Need help understanding the twins

[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.

 

[JesseM]

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.

The twins are imaginary characters in a thought-experiment, we are trying to calculate which twin will be older given the premises of the thought experiment. If we are given specific values for a, BT, and t, then we can calculate twin2's final age using the equation (c/a)*sinh(a*BT/c) + t + BT, yes? But you haven't given us any way to calculate twin1's final age given the same information, and thus no way to calculate whether twin1 will be older or younger than twin2 at the end (and we can define 'the end' for twin1 in terms of taking his age T' when he receives a signal from twin2 and subtracting D/c from T', but this doesn't help in calculating what age he will actually be at the end given specific values for a, BT, and t).

For example, suppose we know that BT=0.5 years, a=2 light years/year^2, and t=5 years. In this case, twin2's elapsed time will be (1/2)*sinh(1) + 5 + 0.5 = 6.0876 years. Does your method give you a way to use this information to calculate a specific numeric value for twin1's elapsed time? Mine does.

 

[cfrogue]

The twins are imaginary characters in a thought-experiment, we are trying to calculate which twin will be older given the premises of the thought experiment. If we are given specific values for a, BT, and t, then we can calculate twin2's final age using the equation (c/a)*sinh(a*BT/c) + t + BT, yes? But you haven't given us any way to calculate twin1's final age given the same information, and thus no way to calculate whether twin1 will be older or younger than twin2 at the end (and we can define 'the end' for twin1 in terms of taking his age T' when he receives a signal from twin2 and subtracting D/c from T', but this doesn't help in calculating what age he will actually be at the end given specific values for a, BT, and t).

For example, suppose we know that BT=0.5 years, a=2 light years/year^2, and t=5 years. In this case, twin2's elapsed time will be (1/2)*sinh(1) + 5 + 0.5 = 6.0876 years. Does your method give you a way to use this information to calculate a specific numeric value for twin1's elapsed time? Mine does.

Point taken.

I will provide a specific example.

 

[cfrogue]

The twins are imaginary characters in a thought-experiment, we are trying to calculate which twin will be older given the premises of the thought experiment. If we are given specific values for a, BT, and t, then we can calculate twin2's final age using the equation (c/a)*sinh(a*BT/c) + t + BT, yes? But you haven't given us any way to calculate twin1's final age given the same information, and thus no way to calculate whether twin1 will be older or younger than twin2 at the end (and we can define 'the end' for twin1 in terms of taking his age T' when he receives a signal from twin2 and subtracting D/c from T', but this doesn't help in calculating what age he will actually be at the end given specific values for a, BT, and t).

For example, suppose we know that BT=0.5 years, a=2 light years/year^2, and t=5 years. In this case, twin2's elapsed time will be (1/2)*sinh(1) + 5 + 0.5 = 6.0876 years. Does your method give you a way to use this information to calculate a specific numeric value for twin1's elapsed time? Mine does.

I have looked at your example.

v = cosh(aBT/c) = cosh(1) = 1.54308063482 c

It seems your acceleration is too high.

 

[JesseM]

I have looked at your example.

v = cosh(aBT/c) = cosh(1) = 1.54308063482 c

It seems your acceleration is too high.

You're using an incorrect equation for v, the correct equation given on the http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html ), which makes sense since you can have an arbitrarily high proper acceleration for arbitrarily long periods of proper time without ever exceeding c.

 

[cfrogue]

You're using an incorrect equation for v, the correct equation given on the http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html ), which makes sense since you can have an arbitrarily high proper acceleration for arbitrarily long periods of proper time without ever exceeding c.

OK, OK, you are right.

Thanks for the calculator link.

Alright v = 0.7615941559557649c.

 

[cfrogue]

You're using an incorrect equation for v, the correct equation given on the http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html ), which makes sense since you can have an arbitrarily high proper acceleration for arbitrarily long periods of proper time without ever exceeding c.

