Need help understanding the twins

[cfrogue]

I specified which post in that same paragraph:

It was the previous post before the one where I wrote that, i.e. post #134.

OK, thanks, please give me some time.

You are very thorough.

 

[cfrogue]

I specified which post in that same paragraph:

It was the previous post before the one where I wrote that, i.e. post #134.

You mean, figure out the time coordinate T in the final rest frame that twin1 will receive the pulse, then subtract D/c from that to get the time coordinate T - D/c in the final rest frame that twin2 stopped accelerating? Of course the time coordinate that twin2 stopped accelerating in this frame is not the same as twin1's age at the moment twin2 stopped accelerating in this frame, because twin1 was not at rest in this frame since t=0 in this frame. But I suppose you can say that in this frame, twin1 was initially accelerating for (c/a)*sinh(a*BT/c) of coordinate time, so that twin1 has been at rest in this frame for (T - D/c) - (c/a)*sinh(a*BT/c). And during the initial period twin1 was accelerating, twin1 aged BT, so twin1's total elapsed time at (T - D/c) in the final frame should be BT + (T - D/c) - (c/a)*sinh(a*BT/c).

This does not make sense to me.

Of course the time coordinate that twin2 stopped accelerating in this frame is not the same as twin1's age at the moment twin2 stopped accelerating in this frame, because twin1 was not at rest in this frame

I do not care of they register the same proper time. I only care that they establish a common stop point in their own proper times.

I am not sure how you could refute this.

 

[JesseM]

You mean, figure out the time coordinate T in the final rest frame that twin1 will receive the pulse, then subtract D/c from that to get the time coordinate T - D/c in the final rest frame that twin2 stopped accelerating? Of course the time coordinate that twin2 stopped accelerating in this frame is not the same as twin1's age at the moment twin2 stopped accelerating in this frame, because twin1 was not at rest in this frame since t=0 in this frame. But I suppose you can say that in this frame, twin1 was initially accelerating for (c/a)*sinh(a*BT/c) of coordinate time, so that twin1 has been at rest in this frame for (T - D/c) - (c/a)*sinh(a*BT/c). And during the initial period twin1 was accelerating, twin1 aged BT, so twin1's total elapsed time at (T - D/c) in the final frame should be BT + (T - D/c) - (c/a)*sinh(a*BT/c).

This does not make sense to me.

Do you disagree, or you don't understand?

Of course the time coordinate that twin2 stopped accelerating in this frame is not the same as twin1's age at the moment twin2 stopped accelerating in this frame, because twin1 was not at rest in this frame

I do not care of they register the same proper time. I only care that they establish a common stop point in their own proper times.

I didn't say anything about registering the same proper time, I was talking about coordinate times. If x,t are the coordinates in the launch frame of twin1 receiving the light signal sent by twin2 at the moment he stopped accelerating, then T = gamma*(t - vx/c^2) is the time coordinate in the final rest frame of twin1 receiving this signal. So, if the final distance between twin1 and twin2 in the final rest frame is D, then that shows that whatever age twin1 was at (T - D/c), this must be the point on his worldline that was simultaneous with twin2 stopping his acceleration, according to the final rest frame's definition of simultaneity. You do want to use the final rest frame's definition of simultaneity to compare their ages, do you not?

If you don't want to use the method above where we figure out the x,t coordinates of twin1 receiving the signal in the launch frame and use it to get the time coordinate T of this event in the final rest frame, what do you want to do? Is T supposed to represent twin1's own proper time when he gets the signal, rather than the time coordinate of his getting the signal in the final rest frame? If so, how do you propose to actually find the value of T if not by dividing his worldline into pieces and calculating the proper time on each one? (which requires that you include the 3a-4a piece)

 

[cfrogue]

Originally Posted by cfrogue
You mean, figure out the time coordinate T in the final rest frame that twin1 will receive the pulse, then subtract D/c from that to get the time coordinate T - D/c in the final rest frame that twin2 stopped accelerating? Of course the time coordinate that twin2 stopped accelerating in this frame is not the same as twin1's age at the moment twin2 stopped accelerating in this frame, because twin1 was not at rest in this frame since t=0 in this frame. But I suppose you can say that in this frame, twin1 was initially accelerating for (c/a)*sinh(a*BT/c) of coordinate time, so that twin1 has been at rest in this frame for (T - D/c) - (c/a)*sinh(a*BT/c). And during the initial period twin1 was accelerating, twin1 aged BT, so twin1's total elapsed time at (T - D/c) in the final frame should be BT + (T - D/c) - (c/a)*sinh(a*BT/c).

This does not make sense to me.

Do you disagree, or you don't understand?

Well, I guess I do not understand.

Perhaps, you could explain it differently.

I didn't say anything about registering the same proper time, I was talking about coordinate times. If x,t are the coordinates in the launch frame of twin1 receiving the light signal sent by twin2 at the moment he stopped accelerating, then T = gamma*(t - vx/c^2) is the time coordinate in the final rest frame of twin1 receiving this signal. So, if the final distance between twin1 and twin2 in the final rest frame is D, then that shows that whatever age twin1 was at (T - D/c), this must be the point on his worldline that was simultaneous with twin2 stopping his acceleration, according to the final rest frame's definition of simultaneity. You do want to use the final rest frame's definition of simultaneity to compare their ages, do you not?

