Bell’s Spaceship Paradox. Does the string break?

[Neddy Bate]

The (people on the trailing) rocket says the other rocket (the leading rocket) moved instantaneously, but she does not say that

(yes, she does say that ... she is standing right there next to that rocket, and she can't help but see it instantaneously move a finite distance away from her).

No, that does not happen in her frame, that is only the trailing rocket changing his line of simultaneity from when she was 50 years old to when she was 80 years old. Him doing so has no effect on how time moves in her frame (she does not jump from 50 to 80 years old in her own frame), and so it also has no effect on the location of the leading rocket in her own frame. The leading rocket simply moves away from her at velocity v for 30 years, getting farther and farther away from her as it goes. By the time she is 80, it is long gone, so it makes sense that when the trailing rocket changes his line of simultaneity from when she was 50 years old to when she was 80 years old, that rocket should be long gone, not still next to her.

So, look at your position in this argument. You are saying the trailing rocket after acceleration thinks she went from 50 to 80 years old, but you want the the leading rocket to still be co-located with her when she is 80, even though it would have had 30 years to move away from her. That is terribly wrong.

 

[Mike_Fontenot]

I (Mike Fontenot) am still going to use the red for my reply to a particular portion of your post, otherwise you can't tell exactly what I'm objecting to. The text in black below is by you, Neddy Bate.

Perhaps you need to remind yourself of the scenario you created. You have the leading rocket next to her (correct, and they are stationary), and then in (you meant "it", the leading rocket) instantly accelerates to a constant velocity v (NO, it doesn't. It instantaneously moves a finite distance away from her [which requires an infinite acceleration for an infinitesimal time ... a Dirac delta function], and then it is stationary thereafter), which causes it to move away from her at constant speed while she ages from 50 to 80. (No, the leading rocket is first stationary with her, then instantaneously changes its position, then stationary thereafter, a fixed distance from her). By the time she is 80, it is no longer next to her anymore, it is a great distance away, right? (She is always in one place ... she doesn't move. The leading rocket has two locations: until he (the traveling twin) instantaneously changes his velocity (from zero to a constant speed toward her), the leading rocket is colocated with her. Then, when he instantaneously changes his velocity, the leading rocket instantaneously moves a finite distance from her (in the direction farther away from him). Thereafter, the leading rocket doesn't move, and neither does she.

 

[Neddy Bate]

I (Mike Fontenot) am still going to use the red for my reply to a particular portion of your post, otherwise you can't tell exactly what I'm objecting to. The text in black below is by you, Neddy Bate.

This makes it very difficult for me to reply, because I would have to separate out the red parts and put them in between quote tags. Since I don't feel like doing that, I will just reply to it all here:

I didn't realize you wanted stop the the rockets after a finite (but small?) distance. I was having them instantaneously accelerate to a constant v and then let them keep moving. My way is better because it demonstrates that by the time she ages from 50 to 80, the front rocket has been moving away from her at constant velocity for 30 years. This explains why the rear rocket says it moved so far forward, because he also says she instantaneously changed from 50 to 80 years old, so of course the leading rocket could not still be located next to her when she is 80. It would be far away.

I don't know what you think you are achieving by stopping the rockets. All that does in make the trailing rocket say that her age went from 50 to 80 and back to 50 again. Since the leading rocket is stopped before it moves a significant distance away from her, then it is still pretty close to her when she is 80, and that is what the trailing rocket will say also, if he knows when the motion is going to stop rather than continue.

You were supposed to be combining the Twin scenario with Bell's scenario (leading and trailing rockets, at constant velocity instead of constant acceleration). It would be more informative to let them keep traveling instead of stopping them. Either way, whatever the trailing rocket assumes as his simultaneity line has nothing to do with her experiences at all. She will only see the leading rocket move away from her at constant velocity, and then stop.

 

[Mike_Fontenot]

This makes it very difficult for me to reply, because I would have to separate out the red parts and put them in between quote tags. Since I don't feel like doing that, I will just reply to it all here:

That's fine.