OK, after the burn of twin1 do you agree
d = (c2/a) [ch(aT/c) - 1]

Or, am I going to have to fight with you over this?

 

[JesseM]

OK, after the burn of twin1 do you agree
d = (c2/a) [ch(aT/c) - 1]

Or, am I going to have to fight with you over this?

In the launch frame where twin1 was at rest before accelerating (and twin2 remains at rest for time t after twin1 finishes accelerating), yes I agree this is the distance at the moment twin1 finishes his burn.

 

[cfrogue]

In the launch frame where twin1 was at rest before accelerating (and twin2 remains at rest for time t after twin1 finishes accelerating), yes I agree this is the distance at the moment twin1 finishes his burn.

So, do you agree from the accelerated frame

d = 1/a * ( cosh(a BT) - 1 )

is the distance between the ships?

 

[JesseM]

So, do you agree from the accelerated frame

d = 1/a * ( cosh(a BT) - 1 )

is the distance between the ships?

The relativistic rocket equations only work in inertial frames...by "accelerated frame" do you just mean the inertial frame where twin1 is at rest after he finishes his burn? If so, then no, this is incorrect. (1/a)*(cosh(a*BT) - 1) would be the distance in this frame between the position where twin1 started his burn and the position where he stopped his burn, but in this frame twin2 is continuing to move at v=tanh(a*BT) throughout the time twin1 is accelerating, and in this frame twin1's acceleration lasts for a time of t=(1/a)*sinh(a*BT). So by the time twin1 finishes his acceleration, twin2 will have moved a distance of vt = (1/a)*sinh(a*BT)*tanh(a*BT) from the position where twin1 started his acceleration, and twin1 will have moved a distance of (1/a)*(cosh(a*BT) - 1) in the opposite direction, so their distance at the time twin1 stops accelerating will be (1/a)*(sinh(a*BT)*tanh(a*BT) + cosh(a*BT) - 1).

 

[cfrogue]

The relativistic rocket equations only work in inertial frames...by "accelerated frame" do you just mean the inertial frame where twin1 is at rest after he finishes his burn? If so, then no, this is incorrect. (1/a)*(cosh(a*BT) - 1) would be the distance in this frame between the position where twin1 started his burn and the position where he stopped his burn, but in this frame twin2 is continuing to move at v=tanh(a*BT) throughout the time twin1 is accelerating, and in this frame twin1's acceleration lasts for a time of t=(1/a)*sinh(a*BT). So by the time twin1 finishes his acceleration, twin2 will have moved a distance of vt = (1/a)*sinh(a*BT)*tanh(a*BT) from the position where twin1 started his acceleration, and twin1 will have moved a distance of (1/a)*(cosh(a*BT) - 1) in the opposite direction, so their distance at the time twin1 stops accelerating will be (1/a)*(sinh(a*BT)*tanh(a*BT) + cosh(a*BT) - 1).

show me a mainstream paper.

 

[JesseM]

show me a mainstream paper.

I don't know of any mainstream paper that specifically considers the problem of how the distance between twins changes when one moves inertially while the other accelerates away at constant proper acceleration, and looks at this from the perspective of a frame where the inertial twin is not at rest. However, most of the reasoning here is a pretty basic application of SR principles:

1. Do you disagree that if twin1 has velocity v=tanh(a*BT) after he finishes accelerating in the launch frame, while twin2 is at rest in this frame, then if we transform into a new inertial frame where twin1 is at rest after he finishes accelerating, then twin2 must have a constant velocity v=tanh(a*BT) in this frame?

2. Do you disagree that if it takes time t1 = (1/a)*sinh(a*BT) in this frame from the beginning to the end of twin1 accelerating, then if twin2 has velocity v=tanh(a*BT) in this frame, he will have moved a distance of v*t1 = (1/a)*sinh(a*BT)*tanh(a*BT) between the time twin1 starts accelerating and the time he stops accelerating?

