Understand Special Relativity and Time paradox

[Dale]

I think that Minkowski diagrams are pretty well established and understood in the field of special relativity, as is the concept of simultaneous spaces.

Yes, but apparently not by everyone. Particularly since we get many novices and students, it is a point that bears mentioning and you didn't so I did.

 

[Austin0]

Hi, Austin0. What I think (if I haven't messed up) is that I've used the Lorentz transformations to develop a Minkowski diagram for the twin's outgoing trip (up to the point of the turnaround) and then, picking up the traveling twin's trip after completion of turnaround, showing the diagram for the return trip.

The diagram shows the intersection of the simultaneous space of the outgoing X1 axis with the 2nd Red worldline right at the start of the outgoing trip. And then it shows the intersection of the outgoing X1 axis with the 2nd Red worldline right after the turnaround is complete. If one would like to nit-pick, they could point to the round off of the total round trip time given as 10 years for the traveling twin (the small turnaround time was not included, which would add some decimal value to the total 10 year number).

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Well I don't think you "messed up" as this is a perfectly standard chart for an instant turnaround. While I am sure all agree this is an accurate charting of two separate inertial frames the question at hand is how this relates to a single extended accelerated frame, yes?

The diagram shows the intersection of the simultaneous space of the outgoing X1 axis with the 2nd Red worldline right at the start of the outgoing trip. And then it shows the intersection of the outgoing X1 axis with the 2nd Red worldline right after the turnaround is complete.

Yes this is self evident but does not address my simple explicit question.

Given a single extended accelerated physical system of clocks and rulers do you think that right after turnaround this system would be congruent with x₁ intersecting event A?
Would there, or not, be a traveler at A with a clock reading of t'=5+(a hair)??

 

[bobc2]

Well I don't think you "messed up" as this is a perfectly standard chart for an instant turnaround. While I am sure all agree this is an accurate charting of two separate inertial frames the question at hand is how this relates to a single extended accelerated frame, yes?

You miss the point here. I explicitly indicated that this is not a chart for an instant turnaround. I emphasized that the curved path portion is so small on this scale that I couldn't represent it with the limited chart space. The Lorentz frames with the simultaneous spaces indicated actually occur before the turnaround and then after the turnaround.

If you don't get this sketch, I could easily select simultaneous spaces much farther away from the start of the trip and from the turnaround. I didn't think this would be such a problem. I'm not showing an accelerated frame at all, so that's not relevant here.

Yes this is self evident but does not address my simple explicit question.

Given a single extended accelerated physical system of clocks and rulers do you think that right after turnaround this system would be congruent with x₁ intersecting event A?
Would there, or not, be a traveler at A with a clock reading of t'=5+(a hair)??

I'm not talking about a single accelerated physical system here. In an earlier post, I analyzed the turnaround using a sequence of incremental inertial frames. We got tangled up with straw men, so now I've simplified the discussion to avoid arguing over single accelerated systems. And the fundamental point illustrated using two frames (outgoing and returning) which are clearly not accelerated is still the same as what I was trying to illustrate all along--just the fundamental concept of simultaneity and the interesting feature of events A, B, and C. Again, those features do not have to be interesting to you or anyone else. It was just a comment in the event anyone else might be interested.

 

[bobc2]

Yes, but apparently not by everyone. Particularly since we get many novices and students, it is a point that bears mentioning and you didn't so I did.

Do you have a problem with Minkowski space-time diagrams in general? Or, is it just when someone refers to the "simultaneous spaces" that show up in the space-time diagrams?

 

[Dale]

Do you have a problem with Minkowski space-time diagrams in general? Or, is it just when someone refers to the "simultaneous spaces" that show up in the space-time diagrams?

Neither. I have a problem when people make mathematically invalid statements and persist in doing so when their error is pointed out to them.

A "sequence of simultaneous spaces" is a simultaneity convention, in this case a non-inertial one. You may try to disguise it all you like, but that is what you are doing. Inventing new terms like "3D worlds" and "simultaneous spaces" doesn't change what it is.

 

[Dale]

And the fundamental point illustrated using two frames (outgoing and returning) which are clearly not accelerated is still the same as what I was trying to illustrate all along--just the fundamental concept of simultaneity and the interesting feature of events A, B, and C.

In neither of those frames does B come before A nor C before B.

 

[bobc2]

In neither of those frames does B come before A nor C before B.

