Solving the Twin Paradox with Lorentz Transformation

[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

 

[matheinste]

If one accepts the principle of SR or GR, two events can be simultaneous only if they are within the lightcone.

Steve Garramone, MD
Melbourne, FL

That should be spacelike separated. There are no restrictions on their locations in spacetime other than that.

Matheinste.

 

[AJ Bentley]

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

Touche.
I allowed myself to get carried away.

However, this points up another facet of SR, namely that although it is possible for one observer to 'lag behind' another in time. It is not possible for an observer to be 'in front' of the other. No?

In which case, the rules of common logic seem not to apply.
Don't get me wrong, SR wins in my book. However, having removed one rule, can we trust any of the others?

 

[yossell]

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.

Consider a distant inhabitant who is (a) at rest and (b) at the origin of an inertial frame moving quickly towards the direction of the earth. If distant enough and quick enough, his plane of simultaneity tips up enough so that, for him, the Earth's world line ends beneath the plane - i.e, for him, the Earth no longer exists.

Is this not coherent?

 

[stevmg]

That should be spacelike separated. There are no restrictions on their locations in spacetime other than that.

Matheinste.

You're right. One Diemnsion - if event at x1, t1 and x2, t2 occur simultaneously, (t1 = t2) there is no way that light could eminate from either one and get to the other "within the speed of light" as each event would have its own light cone and each event would be outside the others light cone and therefore spacelike.

Sorry for being being "spacy" myself.

SMG

When I have time I would like to bust the original supposition apart but I cannot into it now. This is a perfect example of Einstein's train, and an "Ansible" is sort of like a Unicorn - it doesn''t even exist in a make believe world -that's two steps removed from reality.

Now I don't mean to be disrespectful here and all ideas must be entertained. Lord knows, you folks have put up with my idiocy in the past and will do so in the future and my idiocy is far more spacy than this original thread. So thank you all for past and future tolerations of my going down blind alleys and incongruities.

 

[stevmg]

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?

No offense meant to you. I will go over your proposition in closer detail so that I may more clearly undertand it.

 

[matheinste]

Consider a distant inhabitant who is (a) at rest and (b) at the origin of an inertial frame moving quickly towards the direction of the earth. If distant enough and quick enough, his plane of simultaneity tips up enough so that, for him, the Earth's world line ends beneath the plane - i.e, for him, the Earth no longer exists.

Is this not coherent?

Yes, but in this case the non-earth inhabitants will not be aware of the Earth's demise until the light from the event reaches them. They cannot before then say that in their now Earth has vanished. This, for me, and many others is part of the problem with definitions of simultaneity. They have no natural meaning. They are all, probably of necessity, conventional.

Of course there are some serious philosophers and maybe physicists who ague that there is a natural concept of absolute simultaneity. I have yet to study their reasoning. My opinion is that the usual definitions of simultaneity are useful until they lead to contradictions. If a mere convention causes problems in a certain context then don't use it.

Matheinste.

 

[stevmg]

Has it ever been shown either experimentally or through observation of natural phenomena that gravity travels at the speed of light (no faster?)

In other words, if our Sun was to suddenly "go away," it would take eight minutes or so for us to a) lose the light and b) fly off into space.

stevmg

 

[yossell]

Yes, but in this case the non-earth inhabitants will not be aware of the Earth's demise until the light from the event reaches them.

I see and agree.

Of course there are some serious philosophers and maybe physicists who ague that there is a natural concept of absolute simultaneity.

I am aware there is an issue of whether a concept of simultaneity is mere convention or something more. I think this issue is very delicate.

I would distinguish this from the question of whether there is such a thing as absolute simultaneity, which suggests that the notion is not frame-relative. This seems much more implausible, and harder to reconcile with relativity.

 

[matheinste]

I see and agree.




I would distinguish this from the question of whether there is such a thing as absolute simultaneity, which suggests that the notion is not frame-relative. This seems much more implausible, and harder to reconcile with relativity.

I agree, but you know what philosophers are like.

Matheinste.

 

[yossell]

I agree, but you know what philosophers are like.

Matheinste.

Yes. Very well. Do you?

 

[matheinste]

Yes. Very well. Do you?

Not philosophers personally. I do find most philosophical disussions, as regards physics, stimulating, but some of the ideas seem to clash with reality. But then what is reality. That's another question for another day.

Matheinste.

 

[Al68]

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.