If you don't want to use the method above where we figure out the x,t coordinates of twin1 receiving the signal in the launch frame and use it to get the time coordinate T of this event in the final rest frame, what do you want to do? Is T supposed to represent twin1's own proper time when he gets the signal, rather than the time coordinate of his getting the signal in the final rest frame? If so, how do you propose to actually find the value of T if not by dividing his worldline into pieces and calculating the proper time on each one? (which requires that you include the 3a-4a piece)

You like worldlines too much.

They cannot be used to solve unknowns.

This is why I do not like them. I feel caged.

Try to ignore your world line thing for the moment and think about the recursive decision process I described.
We have some unknowns are are trying to decide them.

The process I came up with decides them.

Do you disagree?

 

[JesseM]

You like worldlines too much.

Worldlines are simply functions defining the coordinates an object passes through in a given frame. Are you saying you think the problem can be solved even if we don't know what coordinates the twins pass through?

They cannot be used to solve unknowns.

Sure they can. For example, if you know coordinate position as a function of coordinate time, you can use to solve for the proper time.

Try to ignore your world line thing for the moment and think about the recursive decision process I described.
We have some unknowns are are trying to decide them.

The process I came up with decides them.

Do you disagree?

I have no idea what the process is because you don't answer my questions about it. Again, is T supposed to be the coordinate time in the final rest frame that twin1 receives the signal from twin2, or is it supposed to be twin1's proper time at the moment he receives the signal, or something else? If it's the proper time, then how do you propose to actually decide the value of this variable T?

 

[cfrogue]

Worldlines are simply functions defining the coordinates an object passes through in a given frame. Are you saying you think the problem can be solved even if we don't know what coordinates the twins pass through?

Sure they can. For example, if you know coordinate position as a function of coordinate time, you can use to solve for the proper time.

I have no idea what the process is because you don't answer my questions about it. Again, is T supposed to be the coordinate time in the final rest frame that twin1 receives the signal from twin2, or is it supposed to be twin1's proper time at the moment he receives the signal, or something else? If it's the proper time, then how do you propose to actually decide the value of this variable T?

The process I came up with decides them.

Do you disagree?

O' stops accelerating and immediately sends a light signal to O.

O receives the signal.

O calls this a pseudo end of the experiment.

The problem is that it took time for the light to travel to O from O'.
Thus, O must decide the distance light traveled.
O does the round trip speed of light calculation and is able to decide the distance to O'.
Thus, O then knows the in its own proper time when O' stopped accelerating.
Now, O has BT and t', the relative motion phase.
Since acceleration is absolute motion, O knows (c/a)*sinh(a*BT/c) transpired for the burn of twin2.
Thus, t' is logically decidable from algebra and logic but not from the restrictive logic of world lines.

 

[JesseM]

The process I came up with decides them.

Do you disagree?

I can't agree or disagree if you won't answer my questions about what your "process" actually is. Again, is T the coordinate time when twin1/0 receives the signal, or the proper time of twin1/O, or something else?

 

[cfrogue]

I can't agree or disagree if you won't answer my questions about what your "process" actually is. Again, is T the coordinate time when twin1/0 receives the signal, or the proper time of twin1/O, or something else?

Twin1 receives it at the proper time of twin1.

You know what, I see there is a flaw.

Twin1 needs to then communicate this time logic of its proper time to twin2 for its correct calculation for validation.

No matter, the result is the same.

But, I want to think about this more.

 

[JesseM]

Twin1 receives it at the proper time of twin1.

Duh, I know that. But that still doesn't tell me whether the variable T that you wrote before is supposed to refer to twin1's proper time, or to the coordinate time in their final rest frame.

In any case, my more basic question is this: how do you propose to actually solve for twin1's proper time at the moment he receives the signal? If you don't know the actual value of his proper time when he receives the signal, then this is no use in determining if he is younger or older than twin2 was at the moment twin2 stopped accelerating.

 

[cfrogue]

Duh, I know that. But that still doesn't tell me whether the variable T that you wrote before is supposed to refer to twin1's proper time, or to the coordinate time in their final rest frame.

In any case, my more basic question is this: how do you propose to actually solve for twin1's proper time at the moment he receives the signal? If you don't know the actual value of his proper time when he receives the signal, then this is no use in determining if he is younger or older than twin2 was at the moment twin2 stopped accelerating.

OK you are funny.

 

[cfrogue]

Duh, I know that. But that still doesn't tell me whether the variable T that you wrote before is supposed to refer to twin1's proper time, or to the coordinate time in their final rest frame.

In any case, my more basic question is this: how do you propose to actually solve for twin1's proper time at the moment he receives the signal? If you don't know the actual value of his proper time when he receives the signal, then this is no use in determining if he is younger or older than twin2 was at the moment twin2 stopped accelerating.

No, your logic does not hold.

Twin1 receives the signal and both are in the same frame.

It is a distance calc, no?

 

[JesseM]

No, your logic does not hold.

Twin1 receives the signal and both are in the same frame.