We may not be talking about the same thing. I've made it clear before that in the 2nd Red guy's frame, as the Red guy moves along his worldline, he encounters event A first, then event B, then event C.

However, as the blue guy moves along his worldline, the event B is presented to his outgoing simlultaneous space first (right at the start of the outgoing trip). Then. event A is presented to blue's simultaneous space just after blue completes his turnaround.

Also, just before blue enters his turnaround path, event C is presented to blue's simlultaneous space. Then, event A is not presented to the blue simultaneous space until after the turnaround is complete.

 

[Dale]

We may not be talking about the same thing. I've made it clear before that in the 2nd Red guy's frame, as the Red guy moves along his worldline, he encounters event A first, then event B, then event C.

Not just the red guys frame, but all inertial frames.

However, as the blue guy moves along his worldline, the event B is presented to his outgoing simlultaneous space first (right at the start of the outgoing trip). Then. event A is presented to blue's simultaneous space just after blue completes his turnaround.

Also, just before blue enters his turnaround path, event C is presented to blue's simlultaneous space. Then, event A is not presented to the blue simultaneous space until after the turnaround is complete.

And here you go with your non inertial frame.

A sequence of simultaneous spaces is a simultaneity convention. And I have already told you that the naive simultaneity convention used here cannot cover the red worldline because it violates the few mathematical requirements of a coordinate system.

Your statement is mathematically invalid, as I pointed out well over 100 posts ago. I don't know why you persist in it.

 

[zonde]

I would like to bring up this earlier comment. I am trying to understand bobc2's position.

You should understand that most forum members here would insist that the Andromeda Paradox does not represent anything about physical reality. It could be taken as a pedagogical illustration to help graphically visualize aspects of the mathematics of special relativity. They consider that there are other equally valid interpretations of special relativity, such as the Lorentz Aether theory, that are in conflict with any notion that the Andromeda Paradox illustrates something about reality--thus, in that view, the block universe is not to be taken as a true characterization of external physical reality.

Paradoxes do not represent reality. Paradox reveals contradiction and we do not allow idea that reality could be self contradictory. Only models of reality can be self contradictory (and therefore flawed).

So I would ask in what model Andromeda Paradox is supposed to appear.

 

[PAllen]

However, as the blue guy moves along his worldline, the event B is presented to his outgoing simlultaneous space first (right at the start of the outgoing trip). Then. event A is presented to blue's simultaneous space just after blue completes his turnaround.

Also, just before blue enters his turnaround path, event C is presented to blue's simlultaneous space. Then, event A is not presented to the blue simultaneous space until after the turnaround is complete.

And this is the core of disagreement. You speak of blue's simultaneous space as if it has some physical meaning. Further, since blue, after turnaround, has a different past than the past of the post turnaround inertial frame, any physical procedure defining simultaneity will come out different for the blue observer than for an observer always at rest in the post turnaround inertial frame. Finally, even as a mathematical convention, talking about blue's simultaneous spaces does imply an overall simultaneity convention for the blue world line. For this, there are mathematical requirements - any region where a proposed simultaneity convention for blue has intersecting surfaces is outside the domain of that convention. If you want to talk about a blue simultaneity for such a region, you must adopt a different convention that does not have intersecting surfaces - of which there are many.

 

[Austin0]

Quote by Austin0

Well I don't think you "messed up" as this is a perfectly standard chart for an instant turnaround. While I am sure all agree this is an accurate charting of two separate inertial frames the question at hand is how this relates to a single extended accelerated frame, yes?

You miss the point here. I explicitly indicated that this is not a chart for an instant turnaround. I emphasized that the curved path portion is so small on this scale that I couldn't represent it with the limited chart space. The Lorentz frames with the simultaneous spaces indicated actually occur before the turnaround and then after the turnaround.

If you don't get this sketch, I could easily select simultaneous spaces much farther away from the start of the trip and from the turnaround. I didn't think this would be such a problem. I'm not showing an accelerated frame at all, so that's not relevant here.

yes of course ,instant turnaround is a convenient idealization,
it was understood that this depicted inertial phases before and after turnaround.
When you attribute the x ₁ axis to the traveler you are implicitly applying it to an accelerated frame. I.e. there is no turnaround without acceleration. So whether the traveler frame is accelerating at that time is not relevant , In the context of the overall trip it is non-inertial.
Quote by Austin0

Yes this is self evident but does not address my simple explicit question.

Given a single extended [edit out-accelerated] co-moving physical system of clocks and rulers do you think that right after turnaround this system would be congruent with x1 intersecting event A?
Would there, or not, be a traveler at A with a clock reading of t'=5+(a hair)??