That's right, in this case the rate of Earth's clock is calculated relative to the ship's clock, both wrt to the accelerated frame of the ship.

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??

But there is a "consequence" to the end result. Because the "end result" will vary with the relative velocity of the ship at each point along the way, which is a direct function of acceleration.

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.

This is exactly what Einstein's 1918 resolution did. But of course the frames are not identical.

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

It's not an ad hoc principle that non-inertial motion isn't the same as inertial motion. The math would not identical because the coordinate position of Earth relative to the ship's frame throughout the trip isn't identical to the coordinate position of the ship relative to Earth's frame throughout the trip.

And you can't ignore the consequences of using an accelerated reference frame, such as gravitational time dilation as applied to distant clocks.

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.

Time dilation refers to the relative tick rate of the clocks, not their accumulated elapsed time between events. It's the accumulated time between events, not the tick rate of the clocks, that is not reciprocal during inertial motion. Remember that the distance accumulated during the trip is not the same in each frame, and this factors into the accumulated time on the clocks in addition to their relative tick rates. Also note that the elapsed time between the ship leaving Earth and the ship reaching the destination would be different in each frame whether the ship accelerates or not.

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.

I'd say it's not satisfactorily explained in many resolutions. The fact is that the ship has a clock during its acceleration, and the ship's twin experiences every increment of elapsed time on that clock. Many people correctly point out that there is no need to address that in order to obtain a correct solution. But the typical question is "how does it work from the ship POV?", not "how can we get a correct solution without analyzing the ship POV?"

Judging by this it seems to indicate that G-dilation is exactly equivalent to velocity dilation

Gravitational time dilation isn't "equivalent" to velocity time dilation, it's the same exact phenomenon. All of the gravitational time dilation equations are derived from the lorentz transformations.

But this seems strange if G-dilation is constant but instantaneous velocities are varied.
ANy ideas ?

I'm not sure what you mean, but gravitational time dilation isn't necessarily "constant". It's only constant for certain simplified scenarios. It's not constant for this scenario, since even if the acceleration is constant, the coordinate distance of Earth's clock in the ship's frame isn't constant during the acceleration.

BTW, here's a link to an English translation of Einstein's 1918 paper: http://en.wikisource.org/wiki/Dialog_about_objections_against_the_theory_of_relativity.

 

[stevmg]

Where would I read on gravitational time dilation? I have the 1960s version of Spacetime Physics. Einstein himself in Relativity doesn't really get into the nuts and boltss.

It has taken me long enough just to grasp the basics of SR, much less GR other than understanding what Einstein, Hilbert and Eddington did.

Thanks,

stevmg

 

[stevmg]

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?

By Einstein et al there can be no such thing as an Ansible. Basically, an Ansible literally establishes simultaneity between two events (the event itself and its observation in the Ansible.) This is not possible wrt to all frames of reference. Simultaneity "changes" (or can change) when looking at two distict events from the two different FORs.

The scenario descibed above is literally the Einstein train in which the lightning flashes are perceived as simultaneous in one FOR while in the other FOR, the lead flash occurs before the back flash. [Section IX, The Relativity of Simultaneity, Relativity, Albert Einstein.]

In this case, Bob is traveling towards Granny and sees her event to occur before Alice.

Lorentz time transformation and SR:

t' = \gamma[t - xv/c²]

Set t = 0 (for Alice,) x = whatever (call it x) for the distance between Alice and Granny in a common FOR. v is Bob's speed wrt Alice and going towrds Granny. Do the math... presto! a negative number "'No she isn't, your Granny had her kippers for tea three days ago'

Granny doesn't have to be frozen. All we have to do is "thaw" our brains to allow SR to take over.

Now, if you don't accept the state-of-the-art knowledge of SR as being "real" then that is a different story.

 

[matheinste]

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.

Granny has kippers every day.

Matheinste.

 

[Al68]

Where would I read on gravitational time dilation?

You could start with Einstein's 1907 paper first describing it as a consequence of using an accelerated reference frame: http://physics.uark.edu/publications/volume1/H.M.SCHWARTZ.pdf .

And his 1911 paper on the equivalence principle extended it to gravitational fields: http://einstein.relativitybook.com/Einstein_gravity.html.

Of course a google search will yield more online resources.

 

[AJ Bentley]

Has it ever been shown either experimentally or through observation of natural phenomena that gravity travels at the speed of light (no faster?)