I didn't realize you wanted stop the the rockets after a finite (but small?) distance. I was having them instantaneously accelerate to a constant v and then let them keep moving.

You know that the best way to think about the twin paradox is to assume instantaneous velocity changes by the traveling twin. What we're trying to do here is to incorporate the two rockets into the twin paradox scenario. That means that we need to make the rockets change their velocities instantaneously, just like the traveling twin does.

 

[Neddy Bate]

That's fine.

You know that the best way to think about the twin paradox is to assume instantaneous velocity changes by the traveling twin. What we're trying to do here is to incorporate the two rockets into the twin paradox scenario. That means that we need to make the rockets change their velocities instantaneously, just like the traveling twin does.

Yes, but why stop them? Let them keep moving, at least until HE is back with HER.

 

[Mike_Fontenot]

Yes, but why stop them? Let them keep moving, at least until HE is back with HER.

When he instantaneously changes his velocity wrt her from zero to -0.866 ly/y (toward her), I'm making the trailing rocket make the same instantaneous velocity change at that instant. And the leading rocket also makes that same instantaneous velocity change. But because of the assumption that the separation of the two rockets increases with time, that implies that the leading rocket instantaneously moves away from the home twin. That is absurd, of course, which means that the assumption that the rockets' separation increases is incorrect: two separated rockets with the same accelerometer readings MUST maintain a constant separation, in order to avoid the absurdity.

 

[Neddy Bate]

When he instantaneously changes his velocity wrt her from zero to -0.866 ly/y (toward her), I'm making the trailing rocket make the same instantaneous velocity change at that instant. And the leading rocket also makes that same instantaneous velocity change. But because of the assumption that the separation of the two rockets increases with time, that implies that the leading rocket instantaneously moves away from the home twin. That is absurd, of course, which means that the assumption that the rockets' separation increases is incorrect: two separated rockets with the same accelerometer readings MUST maintain a constant separation, in order to avoid the absurdity.

Why do you have the rockets stop? That means the calculation done by the trailing rocket has to incorporate your pre-planned stop into how far away from her it thinks the leading rocket can get.

 

[Mike_Fontenot]

Why do you have the rockets stop? That means the calculation done by the trailing rocket has to incorporate your pre-planned stop into how far away from her it thinks the leading rocket can get.

I'm not sure what you mean. The trailing rocket does exactly the same thing that the traveling twin does: they each instantaneously change their velocity wrt the home twin, from zero velocity to 0.866 ly/y (directed toward the home twin.) When the leading rocket maintains a constant separation wrt the trailing rocket (contrary to the assumption), the leading rocket will instantaneously change its velocity wrt the home twin (but will NOT instantaneously move away from her, so there is no absurdity in that case). But with the (incorrect) assumption that the separation between the rockets gets larger when their accelerations are equal, the leading rocket WILL instantaneously move away from the home twin (in the direction away from the trailing rocket), which is absurd. So the assumption, that two separated rockets, with equal accelerometer readings, will have a separation that increases as time increases on the trailing rocket, is incorrect. Two equally accelerating separated rockets (as confirmed by their accelerometers) will have a constant separation. That's what this example proves.

 

[Halc]

Halc, I still hope that you will give me your views that I asked you for in my post #71 above.

OK, I will attempt to do this, attempt an 0n-point reply. Post 71 is not yours, so I'm replying to this one, labeled 74

I started interacting with you in the days when you were going on about simultaneity methods and I realize your fundamental disconnect goes at least that far back. You don't know the difference between physical fact and abstraction, as spelled out in the theory of relativity.

Physical facts are things like events (points in spacetime). The rockets changing velocity abruptly is a physical fact. The event where some clock somewhere reads 35.7 is a physical fact. Spacetime intervals between events are facts (frame independent). Relationships between events (being spacelike, timelike, or light-like separated) are facts. Two events cannot be spacelike separated in one frame and timelike in a different one. An accelerometer reading (proper acceleration) is fact. All these facts exist objectively in spacetime and are not dependent on one's choice of frame, making all frames equally valid, per the first premise of SR.