I'm not talking about a single accelerated physical system here. In an earlier post, I analyzed the turnaround using a sequence of incremental inertial frames. We got tangled up with straw men, so now I've simplified the discussion to avoid arguing over single accelerated systems. And the fundamental point illustrated using two frames (outgoing and returning) which are clearly not accelerated is still the same as what I was trying to illustrate all along--just the fundamental concept of simultaneity and the interesting feature of events A, B, and C. Again, those features do not have to be interesting to you or anyone else. It was just a comment in the event anyone else might be interested.

Well you have completely avoided answering my question again.
Whether or not you are talking about a single accelerated system I am asking your thought regarding the x₁ axis as it would apply to such a system (with the edit above).

If there was such a co-moving system at that time after turnaround (inertial) would it correspond (be congruent) to the x₁ axis in your chart?
Would there, or not, be a traveler at A with a clock reading of t'=5+(a hair)??

 

[zonde]

When we speak about reality, do we mean only single moment of space or do we include all past and all future?
I believe that with reality we mean single slice of spacetime i.e. we do not include all past and all future.

 

[PAllen]

When we speak about reality, do we mean only single moment of space or do we include all past and all future?
I believe that with reality we mean single slice of spacetime i.e. we do not include all past and all future.

I value my past and future possibilities perhaps more than you More seriously what slice? Through a given event (e.g. me hitting submit for this post), there are uncountably infinite spacelike slices.

 

[bobc2]

You folks still seem to be troubled by the approximaty of the simultaneous spaces to the accelerated turnaround point in my sketches. So, here is a sketch where we consider the simultaneous spaces far far removed (years and millions of miles) from the turnaround neighborhood. We still have the same interesting feature about the order of events. Event A occurs before event B in the 2nd Black's rest frame. However, for the traveling twin moving along his worldline, event B is presented to his return trip simultaneous space before it is presented to his outgoing simultaneous space.

To emphasize the distinction between the outgoing frame and the return trip frame (not a single acceleration frame), I've colored the outgoing frame blue and the return frame red. The stay-at-home twin has the worldline along the black X4 axis.

 

[Dale]

However, for the traveling twin moving along his worldline, event B is presented to his return trip simultaneous space before it is presented to his outgoing simultaneous space.

Again, as we have already covered over and over again and again, your comments about simultaneous spaces are defining a simultaneity convention, and that convention is non-inertial. All of my previous comments hold. This approach violates the mathematical requirements in the region of the 2nd (now) black observer, so it is not a valid simultaneity convention for that region. If my many posts on this topic were not sufficiently clear, then please read PeterDonis' post 190, which is very well written.

You seem to think that I am having difficulty understanding your point. I understand your point quite clearly. Your point is not unclear, it is wrong.

To emphasize the distinction between the outgoing frame and the return trip frame (not a single acceleration frame), I've colored the outgoing frame blue and the return frame red. The stay-at-home twin has the worldline along the black X4 axis.

In neither the blue frame nor in the red frame does B come before A.

EDIT: oops, it is PAllen's post 190

 

[PeterDonis]

You folks still seem to be troubled by the approximaty of the simultaneous spaces to the accelerated turnaround point in my sketches...We still have the same interesting feature about the order of events.

Bobc2, a while back you did say that this "reversal of time" was just an interesting feature, nothing more. But now you appear to be saying that it *is* more; that there is some genuine physical meaning to the "interesting feature". And the pushback you are getting is because of the obvious paradoxical consequences of such a claim. So which is it?

 

[PAllen]

You folks still seem to be troubled by the approximaty of the simultaneous spaces to the accelerated turnaround point in my sketches. So, here is a sketch where we consider the simultaneous spaces far far removed (years and millions of miles) from the turnaround neighborhood. We still have the same interesting feature about the order of events. Event A occurs before event B in the 2nd Black's rest frame. However, for the traveling twin moving along his worldline, event B is presented to his return trip simultaneous space before it is presented to his outgoing simultaneous space.

To emphasize the distinction between the outgoing frame and the return trip frame (not a single acceleration frame), I've colored the outgoing frame blue and the return frame red. The stay-at-home twin has the worldline along the black X4 axis.