In other words, if our Sun was to suddenly "go away," it would take eight minutes or so for us to a) lose the light and b) fly off into space.

stevmg

That's an important point. (ie. one I wouldn't trust to theorists ).

I'm not aware of any experiments, but then I haven't followed developments for a long time.
I guess for direct proof you would need to look for gravitational waves from a currently exploding supernova. But if it were not so, would we have problems calculating simple parameters like cometary orbits? I'm thinking of the Shoemaker-Levy impact. Where we see the object would be at odds with it's gravitational behaviour?

In other fields, there have been suggestions that EM waves and the quantum particle waves of an electron posses pretty much the same parameters and begin to look like two aspects of the same thing. I have read that Einstein was of that opinion. A book I'm reading at the moment goes into it in detail.
So I would be actually surprised if there was not a one-to-one correspondence between gravity, EM, quantum waves and the whole shebang.

PS You could also pose the related question as to why the centres of both mass and charge properties of a particle appear to coincide absolutely at the same location.

 

[stevmg]

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?

AJ Bentley - did you get my apology for being flippant earlier and my actual explanation of your "humor" kipper scenario? Again, I didn't mean to appear disrespectful.

Below is a schematic depiction of my answer`(the left members of the pairs are the distances, the right elements are the times). v ("approximately the speed of light") and x (the distance from Alice to Proxima Centauri) are arbitrary so are merely noted by v and x. Note that both v and x are positive. x', or the distance in Bob's frame, from him as crosses Alice to Granny is noted but not necessary for the discussion, while t' is calculated by the Lorentz time transformation. v for the Alice-Granny frame of reference is zero (= 0) while we just denote the velocity of the Bob-Granny frame of reference as v. is a positive number but not actually calculated here. t' calculates, by using the above information, to a negative number. This goes along with your supposition as presented.

Bob(0,0)......moving frame(v)......Granny(x',t' - a negative number)
Alice(0,0)......reference frame(v=0)...Granny(x,0)

However, I like the answer matheinstei gave: "Granny has kippers every morning."

Sometimes the obvious eludes us and, using matheinstei's logic - Ansibles and all other theories fit.

Hats off to matheinstei for coming up with an inclusive theory.

 

[stevmg]

PS You could also pose the related question as to why the centres of both mass and charge properties of a particle appear to coincide absolutely at the same location.

How would that be germaine? Not saying the question itself isn't important, but how would it apply here, if we knew the answer?

Again, I think Einstein said that the force field of gravity traveled at the speed of light and I was asking if anyone else out there knew if that were true.

Second point, there is an urban legend that if the Sun did "go away" suddenly and all the planets were to fly off at tangents to their orbits at the time of the loss of gravity that the Earth would fly apart into a zillion pieces. That doesn't make sense to me. However, if the Earth and the other inner planets, besides being held in their near circular orbits by the Sun's gravity, the general "compression" by the force field of the Sun (gravity from the Sun) literally compresses the Earth and all around it together. This is obviously lost the further removed from the Sun (maybe) howver maybe not.

Is there any veracity to this line of thinking? Does the proximate presence of a large mass force a compression on particles and energy to coalesce and form distinct masses such as planets?

Ooooh, I can come up with weird stuff!

 

[Austin0]

Originally Posted by Austin0
Hi Al68
I
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.

That's right, in this case the rate of Earth's clock is calculated relative to the ship's clock, both wrt to the accelerated frame of the ship.

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??

But there is a "consequence" to the end result. Because the "end result" will vary with the relative velocity of the ship at each point along the way, which is a direct function of acceleration.

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,?

The math would not identical because the coordinate position of Earth relative to the ship's frame throughout the trip isn't identical to the coordinate position of the ship relative to Earth's frame throughout the trip.

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

It's not an ad hoc principle that non-inertial motion isn't the same as inertial motion.And you can't ignore the consequences of using an accelerated reference frame, such as ((()))gravitational time dilation as applied to distant clocks.

((())) clock hypothesis, remember?

I specifically recognized there are differences . It was ad hoc because it had no sufficient justification to ascribe the non-reciropcal dilation to acceleration.

"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."

((1))Time dilation refers to the relative tick rate of the clocks, not their accumulated elapsed time between events. It's the accumulated time between events, not the tick rate of the clocks, that is not reciprocal during inertial motion. ((2))Remember that the distance accumulated during the trip is not the same in each frame, and this factors into the accumulated time on the clocks in addition to their relative tick rates. ((3))Also note that the elapsed time between the ship leaving Earth and the ship reaching the destination would be different in each frame whether the ship accelerates or not.