3. Do you disagree that in this frame twin1 moves a distance of (1/a)*(cosh(a*BT) - 1) in this frame between the time he starts accelerating and the time he stops?

4. If you don't disagree with any of the above, then the only remaining question should be whether twin1 and twin2 moved in the same direction from the point where twin1 started accelerating, or in opposite directions. I admit I didn't think this part through very carefully, after further consideration I think the answer should be that they both moved in the same direction (since twin1's initial speed and direction was the same as twin2's, but twin2 continued to move at the same speed and direction while twin1's speed dropped to zero) and that therefore twin1's distance covered should be subtracted from twin2's distance covered rather than added, in which case the distance would be (1/a)*(sinh(a*BT)*tanh(a*BT) - cosh(a*BT) + 1).

 

[cfrogue]

I don't know of any mainstream paper that specifically considers the problem of how the distance between twins changes when one moves inertially while the other accelerates away at constant proper acceleration, and looks at this from the perspective of a frame where the inertial twin is not at rest. However, most of the reasoning here is a pretty basic application of SR principles:

1. Do you disagree that if twin1 has velocity v=tanh(a*BT) after he finishes accelerating in the launch frame, while twin2 is at rest in this frame, then if we transform into a new inertial frame where twin1 is at rest after he finishes accelerating, then twin2 must have a constant velocity v=tanh(a*BT) in this frame?

2. Do you disagree that if it takes time t1 = (1/a)*sinh(a*BT) in this frame from the beginning to the end of twin1 accelerating, then if twin2 has velocity v=tanh(a*BT) in this frame, he will have moved a distance of v*t1 = (1/a)*sinh(a*BT)*tanh(a*BT) between the time twin1 starts accelerating and the time he stops accelerating?

3. Do you disagree that in this frame twin1 moves a distance of (1/a)*(cosh(a*BT) - 1) in this frame between the time he starts accelerating and the time he stops?

4. If you don't disagree with any of the above, then the only remaining question should be whether twin1 and twin2 moved in the same direction from the point where twin1 started accelerating, or in opposite directions. I admit I didn't think this part through very carefully, after further consideration I think the answer should be that they both moved in the same direction (since twin1's initial speed and direction was the same as twin2's, but twin2 continued to move at the same speed and direction while twin1's speed dropped to zero) and that therefore twin1's distance covered should be subtracted from twin2's distance covered rather than added, in which case the distance would be (1/a)*(sinh(a*BT)*tanh(a*BT) - cosh(a*BT) + 1).

What was wrong with my original method of the D/c business?

 

[JesseM]

What was wrong with my original method of the D/c business?

The fact that it doesn't actually give you a way to calculate twin1's final age when twin2 stops accelerating (i.e. the age he is when he receives the light signal, minus D/c) given known values for the variables BT, a, and t that determine twin2's final age. If you think it does, then please show me how you'd calculate a numerical value for twin1's final age given the example values I gave, namely BT=0.5 years, a=2 light years/year^2, and t=5 years.

Again, my method does allow you to calculate twin1's final age given a set of values like this. I can show you the details of the calculation if you're interested.

 

[marxmarvelous]

At Johns Hopkins as a Jr. Instructor the "twins" paradox was never aproached. Likely there has been two generations of teachers who think there is an "answer" to the paradox.

Einstein gave it in 1915; GR. He termed the paradox as "a failure of epistemology." Western 'Greek think" begins with postulates land thru a system of logic arrives at solutions. An unspoken postulate in the twins thing is that the mass environment of neither twin is necessary for fiddling with the SR transformation formulas. Wrong so he thought up GR. If one wants to map wrinkles onto faces of the "twins" one must use the GR equations to the travels of both.
Einstein was largely ignored by the media after he kept repeating that immigrants to palestine must get along with Palestinians and that an international police force must have the power to insure the planet that no nuclear weapons existed anywhere. He also insisted that infinities in formulas weren't physical simply imperfect postulates producing math artifacts. He died before the media was allowed to accept infinite densities here and there.
I expect to be told I am wrong. Could someone carry this century old "twin" thought to a physicist with whom I can discuss this. Thanks

 

[cfrogue]

At Johns Hopkins as a Jr. Instructor the "twins" paradox was never aproached. Likely there has been two generations of teachers who think there is an "answer" to the paradox.