The question is the occurrence of this 'interesting feature' is part of the definition of the applicability of this simultaneity convention to the non-inertial observer. That is, the maximal spacetime region in which this simultaneity convention is applicable is the region in which there are no intersections of simultaneity surfaces. I could describe additional, physical plausibility criteria as well, but it is at least mathematically valid to use such a convention for the non-inertial observer as long as you don't try to cover a region of spacetime including such intersections.

Of course, I remain convinced that, even where applicable, making such statements as 'this is where the distant clock really runs faster than mine' are physically meaningless and conceptually grossly misleading.

 

[bobc2]

Again, as we have already covered over and over again and again, your comments about simultaneous spaces are defining a simultaneity convention, and that convention is non-inertial. All of my previous comments hold.

We will just have to agree to disagree. You concept of non-inertial simply does not apply to the two separate blue and red inertial frames in the above sketch.

This approach violates the mathematical requirements in the region of the 2nd (now) black observer, so it is not a valid simultaneity convention for that region

Neither of the separate individual frames violates the mathematical requirements. If I had been talking about a single non-inertial coordinates, I might have tried using Rindler coordinates or something, but then I would have to explain the Rindler horizen, etc. But, we are confronted with no such situation here.

If my many posts on this topic were not sufficiently clear, then please read PeterDonis' post 190, which is very well written.

There is no PeterDonis post no. 190.

You seem to think that I am having difficulty understanding your point. I understand your point quite clearly. Your point is not unclear, it is wrong.

In that case, your point is also clear--it is just wrong.

In neither the blue frame nor in the red frame does B come before A.

Just look at the space-time diagram. The intersections of the blue and red simultaneous spaces with the 2nd black worldline are there to see. There can be no mistaken about where the intersections are.

 

[bobc2]

The question is the occurrence of this 'interesting feature' is part of the definition of the applicability of this simultaneity convention to the non-inertial observer. That is, the maximal spacetime region in which this simultaneity convention is applicable is the region in which there are no intersections of simultaneity surfaces. I could describe additional, physical plausibility criteria as well, but it is at least mathematically valid to use such a convention for the non-inertial observer as long as you don't try to cover a region of spacetime including such intersections.

Of course, I remain convinced that, even where applicable, making such statements as 'this is where the distant clock really runs faster than mine' are physically meaningless and conceptually grossly misleading.

This makes no sense to me. By that logic you must dismiss the use of Minkowski space-time diagrams entirely, since by that definition there is probably no object in the universe that is inertial. All objects have accelerated at one time or another along the history of its worldline.

 

[bobc2]

Bobc2, a while back you did say that this "reversal of time" was just an interesting feature, nothing more. But now you appear to be saying that it *is* more; that there is some genuine physical meaning to the "interesting feature". And the pushback you are getting is because of the obvious paradoxical consequences of such a claim. So which is it?

Yes, an interesting feature. However, I did say that there is a sense in which one could consider his simultaneous space (as described by a Minkowski space-time diagram) as special. I described the experiences an observer could only have by experiencing these things in a Minkowski space-time frame. For example he measures the speed of light to be c, as do all other observers moving relative to him. And he experiences the laws of physics, as do all other observers living in Minkowski space-time frames. If he did not "live" in a Lorentz-Minkowski-Einstein frame he would not have those experiences, so that is the sense of which I referred.

However you misinterpreted my use of the word "experienced." Webster's dictionary gives two or three definitions for the use of that term. You chose the wrong definition where I thought the context made the definition I was applying very clear.

I'm not insisting anyone embrace that sense of specialness. Of course forum members can consider that observation or dismiss it--whatever their preference.

 

[Dale]

There is no PeterDonis post no. 190.

Oops, my apologies. The excellent post I was referring to was PAllen's.

 

[PeterDonis]

I described the experiences an observer could only have by experiencing these things in a Minkowski space-time frame. For example he measures the speed of light to be c, as do all other observers moving relative to him. And he experiences the laws of physics, as do all other observers living in Minkowski space-time frames.

But that has nothing to do with whether a particular simultaneous space is "special". All of these experiences are local; all they show is that whatever "reality" is, it looks locally like Minkowski spacetime. But a simultaneous space isn't local; that's the whole point.

 

[Dale]

We will just have to agree to disagree.

This isn't a matter of opinion. Your position is mathematically wrong. Try to write it down mathematically and you will see that you violate the one-to-one requirement.

You concept of non-inertial simply does not apply to the two separate blue and red inertial frames in the above sketch.

Neither of the separate individual frames violates the mathematical requirements.

