Solving the Twin Paradox with Lorentz Transformation

[MikeLizzi]

If you read what stevemg read a little more carefully, it is the Twins Paradox. It's just that in the final paragraph quoted here, the phrase "B's FOR" is unclear. What he should have said was something like "the inertial frame of reference in which B is initially (i.e. for the "outward" journey) at rest", and then it all makes sense.

No it is not Dr. Greg.
You don’t get it either. True, they are solving a problem about a round trip. The differential aging is solved for an observer who is always in one inertial reference frame (the initial inertial reference of the astronaut). I haven’t even checked if the calculations are correct because it doesn’t make any difference. Any calculation of differential aging made from one inertial reference frame will give the same results as the calculation made from earth. The paradox will not appear. The paradox appears when you calculate the different ages with the ASTRONAUT as the OBSERVER.

When the astronaut is the observer, the Earth goes away and comes back. The challenge is to correctly calculate the difference in elapsed time between the astronaut and Earth for that observer. That's what the Twins Paradox is about. The solution attempted above does not address that paradox.

 

[Dale]

When the astronaut is the observer, the Earth goes away and comes back. The challenge is to correctly calculate the difference in elapsed time between the astronaut and Earth for that observer. That's what the Twins Paradox is about. The solution attempted above does not address that paradox.

Yes, it does address the paradox because the value calculated is frame invariant. It is the same in ALL frames, inertial or non-inertial.

 

[stevmg]

No it is not Dr. Greg.
You don’t get it either. True, they are solving a problem about a round trip. The differential aging is solved for an observer who is always in one inertial reference frame (the initial inertial reference of the astronaut). I haven’t even checked if the calculations are correct because it doesn’t make any difference. Any calculation of differential aging made from one inertial reference frame will give the same results as the calculation made from earth. The paradox will not appear. The paradox appears when you calculate the different ages with the ASTRONAUT as the OBSERVER.

When the astronaut is the observer, the Earth goes away and comes back. The challenge is to correctly calculate the difference in elapsed time between the astronaut and Earth for that observer. That's what the Twins Paradox is about. The solution attempted above does not address that paradox.

Hey, Paesano -

No, no, no... "When the astronaut is the observer, the Earth goes away and comes back." That ain't true. The Earth does NOT go away and come back. The Earth goes away and keeps on going. The astronaut stays put and then chases the Earth. The way you are looking at it the frame of reference shifts direction, which is not "allowed." The astronaut can himself/herself shift direction but not his/her original frame of reference.

Garramone siempre ha ragione

(By the way, that's what my great great great grandfather said in his defense in Potenza, Lucania Province when he was being sentenced to hanging for stealing horses.)

Sort of like Robin Hood - he didn't steal from the rich and give to the poor... He stole from everybody and kept everything.

 

[DrGreg]

Any calculation of differential aging made from one inertial reference frame will give the same results as the calculation made from earth.

Well that is what the example demonstrates, and you only get an apparent contradiction if you mistakenly treat a non-inertial observer as if they were an inertial observer. So I don't really see what you are objecting to.

 

[AJ Bentley]

Gentlemen, Please!

Humour me, have a go at this restatement of the paradox in a different form.

********************************
Observer Alice, says to Bob ,who is just passing by at nearly the speed of light 'My Granny on Proxima Centauri is just sitting down to kippers for her tea'

Alice knows that because she has an Ansible (which allows her to see what Granny is doing right now without having to wait for the light to arrive)

Bob, who also has an Ansible, takes a quick look and says 'No she isn't, your Granny had her kippers for tea three days ago'

Explain.

You might like to show how Bob's Ansible allows him to travel back in time and re-experience events that have already happened.
What would Alice need to do to 'freeze' her Granny in time so that she is always having tea?
How long can she hold Granny frozen?

 

[stevmg]

LAck of simultaneity

It is 4 AM and I just got up and I can't describe it in detail but Einstein does in his book on Relativity in which he explains that events which are simultaneous from observation in one frame of reference are not when observed from a different frame of reference. His is the example of the lightning strikes on a moving train. Simultaneous when observed by a ground observer, not simultaneous when observed from a train traveler.

Steve Garramone

 

[AJ Bentley]

LAck of simultaneity

That's part way to an answer, but it's basically a mantra.
I'm not asking for a restatement of the principle, or a math derivation of the L transform.