It is a distance calc, no?

If you already knew the age (proper time) that twin1 was when he received it, I agree that by subtracting D/c you could get the age of twin1 at the moment twin2 stopped accelerating in their common rest frame. My point is that you have given no procedure for us to actually calculate twin1's age when he receives the signal in the first place. Or are you suggesting we shouldn't try calculating it at all, but should just determine it by finding some actual twins and performing this as an empirical experiment?

 

[cfrogue]

If you already knew the age (proper time) that twin1 was when he received it, I agree that by subtracting D/c you could get the age of twin1 at the moment twin2 stopped accelerating in their common rest frame. My point is that you have given no procedure for us to actually calculate twin1's age when he receives the signal in the first place. Or are you suggesting we shouldn't try calculating it at all, but should just determine it by finding some actual twins and performing this as an empirical experiment?

No, I am suggesting we deduce the unknowns.

We know BT occurred on the proper time for twin1. We know the start and end points of the proper time of twin1.

We know c/a sinh( a*BT/c ) transpired also when twin2 accelerating into the frame of twin1.
That leaves the elapsed proper time for twin1 of the relative motion phase. That is the only unknown to solve once the correct endpoint is known given the acceleration equations, at least that is what I think.

 

[JesseM]

No, I am suggesting we deduce the unknowns.

We know BT occurred on the proper time for twin1. We know the start and end points of the proper time of twin1.

We know c/a sinh( a*BT/c ) transpired also when twin2 accelerating into the frame of twin1.
That leaves the elapsed proper time for twin1 of the relative motion phase. That is the only unknown to solve once the correct endpoint is known given the acceleration equations, at least that is what I think.

OK, but how exactly do you propose to "solve for" the proper time for twin1 in the phase where both twin1 and twin2 were moving inertially? I don't understand how this business of subtracting D/c is supposed to help with that, if you don't already know the proper time for twin1 at the moment he receives the signal.

 

[cfrogue]

OK, but how exactly do you propose to "solve for" the proper time for twin1 in the phase where both twin1 and twin2 were moving inertially? I don't understand how this business of subtracting D/c is supposed to help with that, if you don't already know the proper time for twin1 at the moment he receives the signal.

OK, first, do you agree we can make the entry point simultaneous for twin1 and twin2 with this D/c business?

In other words, when twin2 stops accelerating, the clock is shut off.

When, twin1 receives the light signal, twin1 shuts off the clock. But, the clock is not yet simultaneous with twin2's clock shut down. Thus, after subtracting the D/c business, their clock shut down and adjusted shut down time for twin1 becomes simultaneous.

Are we agreed at this point?

 

[JesseM]

OK, first, do you agree we can make the entry point simultaneous for twin1 and twin2 with this D/c business?

In other words, when twin2 stops accelerating, the clock is shut off.

When, twin1 receives the light signal, twin1 shuts off the clock. But, the clock is not yet simultaneous with twin2's clock shut down. Thus, after subtracting the D/c business, their clock shut down and adjusted shut down time for twin1 becomes simultaneous.

Are we agreed at this point?

Yes, of course.

 

[cfrogue]

Yes, of course.

OK, so what is left for twin1, the burn time BT for its acceleration, an unknown relative motion time and an known time for the burn of twin2 as c/a sinh( a*BT/c ).

Thus, we can solve for the unknown relative motion time.

 

[JesseM]

OK, so what is left for twin1, the burn time BT for its acceleration, an unknown relative motion time and an known time for the burn of twin2 as c/a sinh( a*BT/c ).

Thus, we can solve for the unknown relative motion time.

How do you "solve for" it if you don't know the total time for twin1? Or if you think there is a procedure that will allow you to figure out the total time before we know the time of the relative inertial motion phase, what is that procedure? This is what I keep asking you, you never give me an answer.

 

[cfrogue]

How do you "solve for" it if you don't know the total time for twin1? Or if you think there is a procedure that will allow you to figure out the total time before we know the time of the relative inertial motion phase, what is that procedure? This is what I keep asking you, you never give me an answer.

We just got through determining the total time.

Remember the D/c business?

I keep telling you the answer.

 

[JesseM]

We just got through determining the total time.

Remember the D/c business?

I keep telling you the answer.

You never determined the total time! You just said that whatever time T that twin1 received the signal, the total time would be T - D/c. But you have given no way to figure out what value T would actually have (as a function of other variables like BT and a), so you don't know the value of T - D/c either.

 

[cfrogue]

You never determined the total time! You just said that whatever time T that twin1 received the signal, the total time would be T - D/c. But you have given no way to figure out what value T would actually have (as a function of other variables like BT and a), so you don't know the value of T - D/c either.

This is not true.

First, you know the start time.

Now, when twin2 enters the frame a light signal is sent.

Wehn twin1 receives this signal, twin1 marks the time.

Twin1 then performs a round trip light distance calc to determine the distance to twins2 and sees this value as D.

Then twin1 subtracts D/c from its written down end of experiment time.

This will be the instant twin1 enters the frame and the end of the burn of twin2 for twin1.

Yes or no.

 

[JesseM]

This is not true.

First, you know the start time.

