Solving the Twin Paradox with Lorentz Transformation

[Aaron_Shaw]

Gentlemen, Please!

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

********************************
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.

 

[Al68]

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.

I've mentioned before that Einstein's 1918 resolution addresses it from the non-inertial frame of the ship. I know most consider the standard resolutions adequate because they provide the correct answer, but Einstein realized full well that a 100% correct resolution isn't necessarily a satisfactory one.

Basically, you can break the period of acceleration into as many segments as you want, and calculate Earth time for each one in the ship's (co-moving inertial) frame. Or just use the equivalent of an infinite series of co-moving inertial frames: gravitational time dilation.

Einstein's resolution just uses the simple gravitational time dilation equation for linear acceleration to calculate elapsed time on Earth's clock in the ship's frame during acceleration. And, unsurprisingly, gets the same answer as the standard resolutions.

 

[Dale]

The only way you resolve that problem is by providing the correct calcultion, or at least explaining the correct calculation, with the ASTRONAUT as the OBSERVER.

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

Pointing out the frame-invariant geometry of the problem (longest interval is a straight line) is a perfect resolution. It gives the student a new way to think about relativistic physics that both clearly demonstrates the mistake in the paradox and helps the student learn more advanced physics.

 

[Austin0]

=Al68;2794254]I've mentioned before that Einstein's 1918 resolution addresses it from the non-inertial frame of the ship. I know most consider the standard resolutions adequate because they provide the correct answer, but Einstein realized full well that a 100% correct resolution isn't necessarily a satisfactory one.

Hi Al68
I don't even remember if I ever read that paper let alone the contents but I have some questions on principle:

1) As I understand it G time dilation in an accelerating frame only has an effect within
the frame itself . A relative dilation between differnt locations in the frame.
It does not have any effect relative to inertial frames {clock hypothesis]

The relationship with other frames is simply derived from the instantaneous relative velocity. As per your statement below ((2))

2) Even in a round trip with only a relatively short acceleration phase compared to total trip length;- m the overall trip time the cumulative diilation is based on both accel. ICMF velocity and inertial velocity
but the inertial phase dilation (from velocity), which would normally be reciprocal [relative] has now become real , actual.
{Catalytic effect}

3) There is no corralation between the relative percentage of the trip that is accelerated and the end result . Quite unusual for a physical phenomenon wouldn't you say?
For a relevant parameter to vary with no consequence to the end result??

IMO The reason many people are unsatisfied with the resolution is:

It seems like you should be able to analyse the picture from either frame in an identical manner. Assume the accelerating frame as at rest and the Earth is accelerating etc.

Draw an Earth worldline that is curved in areas and straight while inertial and apply all the relevant math on that basis. This of course can be easily calculated and in actuality wouldn't the calculations also be identical,?

Then there would be symmetrical Minkowski diagrams [reciprocal mirrow images] and all the analysis that is commonly used in resolutions would be identical.

But this is not allowed. It is denied on the basis of somewhat ad hoc pricipals

a) Acceleration is real as opposed to inertial motion which is purely relative [unreal]

b) Because of a) only inertial frames are considered valid.

c) Because a world line which changes direction makes it longer and accelerated . ANother version of
a) and b)

Regarding :
a) It is true that there are measurable differences between accelerated and inertial motion.

Unquestionable.

But does there simply being a difference mean there must be a specific effect attributable to that difference?

There is no physics principle or concept suggesting that acceleration would result in real dilation or how it might catalytically turn relative dilation into real or explaining how it possibly could effectuate this result.

So it is not because there aren't valid resoluions to the "paradox" that there remains the dissatisfaction [if anything there are too many]

it is because some seemingly valid ways of looking at it are negated on grounds that are themselves not completely satisfactory or consistent.

Basically, ((2)) you can break the period of acceleration into as many segments as you want, and calculate Earth time for each one in the ship's (co-moving inertial) frame. Or just use the equivalent of an infinite series of co-moving inertial frames: gravitational time dilation.

((1))

Einstein's resolution just uses the simple gravitational time dilation equation for linear acceleration to calculate elapsed time on Earth's clock in the ship's frame during acceleration. And, unsurprisingly, gets the same answer as the standard resolutions

I am going to have to read the 1918 paper (again?)
Judging by this it seems to indicate that G-dilation is exactly equivalent to velocity dilation [ICMF's etc.]
COnsistent with clock hypothesis
But this seems strange if G-dilation is constant but instantaneous velocities are varied.
ANy ideas ?
Thanks

 

[AJ Bentley]

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

IMO, the basic problem, and the reason for the astonishing clamour in this thread is that no-one seems to agree on what a 'resolution' is.

Certainly I personally can't agree that simply calculating the ages of the twins correctly is enough. Simply calculating the values correctly gives one an impression of having 'solved it' without striking to the heart of the paradox.

The paradox is deeper than that and for me it hinges on the question of what happens during a velocity change. Not in the sense of how acceleration affect time (General Theory), but what happens to the geometry of space time at that point.

 

[matheinste]

IMO, the basic problem, and the reason for the astonishing clamour in this thread is that no-one seems to agree on what a 'resolution' is.

Certainly I personally can't agree that simply calculating the ages of the twins correctly is enough. Simply calculating the values correctly gives one an impression of having 'solved it' without striking to the heart of the paradox.