Here's where you go awry: The assignment of coordinates to any event is an arbitrary abstraction. There's no factual correct coordinates to assign to any particular event, nor is the spatial distance between the events (the difference of the arbitrary spatial numbers assigned) nor is simultaneity (the difference of the arbitrary temporal number assigned). You seem to treat such things as facts and not abstractions in violation of the first premise, and thus running into contradictions that are merely abstract contradictions, a conflict with beliefs and not a physical problem.

On coordinate systems: There are well behaved coordinate systems where there is a 1-1 correspondence between events and the coordinates assigned to them. An example is a rotating frame in Minkowskian spacetime. There are coordinate systems which do not have this property, including accelerated frames and anything that attempts to map the entire actual universe.
Inertial frames only apply to Minkowskian spacetime, and only inertial frames have properties like nothing-faster-than-c, energy and momentum conservations, etc.

Anyway, the point is, you're treating arbitrary abstraction as fact, and you are in direct denial of the first premise of SR which says any frame is as good as another. When I get in a sports car, I invariably think "wow, this car is fast", and very few choose to think "wow, the road sure is fast", and yet you assert that the guy in the car is obligated to think the latter. This is what I mean when I say you violate the first premise, which states that both statements are valid.

And I would also like to get your critique of my latest proof that when the accelerometers have the same constant readings on each of the two rockets, the distance between the two rockets is constant, according to people on the trailing rocket.

The people on the rocket, or the people not on one, are all free to choose any labeling of the events involved, and thus this statement cannot be true. It falls apart right here. The people on the rocket are probably unconscious during the acceleration phase and are not worrying about assigning coordinates at all.
Relativity theory is not about people. There's no experiment that cannot be done without them, which cannot be described without human participants. What matters is the choice of coordinates assigned to the events involved, and those have not been specified in the above description.

I said this a ways back:

I will state another premise of yours:
When you say "he must conclude", those words mean that he chooses to reference the inertial frame in which he is presently stationary.
...
Kindly let me know if I'm in error with any of these premises, because none of your assertions follow without them.

Well I was in error about that premise, and you didn't kindly let me know, so your 'proof' falls apart since it equates opinions to fact.
Nobody 'must' conclude anything. What somebody concludes is arbitrary, and may or may not be correct. Some people conclude Santa exists. That's a choice, and abstraction, and it cannot be used to demonstrate Santa as a physical fact, buy you are doing exactly that.

My advice is to get rid of the people altogether in any of your examples and just confine it to clocks, cameras, strings, rulers, light-emitters, etc. Both the twins scenario and Bell's thingy can be demonstrated this way, all without dragging anyone's 'conclusion' into the fray. Include frame references in any coordinate statement since your statements (like anything that says 'distance between') require it. The only person that should be concluding anything is the teller of the story.

The characters in your stories seem to all be idealists, and since idealism leads to solipsism, you cannot have more than one character in your stories. So my recommendation is to rid yourself of them all.

 

[Mike_Fontenot]

[...]
[...] you are in direct denial of the first premise of SR which says any frame is as good as another.

It is NOT true that any inertial frame is as good as another, for any arbitrary, but specified, inertial observer.

Einstein showed us how all the inertial observers in any given inertial frame (say frame "Alpha") can construct an array of synchronized clocks separated by yardsticks. The observers start out with only one assumption: that the speed of any light pulse they encounter (or make use of) is always the same: 186,000 miles per second. From that fact alone, with some work and a lot of time, they can synchronize all of their clocks. Once they've done that, they can, at time "T1" in their frame, determine the age of some particular, but arbitrary, distant person ("Person X"). All that is required is that they accept the word of whichever of their fellow inertial observers is colocated with person X at the time "T1" in their inertial frame.