In fact I've deliberately introduced the Ansible so that the standard arguments about L transforms and the speed of light signals don't come into it.

I've fixed the space/time position too, to a single event at the origin with both Bob and Alice in the Here/Now position. The signals from Granny's tea event travel instantaneously to both Bob and Alice so there is no time delay there either, no Doppler shift.

Yet, They can't agree about what Granny is doing right now. Why not?

I'm looking for a simple, common-sense answer. No math, no sound bytes.

 

[MikeLizzi]

Well that is what the example demonstrates, and you only get an apparent contradiction if you mistakenly treat a non-inertial observer as if they were an inertial observer. So I don't really see what you are objecting to.

What I am objecting to is the presentation of solutions as resolutions. If I were given a homework problem to calculate the difference in ages of the twins, I might copy any of a dozen posting in this thread. If I were given the problem of explaining why two different solutions give contradictory results, the only postings worth copying are those like your last one.

Maybe I am being overly sensitive to this issue. This is after all a forum where people are invited to offer opinions and engage in dialogs. But I have had some very negative experiences regarding the Twins Paradox.

 

[Fredrik]

Yet, They can't agree about what Granny is doing right now. Why not?

Because they're using their own motion and a synchronization convention to define what "right now" means. This is what makes the simultaneity lines in Aaron_Shaw's post on the page before this one look the way they do.

In fact I've deliberately introduced the Ansible so that the standard arguments about L transforms and the speed of light signals don't come into it.

Unfortunately an "ansible" can only exist in Galilean spacetime. It can't exist in Minkowski spacetime. It would make SR logically inconsistent (link), so there's no point asking what SR says about anything after you've introduced it.

 

[AJ Bentley]

Because they're using their own motion and a synchronization convention to define what "right now" means.

That's more-or-less correct, but isn't it easier to simply say that the word 'now' has absolultely no meaning (outside of your own narrow world view)?

Unfortunately an "ansible" would make special relativity logically inconsistent, so there's no point asking what SR says about anything after you've introduced it.

Not so. the Ansible merely removes obfuscating factors. It prevents you applying the Lorentz Transform or Doppler shift. Each person carries around their own personal Anisble anyway - it's called 'imagination'.
When we think of Now, we have a very clear image of what that means. In doing so, we Ansible-up our own universe.

The twin paradox comes about because the paradoxee is constantly thinking 'Now Alice is 25 as far as Bob (Now age 35) is concerned' and ' Now Bob is 25 as far as Alice (Now age 35) is concerned.

Remove 'Now' and the paradox is gone with it.

Until the Twins meet up again at journey's end, their ages have no meaning except to themselves. It's very similar to the idea of a quantum state and it's resolution (Don't leap on that observation, it's merely a comparison)

 

[Aaron_Shaw]

Gentlemen, Please!

Humour me, have a go at this restatement of the paradox in a different form.

********************************
Observer Alice, says to Bob ,who is just passing by at nearly the speed of light 'My Granny on Proxima Centauri is just sitting down to kippers for her tea'

Alice knows that because she has an Ansible (which allows her to see what Granny is doing right now without having to wait for the light to arrive)

Bob, who also has an Ansible, takes a quick look and says 'No she isn't, your Granny had her kippers for tea three days ago'

Explain.

You might like to show how Bob's Ansible allows him to travel back in time and re-experience events that have already happened.
What would Alice need to do to 'freeze' her Granny in time so that she is always having tea?
How long can she hold Granny frozen?

That's the old alien invasion from alpha centauri "paradox"?

Lack of simultaneity is the reason of course, but i think that the main cause of "concern" regarding this scenario is implications regarding determinism, fate, free will, and all that stuff.

 

[Austin0]

That's part way to an answer, but it's basically a mantra.
I'm not asking for a restatement of the principle, or a math derivation of the L transform.

In fact I've deliberately introduced the Ansible so that the standard arguments about L transforms and the speed of light signals don't come into it.

I've fixed the space/time position too, to a single event at the origin with both Bob and Alice in the Here/Now position. The signals from Granny's tea event travel instantaneously to both Bob and Alice so there is no time delay there either, no Doppler shift.