((1))This is exactly one of the relevant questions I was referring two.

a) The acceleration explanation says that it , nuf said. Broken symmetry . Earth rules.
The end result complies with the accumulated dilation as calculated in Earth frame.


b)The spacetime pathlength analysis ((2)) is based on the same thing and is in effect simply a graphical representation of the acceleration explanation , without reference to simultaneity.

c)The simultaneity +dilation explanation maintains reciprocity but bases the end result predominantly on relative simultanaeity

They may all produce the same end result but raise questions of consistency and what is really responcible for the end result..

((3)) Here they are both inertial so the cummulative result must totally derive from simultaneity.

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.



Judging by this it seems to indicate that G-dilation is exactly equivalent to velocity dilation

Gravitational time dilation isn't "equivalent" to velocity time dilation, it's the same exact phenomenon. All of the gravitational time dilation equations are derived from the lorentz transformations.

I was referring to G-dilation as applied to an accelerating frame.
If you think this what do you think about Rindler coord system? This is not based on felocity dilation and is an implemention of GR gravity in a moving frame.


But this seems strange if G-dilation is constant but instantaneous velocities are varied.
ANy ideas ?

I'm not sure what you mean, but gravitational time dilation isn't necessarily "constant". It's only constant for certain simplified scenarios. It's not constant for this scenario, since even if the acceleration is constant, the coordinate distance of Earth's clock in the ship's frame isn't constant during the acceleration.

As far as I know in a uniformly accelerating system it is. Whats the relevance of the distance of the Earth's clock?

 

[AJ Bentley]

Does the proximate presence of a large mass force a compression on particles and energy to coalesce and form distinct masses such as planets?

Probably not, but the sudden release of tidal energy would most likely be catastrophic. The planet is just a thin skin on a ball of liquid and it has a hard enough time holding shape as it is. Earthquakes are common enough.

As for the 'related' question.
Since we were discussing the possibility of a variation of the time taken for gravitation to travel, I though immediately of the spatial separation. space-time, relativity - No?

 

[Fredrik]

Many people correctly point out that there is no need to address that in order to obtain a correct solution. But the typical question is "how does it work from the ship POV?", not "how can we get a correct solution without analyzing the ship POV?"

I think the first thing those people should be told is that "the" point of view is something that needs to be defined, and that there's no obvious way to do it. The standard synchronization procedure is natural, but is in no way forced upon us. The question suggests that they think that "the" accelerating reference frame is more significant than it really is, so I think it makes more sense to explain why it isn't very significant than to just answer the question directly.

By the way, if the astronaut twin changes direction three times instead of once, the proper reference frame isn't going to cover a large enough region of spacetime to assign coordinates to all events on the stay-at-home twin's world line.

Gravitational time dilation isn't "equivalent" to velocity time dilation, it's the same exact phenomenon. All of the gravitational time dilation equations are derived from the lorentz transformations.

That last thing is true when we're dealing with Minkowski spacetime (i.e. no gravity), but I don't agree with the first one. The "gravitational" time dilation on a rocket that accelerates at 10 m/s² depends on the fact that clocks at the front and clocks at the rear are attached to the same rigid object. This gives them different speeds in the inertial frame where they both started out at rest, and that makes the one in the front tick faster.

 

[stevmg]

Probably not, but the sudden release of tidal energy would most likely be catastrophic. The planet is just a thin skin on a ball of liquid and it has a hard enough time holding shape as it is. Earthquakes are common enough.

As for the 'related' question.
Since we were discussing the possibility of a variation of the time taken for gravitation to travel, I though immediately of the spatial separation. space-time, relativity - No?

Never looked at the Earth "explosion" from the point of view of a sudden release of energy from the Sun. Actually, my question related to the "non-destruction" of the Sun mass (hence, no concomitant release of "mattergy"), however - even if that were to happen, the collapsing spacetime fabric (I hate to use that word "fabric") would create energy much like an implosion of a cathode ray tube (you know, implosion of an old TV picture tube.)

I think, but I am not saying for sure, if there to be no destruction of the Earth from this horrendous event, there would be an eight minute delay in "feeling" it and seeing it. My intuition is based on the constancy of light speed and its relation to energy, mass velocity, etc.

Even if the electric center of the Earth were "off" from the center of mass, I think it would still be within the 4000-mile radius sphere and the basic calculation would not change very much.