Now, when twin2 enters the frame a light signal is sent.

Wehn twin1 receives this signal, twin1 marks the time.

Yes, but what time will twin1 mark? You haven't given any way to calculate this. Are you proposing that the question could only be determined experimentally, that we'd have to find some real flesh-and-blood twins and send them on a relativistic rocket trip? If not, then we need a way to calculate the value of the time twin1's clock will show at the time he receives the signal from twin2 (as a function of other known variables like BT and a), and you haven't given a way to do this.

 

[cfrogue]

Yes, but what time will twin1 mark? You haven't given any way to calculate this. Are you proposing that the question could only be determined experimentally, that we'd have to find some real flesh-and-blood twins and send them on a relativistic rocket trip? If not, then we need a way to calculate the value of the time twin1's clock will show at the time he receives the signal from twin2 (as a function of other known variables like BT and a), and you haven't given a way to do this.

Are you proposing that the question could only be determined experimentally
Yes. I operate as I have to.

You have not proven my method does not do as advertised.

You are flailing around.

 

[JesseM]

Are you proposing that the question could only be determined experimentally
Yes. I operate as I have to.

You have not proven my method does not do as advertised.

You are flailing around.

Of course your method would work as an experimental procedure. But since we are discussing the predictions of the theory of relativity on a message board, I naturally hoped you were proposing a theoretical method to settle the question of which twin would be older in a thought-experiment (like my own procedure involving dividing twin1's worldline into pieces and calculating the proper time on each piece). I even asked you back in post 152 whether you were proposing the question should be settled by experiment:

If you already knew the age (proper time) that twin1 was when he received it, I agree that by subtracting D/c you could get the age of twin1 at the moment twin2 stopped accelerating in their common rest frame. My point is that you have given no procedure for us to actually calculate twin1's age when he receives the signal in the first place. Or are you suggesting we shouldn't try calculating it at all, but should just determine it by finding some actual twins and performing this as an empirical experiment?

But your answer was "No, I am suggesting we deduce the unknowns". Has your position changed, and now you think only an empirical experiment can determine what age twin1 will be when he receives the signal, that you would not be satisfied with any theoretical procedure which would "deduce" this unknown (even though the age is perfectly decidable in SR using a theoretical analysis like the one I proposed)? I kind of get the feeling you don't have any coherent position at all, and are just arguing with me for the sake of being contrary ('flailing around' to find reasons to disagree with me, one might say).

 

[cfrogue]

Of course your method would work as an experimental procedure. But since we are discussing the predictions of the theory of relativity on a message board, I naturally hoped you were proposing a theoretical method to settle the question of which twin would be older in a thought-experiment (like my own procedure involving dividing twin1's worldline into pieces and calculating the proper time on each piece). I even asked you back in post 152 whether you were proposing the question should be settled by experiment:

But your answer was "No, I am suggesting we deduce the unknowns". Has your position changed, and now you think only an empirical experiment can determine what age twin1 will be when he receives the signal, that you would not be satisfied with any theoretical procedure which would "deduce" this unknown (even though the age is perfectly decidable in SR using a theoretical analysis like the one I proposed)? I kind of get the feeling you don't have any coherent position at all, and are just arguing with me for the sake of being contrary ('flailing around' to find reasons to disagree with me, one might say).

I would hope the theory would provide correct empirical data as predicted by the equations.

I kind of get the feeling you don't have any coherent position at all,
Well, I have deduced the relative motion period of twin1 by solving for it as an unknown using the equation.

T = Bt + t' + c/a sinh( a*BT/c ).

You agree we know T.
You agree we know BT.
You agree we know c/a sinh( a*BT/c ).

Then, I coherently perform a subtraction as
T - BT - c/a sinh( a*BT/c ) = t'.

What is wrong with this?

 

[JesseM]

I would hope the theory would provide correct empirical data as predicted by the equations.

I kind of get the feeling you don't have any coherent position at all,
Well, I have deduced the relative motion period of twin1 by solving for it as an unknown using the equation.

T = Bt + t' + c/a sinh( a*BT/c ).

You agree we know T.

How do we know T? You haven't proposed any way to find it except by empirical experiment. Also, previously you had t' be the relative inertial motion time for twin2, not twin1, and then you wanted to find how long twin1's relative inertial motion time was in relation to that (you incorrectly thought it was t'/gamma before).

 

[cfrogue]

How do we know T? You haven't proposed any way to find it except by empirical experiment. Also, previously you had t' be the relative inertial motion time for twin2, not twin1, and then you wanted to find how long twin1's relative inertial motion time was in relation to that (you incorrectly thought it was t'/gamma before).

OK I can use a posteriori logic to decision problems. Yes, it is the case that t' = t/gamma but I chose to implement an effective procedure within recursion theory to decide this t'.

Now, I am allowed to operate a posteriori within recursion theory to decide an outcome as long as I have an effective procedure.

The subtraction I showed you is this effective procedure.

You and I are different. I am not a caged animal.