Yet, They can't agree about what Granny is doing right now. Why not?

I'm looking for a simple, common-sense answer. No math, no sound bytes.

Hi AJ

Given your imaginary premise of the Ansible then it would seem there would be two possible eventualites:

1) They would not agree.
In this case it might be inferred that relative simultaneity was an actual temporal dislocation. As you put it actually in the past or future.

2) They would agree.

It would seem to follow that this would mean that relative simultaneity would not apply between spatially separated points
but only to observers in the respective frames that were colocated with granny at the time of Ansible observation.

I posted a similar thread a while ago approaching the same question with EPR transmission.

The premise was rejected on the grounds that can be no instantaneous information transmission with EPR due to the neccessity of statisical comparison between the two sites to give meaning to the observations.
At the time I accepted this as a valid criticism but later realized that this was not really the case.
It is not relevant to have real time confirmation of reception. As long as later analysis can confirm reception then their logs give the proper time of reception.
By the time i realized this I was off onto other things and forgot about the problem. Thanks to you I may give it another shot with EPR

 

[matheinste]

If you allow instantaneous transmission of information you do not remove the now but make possible to define the same for everyone, or make time absolute, as in a Galillean or Newtonian universe. If everyone has the same now at every instant then there would be no differential proper times, the now at every spatial location would be the same foer all observers.

I also think that giving objects special properties denied to others is a recipe for many more contradictions somewhere along the line.

Matheinste.

 

[Austin0]

That's the old alien invasion from alpha centauri "paradox"?

Lack of simultaneity is the reason of course, but i think that the main cause of "concern" regarding this scenario is implications regarding determinism, fate, free will, and all that stuff.

If we're on the same page; Penrose looked at the divergent lines of simultaneity as indicating that the observers had an actual relationship with the Alpha C worldline at widely separated points.

Given that the slope of Line's of S in Minkowski spacetime is merely a graphical convention
that represents rulers with clocks [and observers ,real or virtual] that are congruent with and extended along the vector of motion , an alternative view is possible.
Any given event on the AC worldline would also find colocated observers from the respective frames who would disagree on the date.

Does this have any more significance than the difference in respective time [simultaneity]
between the train and track observers?

Isn't it exactly the same situation , just a very long train and tracks?

So if you are going to assume any temporal meaning in one case [which Penrose seemed to do] then to be consistent you should make the same assumption in the other , no?

Not that I have any objection to doing this, in fact consider this question extremely valid and germaine.
I don't see how determinism ,fate or free will is affected in either case??
Just thoughts.

 

[Austin0]

If you allow instantaneous transmission of information you do not remove the now but make possible to define the same for everyone, or make time absolute, as in a Galillean or Newtonian universe. If everyone has the same now at every instant then there would be no differential proper times, the now at every spatial location would be the same foer all observers.

I also think that giving objects special properties denied to others is a recipe for many more contradictions somewhere along the line.

Matheinste.

Would it give everyone the same now or just a standard of evaluation??

A means to determine the difference in "now" at disparate locations?

SR would still apply exactly as it does currently and for the same reasons.
Setting all the clocks to a universal now would make them inoperable for physics or or an invariant measurement of c.

We could right now institute a universal terran time standard but clocks in different parts of the world would be completely out of phase with the sun etc except for a very small region. IMO

thanks

 

[Aaron_Shaw]

I don't see how determinism ,fate or free will is affected in either case??

I think the idea is that one person can see the invading fleet deliberating a potential invasion. He's watching; waiting to discover his fate.
Meanwhile some other bloke moving relatively has already seen the aliens decide on war and launch the fleet.

The first guy is deciding what course of action to take, depending on the outcome of the aliens deliberations which, according to the second guy, is pointless because they've already made their decision.

I think it's just an illusion as the first guy can't have any influence on that outcome at that point anyway due to information transfer speed limit. But i can't find the original problem to refresh my memory.

 

[AJ Bentley]

I thought we were having a sensible discussion. What's all this 'aliens' nonsense?

 

[AJ Bentley]

it would seem there would be two possible eventualites:

1) They would not agree.
In this case it might be inferred that relative simultaneity was an actual temporal dislocation. As you put it actually in the past or future.

2) They would agree.

Case 2 of course means that they are in the same inertial frame and is therefore simply a special case of 1).