 

[MikeLizzi]

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.

Excellent.

If the "standard resolution" you suggest takes acceleration into account I have no problem with it. You don’t actually need to perform a calculation using acceleration. You can use an approximation justified by your understanding of the consequences of acceleration.

You and I (and many others have worked it out that way) Einstein worked it out formally.

I have no issue with making approximations when you know the underlying physics. What I object to is anyone who says the Twins Paradox can be resolved without recourse to the phenomenon of acceleration.

 

[Dale]

What I object to is anyone who says the Twins Paradox can be resolved without recourse to the phenomenon of acceleration.

Since acceleration is a bend in a worldline that is exactly what the geometric approach does.

 

[AJ Bentley]

Since acceleration is a bend in a worldline that is exactly what the geometric approach does.

I'm with Dale.
What matters is the integrated change of velocity, not the rate of change itself.

 

[Al68]

Gravitational time dilation isn't "equivalent" to velocity time dilation, it's the same exact phenomenon. All of the gravitational time dilation equations are derived from the lorentz transformations.

I was referring to G-dilation as applied to an accelerating frame.

So was I.

Whats the relevance of the distance of the Earth's clock?

Because gravitational time dilation is a function of that distance.

 

[Al68]

Many people correctly point out that there is no need to address that in order to obtain a correct solution. But the typical question is "how does it work from the ship POV?", not "how can we get a correct solution without analyzing the ship POV?"

I think the first thing those people should be told is that "the" point of view is something that needs to be defined, and that there's no obvious way to do it. The standard synchronization procedure is natural, but is in no way forced upon us. The question suggests that they think that "the" accelerating reference frame is more significant than it really is, so I think it makes more sense to explain why it isn't very significant than to just answer the question directly.

But the standard resolutions use the standard synch convention, as do virtually all discussions of it. Using the standard synch convention is useful for simple consistency, to avoid comparing apples to oranges, even if it's not forced on us. And importantly here, the standard synch convention is used as a basis for the standard gravitational time dilation equations.

Gravitational time dilation isn't "equivalent" to velocity time dilation, it's the same exact phenomenon. All of the gravitational time dilation equations are derived from the lorentz transformations.

That last thing is true when we're dealing with Minkowski spacetime (i.e. no gravity), but I don't agree with the first one. The "gravitational" time dilation on a rocket that accelerates at 10 m/s² depends on the fact that clocks at the front and clocks at the rear are attached to the same rigid object. This gives them different speeds in the inertial frame where they both started out at rest, and that makes the one in the front tick faster.

Doesn't your explanation substantiate the statement you said you disagree with here? Or am I misreading it?

 

[stevmg]

This post from JesseM was entered earlier in this thread which explains the twins (non) paradox using SR alone. We will just use the Terance inertial, Stella - traveling twin.

No, you can analyze the problem from any inertial frame and all will have the same answer about the age of the inertial twin and the age of the non-inertial twin when they reunite. Let's call the inertial (Earth-bound) twin "Terence" and the traveling twin "Stella", following the Twin Paradox FAQ. First let's look at the numbers in Terence's rest frame. Suppose that in this frame, Stella travels away from Terence inertially at 0.6c for 10 years, at which point she is at a distance of 0.6*10 = 6 light-years from Earth in this frame, then she turns around (i.e. she accelerates, a non-inertial motion which will cause her to experience G-forces that show objectively that she wasn't moving inertially) and heads back towards Terence at 0.6c, finally reuniting with Terence after 20 years have passed since her departure in this frame. Since Terence is at rest in this frame, he has aged 20 years. But since Stella was moving at 0.6c in this frame, the time dilation formula tells us her aging was slowed down by a factor of \sqrt{1 - 0.6^2} = 0.8, so she only aged 0.8*10 = 8 years during the outbound leg of her trip, and another 0.8*10 = during the inbound leg, so she has only aged 16 years between leaving Earth and returning.

By virtue of the "turnaround" that the traveling twin (Stella) does when plotted from the Earthbound twin's (Terence's) frame of reference there is BY DEFINITION a change in the worldline and hence an acceleration as Stella's worldline isn't straight but is composed of two straight "worldlines" meeting at a point. Also, the turn, as depicted, is instantaneous, which would mean an "infinite" deceleration/acceleration.

I really doubt that if one "smooths" this sharp angle out into curve which could be physically tolerated by Stella, that there would be much of a change to merely the summation of the "two" Stella worldlines in terms of proper time.