Einstein gave it in 1915; GR. He termed the paradox as "a failure of epistemology." Western 'Greek think" begins with postulates land thru a system of logic arrives at solutions. An unspoken postulate in the twins thing is that the mass environment of neither twin is necessary for fiddling with the SR transformation formulas. Wrong so he thought up GR. If one wants to map wrinkles onto faces of the "twins" one must use the GR equations to the travels of both.
Einstein was largely ignored by the media after he kept repeating that immigrants to palestine must get along with Palestinians and that an international police force must have the power to insure the planet that no nuclear weapons existed anywhere. He also insisted that infinities in formulas weren't physical simply imperfect postulates producing math artifacts. He died before the media was allowed to accept infinite densities here and there.
I expect to be told I am wrong. Could someone carry this century old "twin" thought to a physicist with whom I can discuss this. Thanks

Good, this is not the normal twins paradox.

 

[cfrogue]

The fact that it doesn't actually give you a way to calculate twin1's final age when twin2 stops accelerating (i.e. the age he is when he receives the light signal, minus D/c) given known values for the variables BT, a, and t that determine twin2's final age. If you think it does, then please show me how you'd calculate a numerical value for twin1's final age given the example values I gave, namely BT=0.5 years, a=2 light years/year^2, and t=5 years.

Again, my method does allow you to calculate twin1's final age given a set of values like this. I can show you the details of the calculation if you're interested.

Yes, I am interested.

No, I cannot give a specific example and you know this.

I would need to actually know the distance between the two when twin2 entered the frame.

For me this does not matter. But for some it does.

So, I am interested how you decide t'.

 

[JesseM]

Yes, I am interested.

OK, I'll get to this soon, but first I want to ask a few more questions about how your own approach is supposed to work:

No, I cannot give a specific example and you know this.

I would need to actually know the distance between the two when twin2 entered the frame.

You can calculate the distance between the two in the final rest frame when twin2 finishes his acceleration given BT, a, and t, but it seems to me that even with this information your approach does not tell us who is older. As I said in post #186, if we are analyzing things from the perspective of the inertial frame where they are at rest after acceleration, then in this frame twin1 moves a distance of (c/a)*(cosh(a*BT/c) - 1) during his acceleration phase, after which he is at rest in this frame. Meanwhile we know that twin2 moves inertially for a proper time of (c/a)*sinh(a*BT/c) + t before beginning to accelerate, and in this frame twin2 is moving at constant velocity v=c*tanh(a*BT/c) before beginning to accelerate, so according to the time dilation equation the coordinate time in this frame before twin2 begins to accelerate must be gamma times the proper time, i.e. gamma*[(c/a)*sinh(a*BT/c) + t], and the coordinate distance twin2 covers in this time is just given by velocity*coordinate time, or gamma*c*tanh(a*BT/c)*[(c/a)*sinh(a*BT/c) + t]. Then during twin2's acceleration phase, twin2 will cover an additional distance of (c/a)*(cosh(a*BT/c) - 1). So, the total distance twin2 travels from the origin (the point where the two twins first separated) in this frame is:

gamma*c*tanh(a*BT/c)*[(c/a)*sinh(a*BT/c) + t] + (c/a)*(cosh(a*BT/c) - 1)

Whereas the total distance twin1 travels from the origin in this frame is:

(c/a)*(cosh(a*BT/c) - 1)

So, subtracting the second from the first shows that the final distance between them will be:

gamma*c*tanh(a*BT/c)*[(c/a)*sinh(a*BT/c) + t]

