sound. So the disturbance propagates at the speed of sound. The other end may not move at all, since the wave loses energy by heating the rod.

It looks like the reason why you asked about this is that you want to know if the observer at the other end will consider the event where your guy started to move, simultaneous with the time when the light from that event reaches him. The answer is no. He will assign time coordinates to these events that differ by L/c, where L is the length he assigns to the rod.

 

[andromeda]

Thanks Nugatory and Fredrik for the response. And you have understood the issue that I have presented in a casual language.

My question in more precise language was like this:

1) Observer is standing next to one end of a very long rod, and at time zero starts moving away from the rod on the straight line designated by the rod with a speed $v$.

2)The rod is rigid in the sense that its length is constant in its own rest frame.

3) The issue is that when visualising close and far ends together and making a step away from them , the observer seems to change the distance to both ends simultaneously no matter how far is the end and I was interested in your opinion about distant simultaneity in such case. The rod would not be disturbed, only the observer moves.

4)I was not yet considering the issue of the rod being moved but that would be the next thing to analyse.

It seems we have no objection to say that the range to every part of the rod or any distant object for that matter changes instantaneously when the observer starts moving.

I find it interesting but not unexpected that to accomplish the same relative motion of the rod by kicking it in one end without touching the observer creates a range of relativity issues including rigid rod contradicting relativity.

I understand the nature of Fredrik's argument such that we have elastic bodies not rigid ones, yet rigid rods are theoretical objects of importance in relativity as can be guessed from Einstein's 1905 paper[1], and declaring rigid rod a contradictory concept may be not right even though a lot of people including Pauli have said that.

I partially question the statement that disturbance propagates at the speed of sound in elastic bodies in classical physics.
Although fully formed disturbance appears to move at the speed of sound, the actual mechanism is that the disturbance builds up rather than moves and all parts of the elastic body act simultneously upon tension force changes as per general solution of the simplified 1D form of the wave equation:
A=F(x-c*t)+G(x+c*t) where A is generalised amplitude and c is the speed of sound.
Tension forces are instantaneous in classical physics and various distant parts of an elastic body slowly accelerate right from the start. Relativistic wave is something I would not go into at this stage.

It looks like the reason why you asked about this is that you want to know if the observer at the other end will consider the event where your guy started to move, simultaneous with the time when the light from that event reaches him. The answer is no. He will assign time coordinates to these events that differ by L/c, where L is the length he assigns to the rod
.

That was not the reason and agree with the above explanation. Many people mix observation delays with simultaneity issues, I do not.
My reason was to get some perspective in the problem of asymmetry in the case when observer moves and the rod moves which plays role in my goal to understand simultaneity. Thanks to you guys I have gained a bit more perspective.

It would be interesting to do some Lorentz transformations in each case to get some definite answers. Acceleration to speed v may be an issue outside of the STR.

______________________________________________________________________________
1.Albert Einstein, “On the Electrodynamics of Moving Bodies” (translation from original Annalen der Physik, 17(1905), pp. 891-921) published on the internet in http://www.fourmilab.ch/etexts/einst...el/specrel.pdf

Just above section I. KINEMATICAL PART:
The theory to be developed is based—like all electrodynamics—on the kinematics
of the rigid body, since the assertions of any such theory have to do
with the relationships between rigid bodies (systems of co-ordinates), clocks,
and electromagnetic processes. Insufficient consideration of this circumstance
lies at the root of the difficulties which the electrodynamics of moving bodies
at present encounters.

 

[Dale]

2)The rod is rigid in the sense that its length is constant in its own rest frame.

3) The issue is that when visualising close and far ends together and making a step away from them , the observer seems to change the distance to both ends simultaneously no matter how far is the end and I was interested in your opinion about distant simultaneity in such case. The rod would not be disturbed, only the observer moves.

Regarding 2. There is a standard sense of the word "rigid" in relativity. It is also called Born rigid, and it describes rigid motion rather than rigid material. So you could say something like "an inertial Born-rigid rod".