If the people in some other inertial frame (frame Beta) tell the people in frame Alpha that they are wrong about the current age of person X, and if the people in frame Alpha try to accept that, then the people in frame Alpha will need to conclude that the speed of light in their frame is NOT 186,000 miles per second, because that was the ONLY assumption that they used to determine the current age of person X. But in that case, they would then be denying the fundamental assumption of special relativity. They would not do that.

 

[phyti]

Mike;
Ann sends time signals periodically using light (blue). Biff receives the signal from clock event At=2 at his reversal. He receives the next signal from clock event At=3.


You state when Biff reverses direction at clock event At=5 (Bt=4), the Line Of Simultaneity jumps forward into the future.

Explain how Biff sees/knows what Ann's clock indicates at clock event At=8, when it hasn't happened yet.

 

[Mike_Fontenot]

Mike;
Ann sends time signals periodically using light (blue). Biff receives the signal from clock event At=2 at his reversal. He receives the next signal from clock event At=3.


You state when Biff reverses direction at clock event At=5 (Bt=4), the Line Of Simultaneity jumps forward into the future.

Explain how Biff sees/knows what Ann's clock indicates at clock event At=8, when it hasn't happened yet.
View attachment 5565

You need to draw his lines of simultaneity, immediately before and immediately after he changes his velocity. Where those lines of simultaneity intersect the vertical axis give her age then, according to him.

 

[Neddy Bate]

I am only replying to the text in this partial quote below. I am not replying to the parts that I removed which are not required for this exercise:

The trailing rocket does exactly the same thing that the traveling twin does: they each instantaneously change their velocity wrt the home twin, from zero velocity to 0.866 ly/y (directed toward the home twin.) The leading rocket will instantaneously change its velocity wrt the home twin (but will NOT instantaneously move away from her, so there is no absurdity in that case).

This is great, because you and I can at least agree on the above. You had set up the scenario in this way in the first place, and basic kinematics tells us that this is what will happen. This is what I have been saying all along. The home twin would simply say the leading rocket moves away from her at v=0.866c. She most certainly would not say the leading rocket teleported to a distant location instantly, regardless of what the trailing rocket might calculate.

Now, do you remember the ages you gave me for the home twin as calculated from the trailing rocket frame? She was 50 years old (before the trailing rocket's acceleration) and 80 years old (after the trailing rocket's acceleration). So to make your scenario even more complete, we can add these details:

"The leading rocket is co-located and stationary with respect to the stay home twin for her time T<50. She can reach over and touch the leading rocket with her hand for her time T<50, but then at the launch event at her time T=50 the rocket takes off and moves away from her at v=0.866c. Now she can not touch it any more, because the leading rocket is not co-located with her for any time T>50.

But with the (incorrect) assumption that the separation between the rockets gets larger when their accelerations are equal, the leading rocket WILL instantaneously move away from the home twin (in the direction away from the trailing rocket), which is absurd. So the assumption, that two separated rockets, with equal accelerometer readings, will have a separation that increases as time increases on the trailing rocket, is incorrect. Two equally accelerating separated rockets (as confirmed by their accelerometers) will have a constant separation. That's what this example proves.

Relativity requires careful analysis, and you did not do that. You guessed that a calculation made by the trailing rocket would somehow affect her frame, even though that same calculation which has her changing from 50 to 80 years old does not affect her frame at all. You cannot argue that one calculation does not affect her but the other one does, it is inconsistent to say the least.

So, let's try a more careful analysis to see if either Fontenot's Solution or Bell's Solution can be shown to be inconsistent with the undeniable facts as established above.