 

[JesseM]

OK I can use a posteriori logic to decision problems. Yes, it is the case that t' = t/gamma

What do t and t' represent? If t represents the proper time of the relative inertial motion phase for twin2 (i.e. the time between twin1 finishing his acceleration and twin2 beginning his own, in the frame where twin1 was at rest between these events), and t' represents the proper time of the relative inertial motion phase for twin1 (i.e. the time between twin1 finishing his acceleration and twin2 beginning his own, in the frame where twin2 was at rest between these events) then it is not the case that t' = t/gamma, this is incorrect reasoning because it ignores the relativity of simultaneity, as I already explained.

but I chose to implement an effective procedure within recursion theory to decide this t'.

Now, I am allowed to operate a posteriori within recursion theory to decide an outcome as long as I have an effective procedure.

The subtraction I showed you is this effective procedure.

What "effective procedure"? Once again you resort to a vague fog of words that have no clear mathematical meaning, I have no idea how "recursion theory" is supposed to tell you how the length of twin2's inertial relative motion phase relates to the length of twin1's inertial relative motion phase, you've never explained this at all! You can't solve the problem with technobabble (and what's more, you have been totally waffling on whether you can find T and t' using a theoretical calculation or whether it requires empirical testing as you suggested in post 163, suggesting even you don't have any clear idea what the hell you are talking about).

If you think that you have an "effective procedure" for theoretically deriving the unknown value of the proper time t' for twin1 during the relative inertial motion as a mathematical function of other variables can be treated as known because they appear in the equation for twin2's total time (like BT, the proper time of acceleration, and a, the value of acceleration, and t, the proper time for twin2 on his own relative inertial motion phase in his frame...note that you cannot treat T as one of the known variables when deriving t', because this variable does not appear in twin2's total time) then show me the actual mathematical derivation, otherwise I'm going to assume you're just bluffing and have no clear idea of a procedure that will give a specific equation for this (and thus you have no theoretical procedure to determine whether twin1 or twin2 will finally have aged more, and what will be the precise ratio of their ages).

 

[cfrogue]

What do t and t' represent? If t represents the proper time of the relative inertial motion phase for twin2 (i.e. the time between twin1 finishing his acceleration and twin2 beginning his own, in the frame where twin1 was at rest between these events), and t' represents the proper time of the relative inertial motion phase for twin1 (i.e. the time between twin1 finishing his acceleration and twin2 beginning his own, in the frame where twin2 was at rest between these events) then it is not the case that t' = t/gamma, this is incorrect reasoning because it ignores the relativity of simultaneity, as I already explained.

This implies all relative motion must apply LT plus an R of S argument. But, R of S is already built into LT.

If you look at the derivation of LT, you will note t + x'/(c+v) + x'/(c-v). This is a direct application of R of S.
http://www.fourmilab.ch/etexts/einstein/specrel/www/

Therefore, I am not following your logic.

What "effective procedure"? Once again you resort to a vague fog of words that have no clear mathematical meaning, I have no idea how "recursion theory" is supposed to tell you how the length of twin2's inertial relative motion phase relates to the length of twin1's inertial relative motion phase, you've never explained this at all! You can't solve the problem with technobabble (and what's more, you have been totally waffling on whether you can find T and t' using a theoretical calculation or whether it requires empirical testing as you suggested in post 163, suggesting even you don't have any clear idea what the hell you are talking about).

effective procedure"? Once again you resort to a vague fog of words

An effective procedure can be a proof or a step by step process/algorithm. The euclidian algorithm for division is an example of a step by step process.

I used a step by step process to decision t'. There are no holes or gaps in the process. There are no undecidables left in the process. The only outcome of the process was t' and that answer is the unique outcome of the process.

If you think that you have an "effective procedure" for theoretically deriving the unknown value of the proper time t' for twin1 during the relative inertial motion as a mathematical function of other variables can be treated as known because they appear in the equation for twin2's total time (like BT, the proper time of acceleration, and a, the value of acceleration, and t, the proper time for twin2 on his own relative inertial motion phase in his frame...note that you cannot treat T as one of the known variables when deriving t', because this variable does not appear in twin2's total time) then show me the actual mathematical derivation, otherwise I'm going to assume you're just bluffing and have no clear idea of a procedure that will give a specific equation for this (and thus you have no theoretical procedure to determine whether twin1 or twin2 will finally have aged more, and what will be the precise ratio of their ages).

I'm going to assume you're just bluffing

No I am not.

If you will simply look at the equation, all variables T = BT + t' + c/a sinh( a*BT/c ) except t' are known.

Now, if you do not believe in the outcome, then you confess SR has a gap in its logic.

Please tell me specifically in the above equation what is not known and why.

 

[JesseM]

This implies all relative motion must apply LT plus an R of S argument. But, R of S is already built into LT.

Yes, the Relativity of Simultaneity is built into the LT. But you didn't use the LT when you said the time of twin1's relative inertial motion phase would be t/gamma! Instead it seems you just used a misapplied version of the time dilation formula. In order to use the LT, you have to pick some specific events with known coordinates in one frame, then the LT will give the coordinates of the same events in the other frame.