Regarding 3. When you use a word like" simultaneously" in relativity, don't forget to specify the reference frame. I assume that you mean the frame of the rod, but it is unclear. In this case you could be asking about the non inertial frame of the observer.

 

[Nugatory]

2)The rod is rigid in the sense that its length is constant in its own rest frame.

That is not physically possible if the rod is being accelerated by pushing on one end. Check out the third paragraph of Fredrik's post 42 in this thread to see the mechanics of what's going on.

Even in classical Newtonian mechanics it's not possible to maintain a constant length of a body that's being pushed at one end. The perfect rigid body that you see in textbook problems is an approximation, but in the real world if you nudge one end of a meter stick it will contract by a nanometer or so before the other end starts to move.

That's small enough that people seldom notice or care, but the effect is still there - and that's one of many reasons why careful authors insist on infinitesimal test particles when describing gravitational and electromagnetic fields.

 

[andromeda]

For he record and convennience of the reader.
A new thread "Confusion over length contraction" that someone has just created:
https://www.physicsforums.com/showthread.php?t=750525
seems to reveal some issues which are relevant to this thread, so my responses there are somewhat complementary.

 

[ghwellsjr]

How about look at it simply, take 4 clocks, 2 in a stationary FOR, 2 in a moving FOR. Let's put these clocks on a train and a platform. One on left side of train one on right, one on left side of platform, one on right. In the platform's FOR, the train and the platform are the same length, the clock distance is also the same. So in the platform's FOR when the left two clocks are at the same point, the right two clocks are also at the same point. Yet in the train's frame the distance between it's clocks and the platforms clocks is different. So in the train's FOR, when the left pair of clocks are in the same place the right pair of clocks can not be in the same place.

So if the left clocks read 0 when they line up, and the right clocks also read 0 when they align. In the train's frame of reference when the left clocks both read 0, the right clocks can't also read 0 since they aren't aligned. Since the math alone isn't convincing you, draw out an actual example of what it would look like in reality, take time dilation, length contraction, and that both frame needs to agree on events. Draw the clocks, draw them moving towards other clocks, look at how time must behave, and what "now" would have to be in different frames.

Thanks. I will look at this example as well. It is good to have different angles on the same problem

Since you have expressed an interest in darkhorror's example, I have decided to make the diagrams he suggested.

The first one is the FOR (Frame Of Reference) in which the platform is at rest and the train is moving to the right at 0.6c. I have depicted the two ends of the platform where there are two clocks in blue and green and the two ends of the train where there are two more clocks in red and black.

At the Coordinate Time of zero, the red and blue clocks are colocated and have a time of zero on them and the black and green clocks are colocated at a different place 8 feet to the right with a time of zero on them. The dots mark off 1 nanosecond increments of Proper Time on all the clocks. You can determine the Proper Time on any clock by counting the number of tick marks above or below the point of colocation. The red and black clocks start at -4 nsec (at the bottom) and go up to +7 nsec (at the top) while the blue and green clocks start at -5 nsec (at the bottom) and go up to +6 nsec (at the top). Please understand that the platform and train existed before and after the brief intervals depicted in the diagram. The speed of light is taken to be 1 foot per nanosecond:

You will note that the Proper Times on the blue and green clocks are synchronized with each other and with the Coordinate Time of the FOR but this cannot be said for the red and black clocks. Note also that the distance between the ends of the train is 8 feet, the same as the distance between the ends of the platform.

Now we transform to the FOR in which the train is at rest and the platform is moving to the left at 0.6c:

Note, as darkhorror mentioned, the distance between the train's clocks is different. It has gone from 8 feet to 10 feet. And the distance between the platform's clocks is also different. It has gone from 8 feet to 6.4 feet. So, he concludes, when the left pair of clocks are in the same place (at the Coordinate Time of 3 nsecs) the right pair of clocks cannot be in the same place (they are separated by 3.6 feet).