Fontenot's Solution
Before the acceleration, the trailing rocket is stationary with respect to the stay home twin for her time T<50. The trailing rocket says she can reach over and touch the leading rocket with her hand for her time T<50, but then we have the launch event at T=50. The trailing rocket calculates that her age has jumped from T=50 to T=80. Yet Fontenot's Solution requires the leading rocket to still be co-located with the home twin at T=80. This means that the trailing rocket has made a calculation that says she can still touch the leading rocket at T=80 because Fontenot's Solution says the trailing rocket should calculate it to still be co-located with her for both T=50 and T=80. Obviously this does not affect her, the reality is that she cannot touch the rocket at T=80, but it does prove Fontenot's Solution is incorrect, since the facts state that the leading rocket is not co-located with her for any time T>50. Fontenot's Solution has created a new phantom-event where the leading rocket is co-located with the home twin at T=80 which is not a real event in the given scenario. This is a real contradiction. So we have proved that Fontenot's Solution cannot be the correct one.

Bell's Solution
Before the acceleration, the trailing rocket is stationary with respect to the stay home twin for her time T<50. The trailing rocket s says she can reach over and touch the leading rocket with her hand for her time T<50, but only until the launch event at T=50. The trailing rocket calculates that her age has jumped from T=50 to T=80. Bell's Solution requires the leading rocket to instantaneously move away from the home twin so that it is no longer co-located with the home twin at T=80. This calculation does not affect her in any way, but it does still fit the facts which are that she can no longer touch the leading rocket at T=80 because it is not co-located with her at that time. The facts state that the leading rocket is not co-located with her for any time T>50. So Bell's Solution is consistent with the all the facts, at least.

 

[Mike_Fontenot]

Neddy, I've read your above post (#113), but it has been so long since we had our interactions that I'm having trouble understanding this last post from you. I'll keep trying to understand your points. My perception was that you and I were on the same page when we last addressed these issues (and you were the one who taught me that the Bell scenario is a DIFFERENT scenario from the one I'm examining). Be aware that what is important to me in these posts is to show that two separated rockets with equal accelerometer readings will NOT get farther apart as the acceleration progresses. And I think I've shown that by showing that the contrary assumption (that the separation will increase) leads to an absurdity. But beware that I am doing that by using instantaneous speed changes (Dirac delta function accelerations, where the acceleration is infinite for an infinitesimal time). Also, to accomplish my objective, I haven't needed to concern myself with what happens after the traveler instantaneously changes his speed (i.e., I haven't concerned myself with his inbound trip, and the reunion ... I've just been interested in what happens as a consequence of his speed change.) (I probably won't continue trying to figure this out until tomorrow morning ... it's about supper time for me.)

 

[Neddy Bate]

Neddy, I've read your above post (#113), but it has been so long since we had our interactions that I'm having trouble understanding this last post from you. I'll keep trying to understand your points. My perception was that you and I were on the same page when we last addressed these issues (and you were the one who taught me that the Bell scenario is a DIFFERENT scenario from the one I'm examining). Be aware that what is important to me in these posts is to show that two separated rockets with equal accelerometer readings will NOT get farther apart as the acceleration progresses. And I think I've shown that by showing that the contrary assumption (that the separation will increase) leads to an absurdity. But beware that I am doing that by using instantaneous speed changes (Dirac delta function accelerations, where the acceleration is infinite for an infinitesimal time). Also, to accomplish my objective, I haven't needed to concern myself with what happens after the traveler instantaneously changes his speed (i.e., I haven't concerned myself with his inbound trip, and the reunion.)

Keep studying it, and ask me if you have any specific questions. I'm sorry but you are not correct on this. Think of it as one of those times when you come back after awhile and say, "I've discovered a flaw in my last proof".

I think you and I agree that the trailing rocket says the home twin (she) changes from 50 years old to 80 years old in an instant. Does that mean you can apply it to her frame and argue, "That would require the stay home twin to see herself immediately change from 50 to 80 in an instant, which is an ABSURDITY"? No, you can't take what the trailing rocket says and apply it to her that way, as I think you and I agree.

The location of the leading rocket is just the same. You are literally saying: "The trailing rocket says the leading rocket changes to a distant location in an instant, and that would require the stay home twin herself to see the location change in an instant, which is an ABSURDITY!" But you can't take what the trailing rocket says and apply it to her that way, for exactly the same reason.