I never said that either the blue or the red frames are non-inertial. I said that the "simultaneous spaces" which you keep talking about define a non-inertial frame. If you restrict your comments to inertial frames and stop discussing the sequence of "simultaneous spaces" then it is clear that A comes before B in every possible inertial frame, including the blue and red ones.

If I had been talking about a single non-inertial coordinates, I might have tried using Rindler coordinates or something, but then I would have to explain the Rindler horizen, etc. But, we are confronted with no such situation here.

Yes, we are talking about non-inertial coordinates. Every time you bring in your sequence of "simultaneous spaces" idea for the traveling twin you are defining a simultaneity convention for a non-inertial frame. It is a perfectly valid simultaneity convention, but it does not cover the entire spacetime. Your problem is that you continue to try to apply it in a region of the spacetime that it cannot cover because it violates the mathematical requirements in that region.

In that case, your point is also clear--it is just wrong.

I can back mine up with math and references if you wish. Can you do the same?

Just look at the space-time diagram. The intersections of the blue and red simultaneous spaces with the 2nd black worldline are there to see. There can be no mistaken about where the intersections are.

And here you go from talking about inertial frames to talking about simultaneous spaces, thereby forming a non-inertial frame which is invalid in the region of the 2nd black worldline.

 

[PAllen]

This makes no sense to me. By that logic you must dismiss the use of Minkowski space-time diagrams entirely, since by that definition there is probably no object in the universe that is inertial. All objects have accelerated at one time or another along the history of its worldline.

Nope. Firstly, anyone can use any inertial frame for any SR analysis of a given scenario, involving any number of objects, distances, times. This is the the most practical approach. Choose the inertial frame for convenience (e.g. COM frame for many kinematic problems). There is no requirement I ever use a frame in which I am at rest.

A different question is what is 'experienced' as a simultaneity surface. You can approach this mathematically or physically.

I prefer physically, and note that there is a well defined 'frame' for an observer when/where different physically reasonable simultaneity definitions agree (to some desired precision). Thus, agreement to some precision between radar simultaneity and Born rigid ruler simultaneity defines the size of a physically meaningful frame for an observer. The longer since your last significant (to desired precision) deviation from inertial, the larger the spatial extent of your physically meaningful simultaneity slice.

Mathematically, along a world line, you can use whatever spacelike surfaces you want as simultaneity slice, for a region of spacetime in which they don't intersect. If you want to cover a region where one choice has intersections, choose a different set of surfaces that don't intersect there.

 

[bobc2]

Bear with me a little more here. I'm trying to get my head into your concept and use of Lorentz frames and simultaneous spaces.

Focus just on the stay-at-home twin. Is he in an inertial frame? Does he exist in a continuous sequence of simultaneous spaces, each one parallel to his X1 axis?

 

[Dale]

Focus just on the stay-at-home twin. Is he in an inertial frame?

He is in all inertial frames. Additionally, he is at rest in an inertial frame.

Does he exist in a continuous sequence of simultaneous spaces, each one parallel to his X1 axis?

Simultaneity is a convention. You can certainly choose that convention if you like, but you don't have to.

Also, there is no empirically discernable sense in which he exists in one simultaneity convention and not in another. If he exists, then he exists regardless of which simultaneity convention you adopt.

 

[ghwellsjr]

Very good graphics, ghwellsjr. Your graphics make it very clear. Good job.

Thanks, I'm glad you like them.
...

The sketch below gets messy, but it illustrates a couple of more details that one may or may not find interesting. Notice that the event A on the 2nd Red stay-at-home guy (displaced from the first red twin) is presented to the returning twin’s trip simultaneous space before it is presented to the outgoing twin’s simultaneous space. Notice that this does not in any way imply that the 2nd Red guy's time is flowing backwards for that Red guy sitting at rest in his own black inertail frame. It's just a feature of special relativity and is no more mysterious than the two twins having different ages after they reunite.

By the way, the blue dots on the traveling twin's worldline are placed with same proper time increments as the black worldline dots (one year intervals of proper time on both worldlines, in accordance with your preference). The hyperbolic calibration curves show the five year lapses.

I have attempted to replicate your drawing without the extra lines:

Now I transform to the IRF in which the traveling twin is stationary during the outbound portion of his trip:

And the transform to the IRF in which the traveling twin is stationary during the inbound portion of his trip:

Now how do you get information from these last two IRF's to draw the extra lines in the first IRF and to come to the conclusions that you do?

 

[bobc2]

Thanks, I'm glad you like them.
...