The lack of simultaneity isn't exactly a dislocation because it's a continuous function, but it'll do.

I would not say that the events for the observers are 'actually in the past or future' - at least, not without specifying who's past/future - but, yes, you get the idea.

I would say that there is no such thing as past or future in any absolute sense as I did for the word 'now'. (Now is just a point in past and future.) These concepts apply to the world view of an individual world line. They have no absolute meaning.
Each F.O.R. sees a different set of events as past/future, depending on the position of those events and his velocity with respect to any opposing view.

In some frames of reference. The birth of christ has not yet happened. (Not that the light hasn't got there yet - I mean literally not happened as viewable by Ansible).

In the same way, in others, you and I are long gone to dust.

Yes. It plays merry havoc with free will - but I don't see that as my problem.

PS That raises the interesting thought of just how far away and how fast moving you would need to be to to be contemporary with JC. (It may be outside the observable universe...)

 

[Fredrik]

That's more-or-less correct, but isn't it easier to simply say that the word 'now' has absolultely no meaning (outside of your own narrow world view)?

You could say that it has no absolute meaning, but maybe that's what you meant. If we want to explain why the incorrect calculation of the stay-at-home twin's age is incorrect, we need to understand the procedures that we use to associate a coordinate system with an observer's world line.

Not so. the Ansible merely removes obfuscating factors.

This is definitely incorrect. As I said, an "ansible" would make SR inconsistent (I included a link to a proof), or simply replace Minkowski spacetime with Galilean spacetime.

It prevents you applying the Lorentz Transform or Doppler shift. Each person carries around their own personal Anisble anyway - it's called 'imagination'.
When we think of Now, we have a very clear image of what that means. In doing so, we Ansible-up our own universe.

Unfortunately one of the "obfuscating factors" you removed is special relativity.

The twin paradox comes about because the paradoxee is constantly thinking 'Now Alice is 25 as far as Bob (Now age 35) is concerned' and ' Now Bob is 25 as far as Alice (Now age 35) is concerned.

Remove 'Now' and the paradox is gone with it.

That's also not correct, because the paradox isn't about what they would be saying before they meet again. It's about two calculations of their ages at the event where they meet when the astronaut twin comes back.

 

[AJ Bentley]

an "ansible" would make SR inconsistent (I included a link to a proof),

The link simply proves the impossibility of such a device - which I freely admit.

My point is that the horizontal axis in the Minkowski diagram is an 'ansible' line - it is a line of instant communication.
I am simply pointing out the significance of that line in terms that anyone can understand.


IMO No absolute meaning is not strong enough. I prefer to say absolutely no meaning and add the rider -except for one very special case. The point needs to be hammered home.

Throughout all of this I am only telling you what I see when I look at a Minkowski diagram. If that isn't SR - then what is?

the paradox isn't about what they would be saying before they meet again. It's about two calculations of their ages at the event where they meet when the astronaut twin comes back

.

The final calculation is simply a bit of kindergarten maths.
MikeLizzi said it:-
"What I am objecting to is the presentation of solutions as resolutions. If I were given a homework problem to calculate the difference in ages of the twins, I might copy any of a dozen posting in this thread".

 

[yossell]

My point is that the horizontal axis in the Minkowski diagram is an 'ansible' line - it is a line of instant communication.

You might like to rethink this. The horizontal line represents a frame's line of simultaneity, the events that are judged simultaneous in that frame, but that doesn't meant that there can be instantaneous signals and communication along these lines.

 

[AJ Bentley]

that doesn't meant that there can be instantaneous signals and communication along these lines.

Did I not just say in words of one syllable that such a device is impossible?

 

[yossell]

uhhh...so it's not a line of instant communication.

 

[AJ Bentley]

uhhh...so it's not a line of instant communication.

Just in case this is a genuine response, and not a wind-up - and in case anyone else has the same problem with what I said:-

'A line of instant communication' does not imply that it can be used in practice for the purposes of instant communication.
It is simply a line on the Minkowski diagram that such signals would travel if it were possible.

 

[Fredrik]

My point is that the horizontal axis in the Minkowski diagram is an 'ansible' line - it is a line of instant communication.

It's an "ansible" line (why not call it a simultaneity line like everyone else?) only for an observer whose world line is a vertical line in the diagram.