Since gamma is 1/sqrt(1 - v^2/c^2), we can plug in v=c*tanh(a*BT/c), showing that gamma = 1/sqrt(1 - tanh^2(a*BT/c)). And making use of the hyperbolic trig identities here, we known tanh(x) = sinh(x)/cosh(x), so gamma = 1/sqrt(1 - sinh^2/cosh^2) = 1/[sqrt(1/cosh^2)*sqrt(cosh^2 - sinh^2)], and since another identity says that cosh^2 - sinh^2 = 1, this reduces to 1/sqrt(1/cosh^2) = cosh(a*BT/c). So with gamma = cosh(a*BT/c), the final distance between them can be rewritten as:

c*cosh(a*BT/c)*tanh(a*BT/c)*[(c/a)*sinh(a*BT/c) + t]

And again making use of the fact that tanh(x) = sinh(x)/cosh(x), this reduces to:

c*sinh(a*BT/c)*[(c/a)*sinh(a*BT/c) + t]

So if you plug in BT=0.5 years, a=2 light years/year^2, and t=5 years, this becomes:

sinh(1)*[sinh(1) + 5] = 1.1752011936438014*[1.1752011936438014 + 5] = 7.25710381376082 light years.

But even with the final distance known, can you calculate a numerical value for the final age of twin1?

 

[cfrogue]

OK, I'll get to this soon, but first I want to ask a few more questions about how your own approach is supposed to work:

You can calculate the distance between the two in the final rest frame when twin2 finishes his acceleration given BT, a, and t, but it seems to me that even with this information your approach does not tell us who is older. As I said in post #186, if we are analyzing things from the perspective of the inertial frame where they are at rest after acceleration, then in this frame twin1 moves a distance of (c/a)*(cosh(a*BT/c) - 1) during his acceleration phase, after which he is at rest in this frame. Meanwhile we know that twin2 moves inertially for a proper time of (c/a)*sinh(a*BT/c) + t before beginning to accelerate, and in this frame twin2 is moving at constant velocity v=c*tanh(a*BT/c) before beginning to accelerate, so the time in this frame before twin2 begins to accelerate must be gamma times the proper time, i.e. gamma*[(c/a)*sinh(a*BT/c) + t], and the distance twin2 covers in this time is just given by velocity*time, or gamma*c*tanh(a*BT/c)*[(c/a)*sinh(a*BT/c) + t]. Then during twin2's acceleration phase, twin2 will cover an additional distance of (c/a)*(cosh(a*BT/c) - 1). So, the total distance twin2 travels from the origin (the point where the two twins first separated) in this frame is:

gamma*c*tanh(a*BT/c)*[(c/a)*sinh(a*BT/c) + t] + (c/a)*(cosh(a*BT/c) - 1)

Whereas the total distance twin1 travels from the origin in this frame is:

(c/a)*(cosh(a*BT/c) - 1)

So, subtracting the second from the first shows that the final distance between them will be:

gamma*c*tanh(a*BT/c)*[(c/a)*sinh(a*BT/c) + t]

Since gamma is 1/sqrt(1 - v^2/c^2), we can plug in v=c*tanh(a*BT/c), showing that gamma = 1/sqrt(1 - tanh^2(a*BT/c)). And making use of the hyperbolic trig identities here, we known tanh(x) = sinh(x)/cosh(x), so gamma = 1/sqrt(1 - sinh^2/cosh^2) = 1/[sqrt(1/cosh^2)*sqrt(cosh^2 - sinh^2)], and since another identity says that cosh^2 - sinh^2 = 1, this reduces to 1/sqrt(1/cosh^2) = cosh(a*BT/c). So with gamma = cosh(a*BT/c), the final distance between them can be rewritten as:

c*cosh(a*BT/c)*tanh(a*BT/c)*[(c/a)*sinh(a*BT/c) + t]

And again making use of the fact that tanh(x) = sinh(x)/cosh(x), this reduces to:

c*sinh(a*BT/c)*[(c/a)*sinh(a*BT/c) + t]

So if you plug in BT=0.5 years, a=2 light years/year^2, and t=5 years, this becomes:

sinh(1)*[sinh(1) + 5] = 1.1752011936438014*[1.1752011936438014 + 5] = 7.25710381376082 light years.