Is this last and italicized statement true?

stevmg

 

[Fredrik]

But the standard resolutions use the standard synch convention, as do virtually all discussions of it. Using the standard synch convention is useful for simple consistency, to avoid comparing apples to oranges, even if it's not forced on us.

I agree with that. I just think it's much more important to understand that it is a convention than to understand how the twin "paradox" is described in the proper reference frame. (That's the term MTW uses). Also, if the astronaut twin takes a more complicated path, the proper reference frame won't assign coordinates to any events on Earth except the ones close to the departure event and the ones close to the return event. (My statement about this in my previous post was too strong). So for most of the trip, the twin on Earth doesn't have an age in the proper reference frame. I don't mind using this coordinate system, but it certainly shouldn't be described as something you have to do to resolve the paradox.

Doesn't your explanation substantiate the statement you said you disagree with here?

I suppose it's mainly a terminology issue. I haven't seen an exact definition of gravitational time dilation, and all the examples I've seen involve two clocks attached to the same rigid object, or two clocks held at fixed by different "altitudes" in a Schwarzschild spacetime, or something like that. I suppose the details of the twin paradox scenario could be considered "something like that". I guess I just don't like the term "gravitational time dilation" in any context, including GR. It just seems even less meaningful here, where there's no gravity and one of the objects is doing geodesic motion. I prefer to describe things in terms of the axiom I keep mentioning in many of my posts here: A clock measures the proper time of the curve in spacetime that represents its motion.

 

[atyy]

Did Einstein really provide a gravitational time dilation resolution? Doesn't that resolution give the wrong answer (infinite) for the standard infinite acceleration turn around version of the twin paradox?

 

[Aaron_Shaw]

I feel compelled to point out, for anyone who hasn't realized yet , that Mr Bentley is fully aware that there cannot be an actual working ansible. He proposed the idea as a way to make it clear that he is giving his explanations of events having already accounted for, and removed, factors such as time taken to observe events at a distance and doppler effects, etc.

Otherwise you say something like "alice sees bobs clock running slow and such happening at this time" and someone invariably replies, unhelpfully irrelavent given the context of the discussion, "no she won't because of doppler effect and this and that...".

Stop telling him there is no such thing ! Lol.

 

[atyy]

I really doubt that if one "smooths" this sharp angle out into curve which could be physically tolerated by Stella, that there would be much of a change to merely the summation of the "two" Stella worldlines in terms of proper time.

Is this last and italicized statement true?

It is always given by the proper time, no matter how weird the curve, as Fredrik says:

I prefer to describe things in terms of the axiom I keep mentioning in many of my posts here: A clock measures the proper time of the curve in spacetime that represents its motion.

 

[Al68]

I don't mind using this coordinate system, but it certainly shouldn't be described as something you have to do to resolve the paradox.

I agree. I was suggesting it was useful to address the questions of many, not that it's needed to resolve the paradox.

I guess I just don't like the term "gravitational time dilation" in any context, including GR. It just seems even less meaningful here, where there's no gravity and one of the objects is doing geodesic motion.

Even in the standard resolutions with instantaneous turnaround, the Earth clock "jumping ahead" meets the definition of gravitational time dilation. But I agree that the term is misleading.

 

[Al68]

Did Einstein really provide a gravitational time dilation resolution?

Yes. Here's an English translation: http://en.wikisource.org/wiki/Dialog_about_objections_against_the_theory_of_relativity

Doesn't that resolution give the wrong answer (infinite) for the standard infinite acceleration turn around version of the twin paradox?

No, it doesn't show the math for any particular example. But the Earth clock "jumping ahead" in standard resolutions is probably the simplest example of gravitational time dilation possible, and they give the correct answer.

Einstein's resolution just points out that if you consider the ship to be stationary in an accelerated reference frame, the concept of gravitational time dilation accounts for Earth's clock "running fast" in this frame. The actual result obtained would equal the result obtained using an inertial reference frame, since this is how gravitational time dilation equations are derived to begin with.

It's not so much an "alternate resolution" as it is an alternate name (gravitational time dilation) for the same effect when using an accelerated frame.

 

[atyy]

Yes. Here's an English translation: http://en.wikisource.org/wiki/Dialog_about_objections_against_the_theory_of_relativityNo, it doesn't show the math for any particular example. But the Earth clock "jumping ahead" in standard resolutions is probably the simplest example of gravitational time dilation possible.