Anyway, the time dilation formula would say that if tA is the time twin1 stops accelerating in the launch frame, and tB is the time twin2 starts accelerating in the launch frame, and the time between tA and tB in the launch frame is t (which is also the proper time twin2 experiences between tA and tB since twin2 is at rest in this frame), then the amount of proper time that elapses on twin1's clock between tA and tB will be t/gamma. But this is not the proper time twin1 experiences during the entire relative inertial motion phase, because the event on twin1's worldline that occurs at time tB, simultaneous with twin2 starting his acceleration in the launch frame (event 3a in my notation from post 122) happens before the event on twin1's worldline that is simultaneous with twin2 starting his acceleration in twin1's own frame (event 4a). So, that's although twin1 has experienced a proper time of t/gamma at 3a, this is not the total proper time experienced by twin1 during the relative inertial motion phase.

effective procedure"? Once again you resort to a vague fog of words

An effective procedure can be a proof or a step by step process/algorithm. The euclidian algorithm for division is an example of a step by step process.

I used a step by step process to decision t'.

Did you? When? What is your equation for t' expressed only in terms of variables that appear in the proper time for twin2, i.e. variables that appear in the equation (c/a)*sinh(a*BT/c) + t + BT?

If you will simply look at the equation, all variables T = BT + t' + c/a sinh( a*BT/c ) except t' are known.

T is not known either, not as a function of variables that appear in the equation for twin2's proper time (t' does not appear in that equation). You need both twins' proper times expressed in terms of the same set of variables if you want to compare their proper times to see whose is larger, and by how much.

 

[cfrogue]

Yes, the Relativity of Simultaneity is built into the LT. But you didn't use the LT when you said the time of twin1's relative inertial motion phase would be t/gamma! Instead it seems you just used a misapplied version of the time dilation formula. In order to use the LT, you have to pick some specific events with known coordinates in one frame, then the LT will give the coordinates of the same events in the other frame.

Anyway, the time dilation formula would say that if tA is the time twin1 stops accelerating in the launch frame, and tB is the time twin2 starts accelerating in the launch frame, and the time between tA and tB in the launch frame is t (which is also the proper time twin2 experiences between tA and tB since twin2 is at rest in this frame), then the amount of proper time that elapses on twin1's clock between tA and tB will be t/gamma. But this is not the proper time twin1 experiences during the entire relative inertial motion phase, because the event on twin1's worldline that occurs at time tB, simultaneous with twin2 starting his acceleration in the launch frame (event 3a in my notation from post 122) happens before the event on twin1's worldline that is simultaneous with twin2 starting his acceleration in twin1's own frame (event 4a). So, that's although twin1 has experienced a proper time of t/gamma at 3a, this is not the total proper time experienced by twin1 during the relative inertial motion phase.

I did not say twin1 will elapse t/gamma in its proper time. If I did I meant twin1 will elapse t/gamma in the time of twin2.

Did you? When? What is your equation for t' expressed only in terms of variables that appear in the proper time for twin2, i.e. variables that appear in the equation (c/a)*sinh(a*BT/c) + t + BT?

False, BT is the elapsed proper time of twin1 since it did the burn.
(c/a)*sinh(a*BT/c) is the elapsed proper time of twin1 while twin2 burns for BT.
T is the calculated proper time of twin1 when twin2 entered the frame.

What I am trying to do is to calculate what transpired in twin1's frame according to twin1.
The whole point of this is that twin1 does not know the relative motion phase elapsed time unless twin1 does a calculation. That is because twin1 does not know when twin2 started the burn. But, by calculation, twin1 knows when twin2 entered the frame ie stopped the burn by the calculation of the D/c business. Then, twin1 knows the start of the relative motion phase because it occurs right after its burn BT. Then, since it knows when twin2 stopped its burn in twin1's proper time, then twin1 subtracts (c/a)*sinh(a*BT/c) from the time twin2 entered the frame and then knows when twin2 started its burn.

T is not known either, not as a function of variables that appear in the equation for twin2's proper time (t' does not appear in that equation). You need both twins' proper times expressed in terms of the same set of variables if you want to compare their proper times to see whose is larger, and by how much.

We are not doing twin2, we are calculating twin1 in the proper time of twin1.

 

[JesseM]

I did not say twin1 will elapse t/gamma in its proper time. If I did I meant twin1 will elapse t/gamma in the time of twin2.

So you don't have a method to calculate the proper time of twin1 in his relative inertial motion phase, in such a way that you can compare his total aging to twin2's? Wasn't comparing their total aging the whole point of what you were asking in post 49, which got this entire lengthy discussion started?

False, BT is the elapsed proper time of twin1 since it did the burn.
(c/a)*sinh(a*BT/c) is the elapsed proper time of twin1 while twin2 burns for BT.
T is the calculated proper time of twin1 when twin2 entered the frame.

But you only "calculated" the unknown variable T in terms of the equally unknown variable t', the proper time of twin1 in his relative inertial motion phase. If you don't know how t' relates to t, the proper time of twin2 in his relative inertial motion phase, then you have no idea which twin is older at the end, which was the question that you were supposedly interested in.