If you did not understand it in post #113: You're idea is that the trailing rocket should keep the leading rocket co-located with the home twin not only when her clock says T=50 and she is 50 years old, but also when her clock says T=80 and she is 80 years old. But it is not true that the rocket is co-located with her at T=80 because it has moved away from her. This proves that your idea is wrong, and Bell's instantaneous re-location of the leading rocket resolves that problem.

None of this stuff is really instantaneous in real life. That artifact arises from the simplification of using instantaneous accelerations.

 

[Halc]

It is NOT true that any inertial frame is as good as another, for any arbitrary, but specified, inertial observer.

That assertion is in direct conflict with the first premise of SR. It is part of what leads you to contradictions, contradictions that don't exist in SR. Furthermore, you've made no attempt to justify this assertion, lacking identification of some empirical test that would falsify the converse.
Secondly, the observer in your 'proof' making the weird claims of instant movement of distant objects is hardly inertial.

Einstein showed us how all the inertial observers in any given inertial frame (say frame "Alpha") can construct an array of synchronized clocks separated by yardsticks.

That's pretty much impossible. If they were inertial, they could not construct anything since construction of extended structures like that usually involves moving about non-inertially. So let's rephrase it. Einstein considered the case of an array of synchronized clocks separated by yardsticks which can be set up in any arbitrary frame (say Alpha). It can be done in any choice of frame, so no set could be identified as a preferred one the presence of an observer doesn't change that.

Notably, all observers, inertial or not, would be 'in' all those frames, not just one of them. Their presence serves no purpose at all, and the arrays could be constructed in their absence.

The observers start out with only one assumption: that the speed of any light pulse they encounter (or make use of) is always the same: 186,000 miles per second. From that fact alone, with some work and a lot of time, they can synchronize all of their clocks.

Not so. They need to arbitrarily select an abstract coordinate system first, and any coordinate system (plus a convention, also abstract) will do. If you think it can be done without this arbitrary selection, or that it cannot be done with the 'wrong' selection, do please elaborate.

If the people in some other inertial frame (frame Beta)

Yet again, everybody is in every frame, so that phrase is meaningless. You need to learn that. It is not possible to exit an inertial frame. We will presume Bob is stationary in Beta, etc, but it's not like Alice (stationary in Alpha) is not in the Beta frame.
Yet again, it is so much easier to describe things without people, since the people seem to only confuse you.

If the people [presumably stationary] in some other inertial frame (frame Beta) tell the people [presumably stationary] in frame Alpha that they are wrong about the current age of person X, and if the people in frame Alpha try to accept that, then the people in frame Alpha will need to conclude that the speed of light in their frame is NOT 186,000 miles per second, because that was the ONLY assumption that they used to determine the current age of person X.

Bob doesn't tell Alice that she's wrong because she isn't. If Bob makes that meaningless statement, Bob is ignorant of relativity theory, which he very much is if he goes on as you describe. There is no 'current age of person X (Xavier)' since that is dependent on the coordinates assigned to the events at which X is present, and those assigned numbers are an arbitrary choice, not physical fact. Bob is wording his assertions as physical fact, which is his mistake (among others). Light moving at c in one frame does not imply that it doesn't in another. That would also be a direct violation of the premises of SR. No conflict with this has been identified.
You've not backed any of your assertions, and since your assertions lead to self-admitted absurdities, your assertions have been self-falsified.

 

[Mike_Fontenot]

These "back-and-forth's", between me and both of you, have become so contorted that I think further communication is futile. I still stand by EVERYTHING I've posted. I'm certain that I am correct.

 

[phyti]

Mike;

The clock synch method was intended for a local inertial ref. frame with its elements moving with a constant velocity. An expansive astronomical system of clocks would not be logistically possible with the motion of distant bodies influenced by other sources of gravity in their region.

Transit times for light over great distances like 100 ly would have no value or benefit.