Also note, as darkhorror advised, that we have properly taken into account Time Dilation, Length Contraction, and the Relativity of Simultaneity (or as he said, what "now" would have to be in different frames).

He also said that both frames need to agree on events and they do, they just have different coordinates for the events in the two frames as determined by the Lorentz Transformation process.

Now I want to show you another way in which frames need to agree on events and that is how light travels between each pair of events. This is the real significance of doing the transformation between frames. I have drawn in a whole bunch of random light signals going from an event on one line to an event on another line. Note that each of these thin lines (in the color of the thick source line) travels upwards to the left or to the right on a 45-degree diagonal between events. Here is the first FOR with those light signals drawn in:

And here is the second FOR with the same light signals drawn in going at 45-degree angles. You should confirm that both frames depict exactly the same pattern of lines going between the same pairs of events at 45-degree angles:

And there is nothing magic about rest frames. Here is a frame in which both the platform and the train are traveling at the same speed (0.333c) but in opposite directions:

Same results for light traveling between events.

And let's do one for a frame traveling at -0.333c with respect to the first one:

One other aspect of these light signals is to show that an observer located with anyone of the clocks sees exactly the same thing in all frames. The light signal carries the information of how each observer sees the other clocks. So, for example, when the red clock is at +4 nsec, it sees the black and green clocks colocated with zero on both of them. Another example: when the blue clock is at +4 nsecs, it sees the red clock at +2 nsec, the black clock at -2 nsecs and the green clock at -4 nsecs. Veryify that this is true in all frames.

Does this all make sense to you? Does it help? Any questions?

 

[andromeda]

Does this all make sense to you? Does it help? Any questions?

I will respond as soon as I can.

Thank you very much for help and for going to such length in your presentations here and in another thread.

The pictures are worth thousand words. It will take me a while to come up with a meaningful response to this and the other related thread posts. Unfortunately I cannot do relativity for living and right now I need to attend my less glamorous yet still interesting daily job, where distant things appear simultaneous no matter what (small) speed, where twins talk over radio with no delay

 

[PAllen]

I will respond as soon as I can.

Thank you very much for help and for going to such length in your presentations here and in another thread.

The pictures are worth thousand words. It will take me a while to come up with a meaningful response to this and the other related thread posts. Unfortunately I cannot do relativity for living and right now I need to attend my less glamorous yet still interesting daily job, where distant things appear simultaneous no matter what (small) speed, where twins talk over radio with no delay

But every magnet you have ever held demonstrates a relativistic effect ...

 

[Nugatory]

where twins talk over radio with no delay

As long as they aren't Apollo astronauts... Dunno if you were around for that, but the time delay in in radio conversations between Earth and moon was very clearly perceptible. No relativistic effects were involved here, but the demonstration that radio doesn't travel instantaneously was very convincing.

 

[ghwellsjr]

where distant things appear simultaneous no matter what (small) speed, where twins talk over radio with no delay

But when they talk on their cell phones, there is a small delay but it is not generally noticeable unless they are in the same room and can hear each other directly plus on the phone and then it is very disturbing. My point is that "small delay" is not "no delay".

Also, simultaneity is not an appearance. You cannot determine simultaneity of distant events simply by observing. That's one of the things I tried to point out in my previous post: No matter what frame you use to depict a scenario, all appearances remain the same even though the simultaneity of distant events has changed enormously.

Consider my last two diagrams: the order of the two colocations is different but nobody can tell, can they, just by looking? And in the first two diagrams where the two events are simultaneous in the rest frame of the platform but not in the rest frame of the train, observers on the platform or on the train still cannot determine the simultaneity of those two events. Everything appears identically to each of them no matter what frame we use to describe the scenario.

When you get a chance, please study those diagram, they could help you get over some of your misconceptions.

EDIT: I guess I'm going to have to make some more diagrams showing how observers on the platform or on the train can establish simultaneity in their own rest frames.

 

[andromeda]

But every magnet you have ever held demonstrates a relativistic effect ...