But even with the final distance known, can you calculate a numerical value for the final age of twin1?

I am going to need some time to follow your argument.


But even with the final distance known, can you calculate a numerical value for the final age of twin1?

Yes, why not?

It is the same type of problem.

I sub off the distance of the twin1 accel phase and then I sub the distance of the twin2 accel phase.

I am left with the distance of the relative motion phase and thus, d/v = t'.

No?

 

[JesseM]

I am going to need some time to follow your argument.But even with the final distance known, can you calculate a numerical value for the final age of twin1?

Yes, why not?

It is the same type of problem.

I sub off the distance of the twin1 accel phase and then I sub the distance of the twin2 accel phase.

I am left with the distance of the relative motion phase and thus, d/v = t'.

No?

Sure, that works. But I thought your calculation method was supposed to involve subtracting D/c from the time twin1 receives the signal--that doesn't appear to happen anywhere in the method above.

My method for calculating twin1's time from the perspective of the final rest frame was basically similar but without the need to consider distances. I would just say that since we know twin2 moved inertially at v=c*tanh(a*BT/c) before beginning to accelerate, and we know twin2 experienced a proper time of [(c/a)*sinh(a*BT/c) + t] before beginning to accelerate, the according to the time dilation equation the coordinate time in this frame before twin2 accelerates would just be gamma times the proper time, i.e. gamma*[(c/a)*sinh(a*BT/c) + t]. As I showed in my previous post, if v=c*tanh(a*BT/c) then gamma=cosh(a*BT/c), so the coordinate time before twin2 accelerates can also be written as cosh(a*BT/c)*[(c/a)*sinh(a*BT/c) + t]. Then the coordinate time for twin2 to accelerate in this frame is (c/a)*sinh(a*BT/c), so the total coordinate time from start to finish is cosh(a*BT/c)*[(c/a)*sinh(a*BT/c) + t] + (c/a)*sinh(a*BT/c).

Now, we know that in this frame it took twin1 a coordinate time of (c/a)*sinh(a*BT/c) to do his own acceleration, so the coordinate time from the end of twin1's acceleration to the end of twin2's acceleration must be:

[cosh(a*BT/c)*[(c/a)*sinh(a*BT/c) + t] + (c/a)*sinh(a*BT/c)] - (c/a)*sinh(a*BT/c)

Which just reduces to cosh(a*BT/c)*[(c/a)*sinh(a*BT/c) + t]...since twin1 was at rest throughout this period, this would also be the proper time for twin1 from the end of his acceleration to the moment when twin2 stopped his own acceleration. And we know the proper time for twin1 during his acceleration was BT, so add them together and we have twin1's total proper time from start to finish:

BT + cosh(a*BT/c)*[(c/a)*sinh(a*BT/c) + t]

Since gamma = cosh(a*BT/c) is greater than 1, this total proper time will naturally be larger than twin2's total proper time from start to finish, i.e. BT + (c/a)*sinh(a*BT/c) + t. So, twin1 will be older than twin2 at the moment twin2 stops his acceleration and we compare their ages in the final rest frame.

 

[cfrogue]

Sure, that works. But I thought your calculation method was supposed to involve subtracting D/c from the time twin1 receives the signal--that doesn't appear to happen anywhere in the method above.