On the left is your example with a 'helper' A2 synchronized to the A clock.
A signals (blue) to A2 for the time on the B clock, which A2 immediately signals back to A with Bt=4. Since A and B were together at t=0, A can calculate the rate of the B clock as 4/5 or .80, and a corresponding speed of .6c

On the right is your example with no 'helper'.
A signals (blue) to B for the time on the B clock, which B immediately signals back to A with Bt=4. Since A and B were together at t=0, A can calculate the rate of the B clock as 4/5 or .80,
and a corresponding speed of .6c.

Same result with fewer people.

Does A know the current age of B?
It required 8 units of time for A to get the measurements.
Now, At=8 A knows (has factual evidence) that the B time WAS 4 when the A time WAS 5.
Now, A does not know what the B time IS.
She can speculate, IF nothing changes, the B time would be 6.4, but that's a conditional statement, NOT a certainty.
There is no instant knowledge.

 

[phyti]

Mike;

You need to draw his lines of simultaneity, immediately before and immediately after he changes his velocity. Where those lines of simultaneity intersect the vertical axis give her age then, according to him

You have received some good suggestions to simplify and sort out facts from fiction.

With the A frame as the ref. frame, consider the moving B frame. Its ct axis is the spacetime path/line labeled B (since the clock moves with the observer).
Its physical Bx axis is parallel to Ax. I.e. all motion occurs on the x axis.
The spacetime graphic plots object motion to light motion vt/ct. That's why blue light lines have an angle of 45º and object lines have an angle with a slope of v/c.

The green Bx axis (aka line of simultaneity) is an abstract mathematical line connecting an A-clock event with a corresponding B-clock event. It is NOT a magical beacon that points to something. It's a calculation resulting from a round trip measurement using light.
Instead of a continuous curve transitioning from the outbound to inbound path, there is an instantaneous reversal at Bt=4. The last strictly outbound los before the reversal was At=2 to Bt=2.5, which was calculated at Bt=4.
The green los continues to rotate cw since the roundtrip light transit time is remaining constant resulting from emission outbound and detection inbound. At reversal the los will be At=5 to Bt=4 when calculated at Bt=5.5. Thus the los is the same for a blink during the reversal. Continuing, the los rotation is a mirror image of the outbound path.

The next los strictly for the inbound path will be At=8 to Bt=5.5, when calculated at Bt=7.
There is a los for each A-clock event for the history, but no jump at Bt=4. There is no los forward since the measurements needed are still in the future.

There is no instant knowledge.

The spacetime graphics are histories of events, so you have to consider the sequence of events as time accumulates, which you seem to ignore.

 

[Mike_Fontenot]

Phyti, the twin paradox has NOTHING to do with light signals being sent back and forth between the twins. It's about the time dilation equation, which says that any person (she) who isn't accelerating (i.e., who is inertial) must conclude that any other person (he) who is moving with respect to her is ageing more slowly than she is, by the factor "gamma", where

gamma = 1 / (sqrt [ 1 - { v * v } ] ).

For example, if v = 0.866 lightyears per year, gamma = 2.0.

So the home twin (she), who never accelerates, must conclude that he (who is also inertial except for the one instant when he instantaneously reverses his course) is ageing only half as fast as she is ageing. So she knows that, if she is 80 years old when he returns, he will be 40 years old then.

So what is the (apparent) paradox? He is inertial during the two legs of his trip, so he likewise concludes that she will be only half his age when they are reunited. We know that he is 20 years old when he reverses course, because that is an "event" that everyone must agree about. And we know that he ages by another 20 years during his return trip. So, since he is 40 years old when they are reunited, he expects that she will be only 20 years old then. But he finds her to be 80 years old at their reunion. Why was he wrong in his expectations of her age at their reunion? The ONLY answer is that, when he instantaneously reverses course, he MUST conclude that she instantaneously gets 60 years older during his course reversal. THAT is what resolves the twin paradox.

You're welcome.