That was only a joke as you can see from the smiley face. I don't challenge SR - only interpret it to find consistency with the common sense.

 

[Dale]

Again common sense is not relevant to relativistic physics. It evolved for different purposes, like fighting and fleeing.

 

[nearlynothing]

If i understand the scenario you described in the first post correctly, you¿re asking why both ends of the rod would cross the x-axis simultaneously and also the x' axis simultaneously if simutaneity is a relative thing.

The thing is that the EVENTS that represent the crossing of the x-axis simutaneously in the unprimed coordinate system are in fact different from those events that represent the crossing of the primed axis simultaneously in the primed system of coordinates.

If instead of just the rod you had a rod with lights at both its ends, and you arranged things so that they flashed when they crossed the x-axis in the unprimed system, they would both flash simultaneously in that system, now go to the other system, they will NOT flash simultaneously.

Those events are no longer simultaneous in the primed system. Of course you can always find a pair of events in the world-line of the ends of the rods that happen sumiltneously on the x' axis or on the x axis, but these can't be the same two events for both coordinate systems if the systems are moving relative to each other.

 

[andromeda]

Of course you can always find a pair of events in the world-line of the ends of the rods that happen sumiltneously on the x' axis or on the x axis, but these can't be the same two events for both coordinate systems if the systems are moving relative to each other.

There will always be two real physical events in the case you describe. One flash per one end of the rod. They will only be interpreted separately by two systems.

 

[nearlynothing]

There will always be two real physical events in the case you describe. One flash per one end of the rod. They will only be interpreted separately by two systems.

Yeah they are two different events, but the fact that there are flashes changes nothing, you had 2 different physical events too, namely the crossing of the rod´s ends with the x axis.
And yeah relative simultaneity refers to how different observers measure it.
Or how was my example any different from your scenario? or did i just answer the wrong question

Edit: I made a mistake when i said you could always find a pair of events on the world-lines of the ends of the rod that are simultaneous on the x' axis. Well, you cant, my bad lol

But yeah the rest of my argument is still valid.

 

[andromeda]

Or how was my example any different from your scenario? or did i just answer the wrong question
...
But yeah the rest of my argument is still valid.

Your example is not different. Light flashes coinciding with line crossing is another pair of events that can be useful because light can be captured elsewhere why line crossing is only where it happens.

In the future I intend to respond to summarise all arguments and at this stage I do not exactly know what that response's outcome will be, but it should be conclusive since every discussion should have its end.

More references to your post below:

If i understand the scenario you described in the first post correctly, you¿re asking why both ends of the rod would cross the x-axis simultaneously and also the x' axis simultaneously if simutaneity is a relative thing.

Your understanding of my problem is basically correct because intuitively the rod fully aligns with x at some stage, then because x' is aligned with x. There is no doubt Lorentz transformation shows two successive events on x' while rod is crossing the line. I do not question that, but what does it really mean? This has triggered the discussion in which I still have to give my response after considering all arguments and it may take some time.

The thing is that the EVENTS that represent the crossing of the x-axis simutaneously in the unprimed coordinate system are in fact different from those events that represent the crossing of the primed axis simultaneously in the primed system of coordinates.
.

True. The two distant events in the primed for the same clock indication t' will be successive in the unprimed.

If instead of just the rod you had a rod with lights at both its ends, and you arranged things so that they flashed when they crossed the x-axis in the unprimed system, they would both flash simultaneously in that system, now go to the other system, they will NOT flash simultaneously.

This is a conclusion from analysing event time after Lorentz transformation and when simultaneity is judged by the same clock time. But the clocks can be synchronised in many different ways and the time of events on each end of the rod aligning with x can be anything. Lorentz transformation is derived by Einstein in his 1905 paper based on a particular synchronisation method, and this is not the only one possible method while still being rational and consistent.
As I said, I am working towards a summary response to all arguments presented in this thread.