What I am trying to do is to calculate what transpired in twin1's frame according to twin1.
The whole point of this is that twin1 does not know the relative motion phase elapsed time unless twin1 does a calculation. That is because twin1 does not know when twin2 started the burn. But, by calculation, twin1 knows when twin2 entered the frame ie stopped the burn by the calculation of the D/c business. Then, twin1 knows the start of the relative motion phase because it occurs right after its burn BT. Then, since it knows when twin2 stopped its burn in twin1's proper time, then twin1 subtracts (c/a)*sinh(a*BT/c) from the time twin2 entered the frame and then knows when twin2 started its burn.

Yes, if this were an actual empirical experiment twin1 could just find the time he received the signal by observation, then do the subtraction of D/c and BT and (c/a)*sinh(a*BT/c) to find t', the time of his own relative inertial motion phase in his frame. But if we are supposed to be calculating t' rather than doing an empirical experiment, then since you don't know how either T or t' relate to twin2's total elapsed time (BT + t + (c/a)*sinh(a*BT/c)), then you don't have an actual method to calculate which twin is older at the end (even though this question is completely answerable in SR theoretically).

We are not doing twin2, we are calculating twin1 in the proper time of twin1.

You aren't really "calculating" anything helpful to the problem though, you're just defining one unknown variable in terms of another unknown variable (either defining T in terms of t' or vice versa)...you haven't given any non-empirical procedure to actually find the value of either T or t' if we have known values for the variables a, BT, and t.

 

[cfrogue]

So you don't have a method to calculate the proper time of twin1 in his relative inertial motion phase, in such a way that you can compare his total aging to twin2's? Wasn't comparing their total aging the whole point of what you were asking in post 49, which got this entire lengthy discussion started?

Yes, this is the context and yes, I do have a method. I have showed it over and over.
There is nothing wrong with it either.

But you only "calculated" the unknown variable T in terms of the equally unknown variable t', the proper time of twin1 in his relative inertial motion phase. If you don't know how t' relates to t, the proper time of twin2 in his relative inertial motion phase, then you have no idea which twin is older at the end, which was the question that you were supposedly interested in.

No I did not.

I calculaterd T as the time twin2 stopped accelerating. You have agreed to this over and over.

Yes, if this were an actual empirical experiment twin1 could just find the time he received the signal by observation, then do the subtraction of D/c and BT and (c/a)*sinh(a*BT/c) to find t', the time of his own relative inertial motion phase in his frame. But if we are supposed to be calculating t' rather than doing an empirical experiment, then since you don't know how either T or t' relate to twin2's total elapsed time (BT + t + (c/a)*sinh(a*BT/c)), then you don't have an actual method to calculate which twin is older at the end (even though this question is completely answerable in SR theoretically).

I have already agreed I cannot do this under theory. This is why I am using a thought experiment according to the rules of the theory.

Do you realize the conclusions of the normal twin's experiment must also rely on results from the thought experiment.
There is nothing illegal in what I did.

You aren't really "calculating" anything helpful to the problem though, you're just defining one unknown variable in terms of another unknown variable (either defining T in terms of t' or vice versa)...you haven't given any non-empirical procedure to actually find the value of either T or t' if we have known values for the variables a, BT, and t.

That is false and I have had you agree with all the terms of the equations and all of their values except you are unable to agree to a simple math subtraction.

And, no I am defining T in terms of t' but arriving at its answer not by using t' but by using a light pulse and round trip speed of light calculation. I am not in a circular issue here and my reasoning is coherent and sound.

 

[JesseM]

So you don't have a method to calculate the proper time of twin1 in his relative inertial motion phase, in such a way that you can compare his total aging to twin2's? Wasn't comparing their total aging the whole point of what you were asking in post 49, which got this entire lengthy discussion started?

Yes, this is the context and yes, I do have a method. I have showed it over and over.

You have a theoretical method to calcuate the proper time of twin1 in such a way that you can compare his total aging to twin2's? If so you haven't explained this method. Your equation for twin2's total time involved the variable t, and your equation for twin1's total time involved the variables T and t', but you never showed how to theoretically derive the relationship between t and T/t'. Without knowing the relationship, how do you expect to determine which twin has aged more? If t is much larger than t' then twin2 will have aged more in total, while if t' is much larger than t then twin1 will have aged more in total.

No I did not.

I calculaterd T as the time twin2 stopped accelerating. You have agreed to this over and over.

You didn't calculate it in a way that allows us to determine how the value of T relates to the total elapsed time for twin2, i.e. you don't know whether T is larger than or smaller than (c/a)*sinh(a*BT/c) + t + BT.

I have already agreed I cannot do this under theory.

Sure you can, it would be a pretty poor theory that couldn't answer questions about proper time in a well-defined thought-experiment like this one! I already explained the theoretical method to determine the total proper time for twin1 as a function of a, BT and t (the variables which appear in the equation for the total proper time of twin2), that was what posts 122 and 134 were all about. Again, the total time for twin1 would be the sum of these pieces:

1a to 2a: BT

2a to 3a: t/gamma

3a to 4a: here we use the formula gamma*d*v/c^2 found in post 134, where d is the distance between twin1 and twin2 in the launch frame at the moment twin2 begins to accelerate, and v is twin1's velocity in the launch frame at that moment. And d and v can themselves be found as functions of a and BT and t using the relativistic rocket equations, v (twin1's final velocity in the launch frame) should be c*tanh(a*BT/c), while d should be (c^2/a)*[cosh(a*BT/c) - 1] + v*t. Alternately, if t1 = the time twin1 stops accelerating in the launch frame = (c/a)*sinh(a*BT/c), then v = a*t1/sqrt[1 + (a*t1/c)^2], and d would be (c^2/a)*(sqrt[1 + (a*t1/c)^2] - 1) + v*t.