I have attempted to replicate your drawing without the extra lines:

Now how do you get information from these last two IRF's to draw the extra lines in the first IRF and to come to the conclusions that you do?

Once again a very good job. You have presented all of the information in an easily understood picture that accounts very well for the difference in the twins' ages at the reunion. And this of course is exactly what was originally requested at the beginning of this thread. No additonal comments were really needed at that point.

I had originally simply tried to give the discussion a larger context by expanding the picture by adding frame coordinates for the outgoing and return trip frames. Of course subsequent comments (beginning with an observation by Vandam) led to the addition of a second Red guy in the rest frame with events A and B and the intersections of the blue X1 and X1' axes with that 2nd Red guy's worldline. My picture of the blue guy jumping frames (to use a phrase from Rindler's textbook) then generated a series of push-backs from others.

But, to answer your question I've displayed my construction, sketch b) below, to show how my additional features would be added to your sketch a). I just wanted to show the coordinates associated with your frames (Rindler makes a distinction between "frames" and "frame coordinates"). So, I began by establishing the X4 and X4' coordinates along the direction of the inertial worldlines. Then, I used 45-degree green lines to represent photon worldlines. The simultaneous spaces for the two coordinate systems are then established by adding in the X1 and X1' axes. These X1 and X1' axes are of course placed such that the photon worldlines bisect the angles between the X4-X1 pairs.

The 2nd Red guy worldline was just added into illustrate the interesting feature mentioned by Vandam.

[edit: Note the slight discrepancy in my sketch b). The green photon worldline is not exactly 45-degrees. I hope this does not distract from illustrating the basic concepts.

 

[ghwellsjr]

Once again a very good job. You have presented all of the information in an easily understood picture that accounts very well for the difference in the twins' ages at the reunion. And this of course is exactly what was originally requested at the beginning of this thread. No additonal comments were really needed at that point.

And once again, thanks.

I had originally simply tried to give the discussion a larger context by expanding the picture by adding frame coordinates for the outgoing and return trip frames. Of course subsequent comments (beginning with an observation by Vandam) led to the addition of a second Red guy in the rest frame with events A and B and the intersections of the blue X1 and X1' axes with that 2nd Red guy's worldline. My picture of the blue guy jumping frames (to use a phrase from Rindler's textbook) then generated a series of push-backs from others.

Maybe you're thinking of a different thread since Vandam got himself banned before this thread was started.

But, to answer your question I've displayed my construction, sketch b) below, to show how my additional features would be added to your sketch a). I just wanted to show the coordinates associated with your frames (Rindler makes a distinction between "frames" and "frame coordinates"). So, I began by establishing the X4 and X4' coordinates along the direction of the inertial worldlines. Then, I used 45-degree green lines to represent photon worldlines. The simultaneous spaces for the two coordinate systems are then established by adding in the X1 and X1' axes. These X1 and X1' axes are of course placed such that the photon worldlines bisect the angles between the X4-X1 pairs.

The 2nd Red guy worldline was just added into illustrate the interesting feature mentioned by Vandam.

I take it that you are applying angles to create your lines rather than having a computer program do it for you. That would explain why my diagram didn't line up with the axes as you had intended. But now that I understand what your intent was, I can redraw my diagrams to portray what you want. However, I need to move the second red guy out a little further because he is too close to the intersection of the X₁ and X'₁ axes to show what you want. So here is another set of drawings starting with the original and final rest frame of all the participants:

Now the rest frame for the traveler during the outbound:

You should know that I adjusted the coordinate time of event B in the first frame so that the event appears simultaneous with the common origin of all the frames. That is another way of forcing event B to be simultaneous with your X₁ axis.

And the rest frame for the traveler during the inbound:

Again, I adjusted the coordinate time of event A in the first frame so that the event appears simultaneous with the traveling twin's turn-around event. That is another way to force event A to be simultaneous with your X'₁ axis.

Now I think your interesting observation was that in the traveler's simultaneous space, event A occurs after event B. However, I think you can see that following your definition of simultaneous space, you really should say that the interesting observation is that in the traveler's simultaneous space, event A occurs both before and after event B. And now that I've pointed that out, you can easily go back to your original sketch or mark up mine to show this interesting observation.

 

[Dale]

I think you can see that following your definition of simultaneous space, you really should say that the interesting observation is that in the traveler's simultaneous space, event A occurs both before and after event B.

Which is precisely why that particular simultaneity convention is not valid in that region.