 

[Dale]

Of course you can always find a pair of events in the world-line of the ends of the rods that happen sumiltneously on the x' axis or on the x axis

Most of this post is correct, but this point is a small mistake. The rod is not parallel to the x' axis, so only one end crosses at a time in the primed frame.

Edit: I think that is what you were referring to in your edit of post 56.

 

[nearlynothing]

Most of this post is correct, but this point is a small mistake. The rod is not parallel to the x' axis, so only one end crosses at a time in the primed frame.

Edit: I think that is what you were referring to in your edit of post 56.

yup that's what i meant in my edit to that post, I was thinking about the rod as parallel to the x' axis for some reason. Too much time ignoring 2 of the 3 space directions while making space-time diagrams i guess.
Also cause of the same wrong way of thinking about the positioning of the rod I said that the events representing the crossings in both inertial frames are different, that's also very wrong. Both events are the same events of course.

 

[Dale]

No worries it seems like you are starting from a decent foundation, and such minor errors will be somewhat self correcting

 

[ghwellsjr]

Also, simultaneity is not an appearance. You cannot determine simultaneity of distant events simply by observing. That's one of the things I tried to point out in my previous post: No matter what frame you use to depict a scenario, all appearances remain the same even though the simultaneity of distant events has changed enormously.

Consider my last two diagrams: the order of the two colocations is different but nobody can tell, can they, just by looking? And in the first two diagrams where the two events are simultaneous in the rest frame of the platform but not in the rest frame of the train, observers on the platform or on the train still cannot determine the simultaneity of those two events. Everything appears identically to each of them no matter what frame we use to describe the scenario.

When you get a chance, please study those diagram, they could help you get over some of your misconceptions.

EDIT: I guess I'm going to have to make some more diagrams showing how observers on the platform or on the train can establish simultaneity in their own rest frames.

Remember, we are taking darkhorror's scenario from post #18:

How about look at it simply, take 4 clocks, 2 in a stationary FOR, 2 in a moving FOR. Let's put these clocks on a train and a platform. One on left side of train one on right, one on left side of platform, one on right. In the platform's FOR, the train and the platform are the same length, the clock distance is also the same. So in the platform's FOR when the left two clocks are at the same point, the right two clocks are also at the same point.
...
So if the left clocks read 0 when they line up, and the right clocks also read 0 when they align. In the train's frame of reference when the left clocks both read 0, the right clocks can't also read 0 since they aren't aligned.

I already drew two diagrams in post #47, the first one for the rest frame of the platform where each pair of clocks are colocated and display their Proper Times of zero at the Coordinate Time of zero:

...and the second one transformed to the rest frame of train where none of the colocated clocks displaying Proper Times of zero are aligned with the Coordinate Time of zero:

Now I want to pick up in the rest frame of the train where I will drawn in outgoing radar signals from the Red and Black observers at either end of the train and the reflections off of a couple objects to show how they can establish which remote events are simultaneous to the events of the Proper Times on their own clocks.

It's important to realize that an observer is constantly and continually sending out radar signals but we don't want to draw every one of them in our diagrams because it would be far too cluttered. Instead, I'm just going to draw in a few that will illustrate how the observer establishes simultaneity of distant events to events at his own local clock. The observer has to wait for the radar signals coded with his sent time to bounce off an object and return an echo signal, along with an image of the object identifying a specific event. In our case, we will use the Proper Time on a particular remote clock. Please realize that we are not concerned with the actual Proper Time on the remote clock, only with the fact that it identifies separate events at the location of the remote clock. When the observer receives an echo along with the image of the remote event, he logs the coded sent time, the received time and the time he sees on the remote clock. After he collects a lot of data, he goes back and looks at his logs, makes an assumption based on Einstein's second postulate and does some calculations to establish the simultaneous events. Hopefully, this will make sense as we work through the examples.