4a to 5a: (c/a)*sinh(a*BT/c)

So, summing those five terms will give you twin1's total proper time T as a function of a, BT and t. Note that I also gave a different but equally valid method for calculating twin1's total proper time in the last two paragraphs of post 134.

Do you realize the conclusions of the normal twin's experiment must also rely on results from the thought experiment.

If the velocities and time intervals are known than you can calculate how much each twin ages, you don't have to include unknown variables which would require an empirical experiment to determine.

You aren't really "calculating" anything helpful to the problem though, you're just defining one unknown variable in terms of another unknown variable (either defining T in terms of t' or vice versa)...you haven't given any non-empirical procedure to actually find the value of either T or t' if we have known values for the variables a, BT, and t.

That is false and I have had you agree with all the terms of the equations and all of their values except you are unable to agree to a simple math subtraction.

I agreed with your equations, but none of your equations give a purely theoretical procedure for calculating T or t' as a function of a, BT, and t (as mine did above).

And, no I am defining T in terms of t' but arriving at its answer not by using t' but by using a light pulse and round trip speed of light calculation.

I was referring to the equation you wrote down earlier, namely T = BT + t' + c/a sinh( a*BT/c ). If you want to include the whole pointless business of light signals, that just adds a third unknown, the proper time of twin1 at the moment he receives the signal...call that T'. Then your equations would be T = T' - D/c and T = BT + t' + c/a sinh( a*BT/c ). And you don't have any theoretical procedure for calculating T, t' or T as a function of a, BT and t, so you don't have a theoretical procedure to determine which twin is older or by how much, which was the original point of this whole discussion.

 

[cfrogue]

You have a theoretical method to calcuate the proper time of twin1 in such a way that you can compare his total aging to twin2's? If so you haven't explained this method. Your equation for twin2's total time involved the variable t, and your equation for twin1's total time involved the variables T and t', but you never showed how to theoretically derive the relationship between t and T/t'. Without knowing the relationship, how do you expect to determine which twin has aged more? If t is much larger than t' then twin2 will have aged more in total, while if t' is much larger than t then twin1 will have aged more in total.

You didn't calculate it in a way that allows us to determine how the value of T relates to the total elapsed time for twin2, i.e. you don't know whether T is larger than or smaller than (c/a)*sinh(a*BT/c) + t + BT.

Sure you can, it would be a pretty poor theory that couldn't answer questions about proper time in a well-defined thought-experiment like this one! I already explained the theoretical method to determine the total proper time for twin1 as a function of a, BT and t (the variables which appear in the equation for the total proper time of twin2), that was what posts 122 and 134 were all about. Again, the total time for twin1 would be the sum of these pieces:

1a to 2a: BT

2a to 3a: t/gamma

3a to 4a: here we use the formula gamma*d*v/c^2 found in post 134, where d is the distance between twin1 and twin2 in the launch frame at the moment twin2 begins to accelerate, and v is twin1's velocity in the launch frame at that moment. And d and v can themselves be found as functions of a and BT and t using the relativistic rocket equations, v (twin1's final velocity in the launch frame) should be c*tanh(a*BT/c), while d should be (c^2/a)*[cosh(a*BT/c) - 1] + v*t. Alternately, if t1 = the time twin1 stops accelerating in the launch frame = (c/a)*sinh(a*BT/c), then v = a*t1/sqrt[1 + (a*t1/c)^2], and d would be (c^2/a)*(sqrt[1 + (a*t1/c)^2] - 1) + v*t.

4a to 5a: (c/a)*sinh(a*BT/c)

So, summing those five terms will give you twin1's total proper time T as a function of a, BT and t. Note that I also gave a different but equally valid method for calculating twin1's total proper time in the last two paragraphs of post 134.


If the velocities and time intervals are known than you can calculate how much each twin ages, you don't have to include unknown variables which would require an empirical experiment to determine.


I agreed with your equations, but none of your equations give a purely theoretical procedure for calculating T or t' as a function of a, BT, and t (as mine did above).

I was referring to the equation you wrote down earlier, namely T = BT + t' + c/a sinh( a*BT/c ). If you want to include the whole pointless business of light signals, that just adds a third unknown, the proper time of twin1 at the moment he receives the signal...call that T'. Then your equations would be T = T' - D/c and T = BT + t' + c/a sinh( a*BT/c ). And you don't have any theoretical procedure for calculating T, t' or T as a function of a, BT and t, so you don't have a theoretical procedure to determine which twin is older or by how much, which was the original point of this whole discussion.

You are confused.

Twin1 is trying to calculate.

Twin2 knows what to do.

It needs to be established in twin1 all the timing.

This has been done.

Once that is done, twin1 can calculate twin2.

We have done that.