My method for calculating twin1's time from the perspective of the final rest frame was basically similar but without the need to consider distances. I would just say that since we know twin2 moved inertially at v=c*tanh(a*BT/c) before beginning to accelerate, and we know twin2 experienced a proper time of [(c/a)*sinh(a*BT/c) + t] before beginning to accelerate, the according to the time dilation equation the coordinate time in this frame before twin2 accelerates would just be gamma times the proper time, i.e. gamma*[(c/a)*sinh(a*BT/c) + t]. As I showed in my previous post, if v=c*tanh(a*BT/c) then gamma=cosh(a*BT/c), so the coordinate time before twin2 accelerates can also be written as cosh(a*BT/c)*[(c/a)*sinh(a*BT/c) + t]. Then the coordinate time for twin2 to accelerate in this frame is (c/a)*sinh(a*BT/c), so the total coordinate time from start to finish is cosh(a*BT/c)*[(c/a)*sinh(a*BT/c) + t] + (c/a)*sinh(a*BT/c).

Now, we know that in this frame it took twin1 a coordinate time of (c/a)*sinh(a*BT/c) to do his own acceleration, so the coordinate time from the end of twin1's acceleration to the end of twin2's acceleration must be:

[cosh(a*BT/c)*[(c/a)*sinh(a*BT/c) + t] + (c/a)*sinh(a*BT/c)] - (c/a)*sinh(a*BT/c)

Which just reduces to cosh(a*BT/c)*[(c/a)*sinh(a*BT/c) + t]...since twin1 was at rest throughout this period, this would also be the proper time for twin1 from the end of his acceleration to the moment when twin2 stopped his own acceleration. And we know the proper time for twin1 during his acceleration was BT, so add them together and we have twin1's total proper time from start to finish:

BT + cosh(a*BT/c)*[(c/a)*sinh(a*BT/c) + t]

Since gamma = cosh(a*BT/c) is greater than 1, this total proper time will naturally be larger than twin2's total proper time from start to finish, i.e. BT + (c/a)*sinh(a*BT/c) + t. So, twin1 will be older than twin2 at the moment twin2 stops his acceleration and we compare their ages in the final rest frame.

Since you said I am stupid in the other thread, what would make you think I could follow this unless you are stupid.

You are inconsistent.

 

[JesseM]

Since you said I am stupid in the other thread, what would make you think I could follow this unless you are stupid.

You are inconsistent.

I didn't say you were stupid, I said your argument was stupid. An intelligent person can make a stupid argument if they are too confident of their own correctness and are too quick to be snidely dismissive of the counterarguments made by others and not really pay attention to what people tell them, as seems to be the case with you on all of these threads.

 

[cfrogue]

I didn't say you were stupid, I said your argument was stupid. An intelligent person can make a stupid argument if they are too confident of their own correctness and are too quick to be snidely dismissive of the counterarguments made by others and not really pay attention to what people tell them, as seems to be the case with you on all of these threads.

Yes, maybe read this to yourself.

You can already tell I understand SR.

Have you considered yet I might be seeing something?

Answer the time dilation in the other thread and I will run you into a contradiction.

I think you can see this though.


Anyway, you said I was stupid. No matter, I am not.

But, I am going to look through your length argument here.

I will confess, you seem to have something with it.

Note how I am not so arrogant as to assume I know everything.

 

[Al68]

You can already tell I understand SR.

LOL. Are you aware that length contraction is a major part of SR?

 

[cfrogue]

LOL. Are you aware that length contraction is a major part of SR?

LOL, what is that?

 

[JesseM]

Yes, maybe read this to yourself.

You can already tell I understand SR.

No, I certainly wouldn't agree with that, you seem to understand some things but then you make really basic mistakes, like arguing that the past light cone would not look like a contracting light sphere if you plotted it over time. If you want to understand SR, I think you really need to find a text that develops it in a step-by-step manner (http://www.oberlin.edu/physics/dstyer/Einstein/SRBook.pdf . Your posts on relativity seem to be a fine example of this.

Anyway, you said I was stupid.

No, in fact I did not, I said your argument was stupid and that's all that I meant (if you want my real opinion of you, I think you are probably fairly intelligent but suffering from the type of overselfconfidence issue discussed above). But you love to tell me I'm wrong about the content of my own statements and opinions, apparently.