The paradox is deeper than that and for me it hinges on the question of what happens during a velocity change. Not in the sense of how acceleration affect time (General Theory), but what happens to the geometry of space time at that point.

As Dalespam pointed out earlier, differing path lengths is the simplest answer. The path length (proper time) is frame independent, that is, everyone agrees upon it. The main problem is that because of the strangeness of the result the problem is introduced to whet the readers appetite for relativity before the reader is equipped with the tools to resolve it. Once the concept of proper time is understood the problem goes away. The twins scenario is no "deeper" than anything else in SR.

Matheinste.

 

[Austin0]

IMO, the basic problem, and the reason for the astonishing clamour in this thread is that no-one seems to agree on what a 'resolution' is.

Certainly I personally can't agree that simply calculating the ages of the twins correctly is enough. Simply calculating the values correctly gives one an impression of having 'solved it' without striking to the heart of the paradox.

The paradox is deeper than that and for me it hinges on the question of what happens during a velocity change. Not in the sense of how acceleration affect time (General Theory), but what happens to the geometry of space time at that point.

I agree the paradox is deeper and has nothing to do with the twins final ages really.

There are deeper questions intrinsic to the problem.

It involves the meaning of simultaneity. And the relationship between simultaneity and time dilation.

The meaning and reality of acceleration.

Certain inconsistencies between different valid methods of resolution which derive the same results.

On the GR and how acceleration effects the geometry of space ; from what I have gathered SR says there is no change in the geometry but GR and Rindler coordinates seem to imply there would be. My study of bGR and Rindler is just beginning so I would like to know the answer my self.

 

[AJ Bentley]

I agree the paradox is deeper and has nothing to do with the twins final ages really.

There are deeper questions intrinsic to the problem.

It involves the meaning of simultaneity. And the relationship between simultaneity and time dilation.

The meaning and reality of acceleration.

Certain inconsistencies between different valid methods of resolution which derive the same results.

On the GR and how acceleration effects the geometry of space ; from what I have gathered SR says there is no change in the geometry but GR and Rindler coordinates seem to imply there would be. My study of bGR and Rindler is just beginning so I would like to know the answer my self.

My point exactly.

I 'solved' this problem decades ago. Then suddenly realized much later that what I had wasn't a resolution at all - just a maths question with a textbook solution.

 

[matheinste]

I agree the paradox is deeper and has nothing to do with the twins final ages really.

No. The differeing ages of the twins is put forward as the paradox.

Matheinste

 

[AJ Bentley]

No. The differeing ages of the twins is put forward as the paradox.

Matheinste

Hmm... not a very good rolleyes smilie is it?

 

[matheinste]

As far as a "deeper" meaning goes; Differential ageing is a direct logical consequence of the axioms of SR. That is where any deeper meaning lies and when we find that deeper meaning we can start looking for a still deeper one.

Matheinste.

 

[AJ Bentley]

As far as a "deeper" meaning goes; Differential ageing is a direct logical consequence of the axioms of SR. That is where any deeper meaning lies and when we find that deeper meaning we can start looking for a still deeper one.

Matheinste.

Has it never occurred to you to wonder how it is possible for JC to be still alive (or not even yet born) in some F.O.R.?

Or that in others, the Earth no longer exists?

And that all that separates these frames is the velocity of the inhabitants?

 

[Fredrik]

The differeing ages of the twins is put forward as the paradox.

I disagree. The age difference is a paradox in the sense "a counterintuitive result", but a lot of people have used this scenario to argue that SR is logically inconsistent, not just that it's counterintuitive. All of them are using the word "paradox" in the sense "logical contradiction". You can't prove them wrong by showing that the astronaut twin will be younger than his brother at the reunion.

 

[matheinste]

Or that in others, the Earth no longer exists?

And that all that separates these frames is the velocity of the inhabitants?

I would be interested to know which particular argument you use to come to the conclusion that there are frames of reference for whom the inhabitants at rest in them the Earth no longer exists, even allowing your implicit assumption that the Earth will eventually cease to exist in the reference frame in which it is at rest.

Idon't wish to get into discussions about determinism as the subject carries too much baggage with it.

Matheinste.

 

[matheinste]

I disagree. The age difference is a paradox in the sense "a counterintuitive result", but a lot of people have used this scenario to argue that SR is logically inconsistent, not just that it's counterintuitive. All of them are using the word "paradox" in the sense "logical contradiction". You can't prove them wrong by showing that the astronaut twin will be younger than his brother at the reunion.

I hate the use of the word paradox to describe it because as you say, as others have said and as I have said ad nauseam, it follows logically from the axioms of SR. I cannot see how the result can be used to prove SR inconsistent.

I have seen more than one author express the opinion that all the time spent arguing about the resolution would be better spent actually learning some basic SR.

Matheinste

 

[stevmg]

What is an "Ansible?" Is it from the same world as the old chemical "phlogisten?"

If one accepts the principle of SR or GR, two events can be simultaneous only if they are within the lightcone. Now, if you don't accept that premise (i,e,, don't "believe" in relativity) well that's your "bag." But, so far, all experimental evidence and natural phenomenon such as the mu-mesons, etc. support relativity.

Now, I am not one to challenge Einstein, Hilbert, Eddington and all those giants of the past.

If they are wrong - as they say in Missouri - show me.

Steve Garramone, MD
Melbourne, FL