IMO No absolute meaning is not strong enough. I prefer to say absolutely no meaning and add the rider -except for one very special case. The point needs to be hammered home.

I can't agree with that. Every coordinate system gives meaning to the concept of simultaneity. You could argue that it's not "natural" enough, but the standard synchronization procedure is definitely natural enough. It just isn't absolute.

Throughout all of this I am only telling you what I see when I look at a Minkowski diagram. If that isn't SR - then what is?

You can't tell just by looking at the diagram if it's a diagram of something moving in Galilean spacetime (the one used in pre-relativistic theories) or of something moving in Minkowski spacetime.

The final calculation is simply a bit of kindergarten maths.

Yes, it's not hard to calculate the final age. But to resolve the paradox, you need to explain what's wrong with the incorrect calculation that just uses the time dilation formula twice.

Just in case this is a genuine response, and not a wind-up - and in case anyone else has the same problem with what I said:-

'A line of instant communication' does not imply that it can be used in practice for the purposes of instant communication.
It is simply a line on the Minkowski diagram that such signals would travel if it were possible.

It was very easy to misunderstand you because you defined an "ansible" to be a machine that does instantaneous communication and called these lines "ansible lines". Everyone else calls them simultaneity lines.

 

[AJ Bentley]

To save laying out a lot of quotes Fredrik, your points in order.

1/ I'm using the word Ansible to make people think 'what does he mean?' rather than assuming they know what it means because they've seen the phrase before. In the context, I was pointing out that line as an example - there are of course an infinite number of such lines - each point on the diagram has an infinite number of them passing through it. Each corresponding to a different 'now'. Each is no more important than the other.

2/ The resolution of the paradox hinges on knocking down the concept of simultaneity - why do you keep trying to prop it up?

3/http://en.wikipedia.org/wiki/Minkowski_diagram"

4/ What can I tell you? It's a conceptual device - a thought tool - something to use in a thought experiment. You actually want me to build one for you?

 

[yossell]

It's not possible to combine Minksowski space-time with the possibility of instant communication. The two aren't compatible, so I don't understand how you're combining them when `a line on the Minkowski diagram that such signals would travel if it were possible'

I agree that sometimes we can define things in terms of counterfactuals - the path an object would travel if it were unacted on by forces; the force a unit charge would feel if it were at a certain point - even if it's practically impossible to get a unit charge to that point. But in this case, instant communication and Minkowski spacetime are incompatible with each other, so I think it's incoherent to talk of a line in a *Minkowski* space that a signal would travel were it possible.

 

[AJ Bentley]

OK, that's enough.
This thread has become virtually a monologue, My bad.

Anyone wants to PM me on the subject is welcome.

Over and out.

 

[Fredrik]

2/ The resolution of the paradox hinges on knocking down the concept of simultaneity - why do you keep trying to prop it up?

No, to resolve the paradox, you have to explain what's wrong with the incorrect calculation, and to do that you need to understand the precise nature of relative simultaneity. It's not sufficient to just "knock down" absolute simultaneity.

3/http://en.wikipedia.org/wiki/Minkowski_diagram"

Is that supposed to refute what I said? It doesn't. You can draw spacetime diagrams for Galilean spacetime too. I don't like the term "Minkowski diagram" for precisely this reason.

4/ What can I tell you? It's a conceptual device - a thought tool - something to use in a thought experiment. You actually want me to build one for you?

I was just explaining to you why it was your fault that yossell misunderstood you. I have no idea why you're saying the things you're saying now. They seem completely unrelated to what we were talking about.

 

[MikeLizzi]

Yes, it does address the paradox because the value calculated is frame invariant. It is the same in ALL frames, inertial or non-inertial.

No it doesn’t address the problem. And now we are getting close to the source of your misunderstanding.

You wrote: “Because the value calculated is frame invariant”

So why calculate the value at all? We already know the value of the difference in ages by calculation from the point of view of earth. So what is the point of your exercise?

The point of the paradox is that a superficial calculation made with the astronaut as the observer gives a contradictory answer. The only way you resolve that problem is by providing the correct calculation, or at least explaining the correct calculation, with the ASTRONAUT as the OBSERVER.

You have not done that. You have not resolved the paradox.