We'll start with the Red observer at the rear of the train sending radar signals to the Blue clock at the left end of the platform:

Even though the first radar signal he sent out was at his Proper Time of -14 nsec, he doesn't detect the echo until his Proper Time of -3.5 nsec. Here is his log of the data going from his Proper Time of -3.5 nsec to his Proper Time of -1 nsec:

Sent	Rcvd	Blue's
Time	Time	Time
-14	-3.5	-7
-12	-3	-6
-10	-2.5	-5
-8	-2	-4
-6	-1.5	-3
-4	-1	-2

His next step is to average the Sent Time and the Received Time and make a new column which identifies the established time of the measurement. This process is applying Einstein's convention that the radar signal takes the same amount of time to get to a target as the echo signal takes to get back. So here is a new table with Red's Time added that he established for each radar sent/received signal:

Sent	Rcvd	Blue's	Red's
Time	Time	Time	Time
-14	-3.5	-7	-8.75
-12	-3	-6	-7.5
-10	-2.5	-5	-6.25
-8	-2	-4	-5
-6	-1.5	-3	-3.75
-4	-1	-2	-2.5

Now he's got a list of simultaneous events according to his established rest frame in the last two columns. For example, the event of his own clock displaying -8.75 is simultaneous with the event of Blue's clock displaying -7. And if you look at the above diagram, you can see that they both have the Coordinate Time of -5.75 nsecs in the train's rest frame. But note that the Red observer has no awareness of the Coordinate Time, he is basing this on his own Proper Time.

OK, does this all make sense? Now let's show another diagram where the Red observer is sending radar signals to the Black clock at the front end of the train. You should be aware that he is sending both sets of signals at the same time (the ones shown in the previous diagram and this one) but we're just showing them on two separate diagrams to avoid clutter:

And here is his completed list including his calculated average for his established time of the measurements:

Sent	Rcvd	Black's	Red's
Time	Time	Time	Time
-16	4	0	-6
-15	5	1	-5
-14	6	2	-4
-13	7	3	-3
-12	8	4	-2

Now what the Red observer can do is look in both lists and find examples where a time in the last column from one list matches the time in the last column of the other list and that will allow him to identify two remote events for different objects that are simultaneous. Keep in mind that in a real situation, his list would be vastly longer and include matches for every row but in our very sparse example, we can identify one example where the Red's Time of -5 is simultaneous with Blue's Time of -4 and Black's Time of 1.

Now we can combine the important signals from the above two diagrams and show how the Red observer establishes the simultaneity of those three events:

As a side note, we can also show how the Red observer establishes that Black's Time of zero is not simultaneous with his own time of zero because that remote event occurred at his time of -6 based on the information from the previous list.

Next I want to illustrate a very important characteristic of simultaneous events: all inertial observers at rest with each other will establish the same set of simultaneous events no matter how their individual clocks are set (or even if their own clocks tick at different rates, which I will not show). In other words, simultaneity, as established by an observer, has nothing to do with the synchronization of clocks or even with the existence of any clocks beyond his own individual clock.

I'm going to now do a similar thing with the Black observer to show the radar measurements he makes to establish the same set of simultaneous events that the Red observer established since they are mutually at rest. We start with the same rest frame of the train but with radar signals emitted by the Black observer to reflect off the Red clock:

And here is his list:

Sent	Rcvd	Red's	Black's
Time	Time	Time	Time
-10	10	-6	0
-9	11	-5	1
-8	12	-4	2
-7	13	-3	3
-6	14	-2	4

Another diagram showing the Black observer's signals bouncing off the Blue clock:

And the corresponding list:

Sent	Rcvd	Blue's	Black's
Time	Time	Time	Time
-8	0	-8	-4
-7	4	-6	-1.5
-6	8	-4	1
-5	12	-2	3.5

By comparing both lists, we see that Black's Time of 1 is simultaneous with Red's Time of -5 and Blue's Time of -4, the same as Red established.

And here is a diagram showing just the significant radar signals the Black sends and receives:

I hope this shows in a clear and understandable way how simultaneity is established by an inertial observer and that it takes far more than simply observations. It takes sending and receiving radar signals, logging their times, the assumption of Einstein's second postulate when calculating average times and the comparison of results.