Yes, I know about that from your earlier posts some years ago - I meant - is the resolution completely correct, ie. we can use it to calculate correct numbers, or is it merely heuristic?

 

[atyy]

Ok, my naive attempt to do gravitational time dilation.

From Wikipedia: T=exp(gh/c²)

Question 1: Why isn't g infinite, since acceleration is infinite?

Question 2: What h do I use? Do I need a definition of simultaneity here to define the spatial separation between the twins at the time corresponding to the infinite acceleration event?

 

[Al68]

Yes, I know about that from your earlier posts some years ago - I meant - is the resolution completely correct, ie. we can use it to calculate correct numbers, or is it merely heuristic?

The resolution doesn't actually provide any means to calculate an answer. It assumes the reader can do that on their own. But it's logically impossible to get a different answer using the accelerated frame than from using an inertial frame, if each calculation is itself correct.

Ok, my naive attempt to do gravitational time dilation.

From Wikipedia: T=exp(gh/c²)

Question 1: Why isn't g infinite, since acceleration is infinite?

Question 2: What h do I use? Do I need a definition of simultaneity here to define the spatial separation between the twins at the time corresponding to the infinite acceleration event?

Sure, the rate of Earth's clock is technically "infinite" if it "jumps" ahead instantaneously. But you can't use that equation that way. That's like trying to use the equation v=at to calculate the relative velocity of the ship after the same instantaneous turnaround with infinite acceleration.

An instantaneous turnaround simplifies the calculations from the ship's accelerated frame as well as in inertial frames, since simple lorentz transformations provide the answer for both.

 

[atyy]

The resolution doesn't actually provide any means to calculate an answer. It assumes the reader can do that on their own. But it's logically impossible to get a different answer using the accelerated frame than from using an inertial frame, if each calculation is itself correct.
Sure, the rate of Earth's clock is technically "infinite" if it "jumps" ahead instantaneously. But you can't use that equation that way. That's like trying to use the equation v=at to calculate the relative velocity of the ship after the same instantaneous turnaround with infinite acceleration.

An instantaneous turnaround simplifies the calculations from the ship's accelerated frame as well as in inertial frames, since simple lorentz transformations provide the answer for both.

OK, I think I understand. Doing the calculation in an accelerated frame doesn't always have the interpretation of gravitational time dilation - it has one only when the metric in the accelerated frame is static. I imagine that no accelerated frame metric in the instantaneous turn around case is static?

 

[Al68]

OK, I think I understand. Doing the calculation in an accelerated frame doesn't always have the interpretation of gravitational time dilation - it has one only when the metric in the accelerated frame is static. I imagine that no accelerated frame metric in the instantaneous turn around case is static?

If I understand your question correctly, time dilation of a distant clock wrt an accelerated frame would still be "gravitational time dilation". But an instantaneous turnaround isn't a real metric, it's a shortcut for a real metric used to avoid the otherwise complicated math.

If we wanted to do the actual calculations, we wouldn't use that shortcut, we would specify an actual acceleration profile instead.

 

[atyy]

If I understand your question correctly, time dilation of a distant clock wrt an accelerated frame would still be "gravitational time dilation". But an instantaneous turnaround isn't a real metric, it's a shortcut for a real metric used to avoid the otherwise complicated math.

If we wanted to do the actual calculations, we wouldn't use that shortcut, we would specify an actual acceleration profile instead.

How about the metric on p8 of http://arxiv.org/PS_cache/gr-qc/pdf/0104/0104077v2.pdf ?

I think it integrates the proper time correctly, but it's not obvious to me that we can extract a gravitational potential from it.

 

[Al68]

How about the metric on p8 of http://arxiv.org/PS_cache/gr-qc/pdf/0104/0104077v2.pdf ?

I think it integrates the proper time correctly, but it's not obvious to me that we can extract a gravitational potential from it.

I'm not sure what to make of it in those defined coordinates, since I only glanced at it, and "radar time" and "radar distance" aren't familiar enough to me. I did notice that the rate of acceleration doesn't appear to be part of the metric in the applicable regions, which might? make sense, if it's a shortcut for a real acceleration profile.

But I'm definitely not the right person to discuss that paper with. Maybe someone much more knowledgeable than I am could address it?

 

[Austin0]

Originally Posted by MikeLizzi
What I object to is anyone who says the Twins Paradox can be resolved without recourse to the phenomenon of acceleration.

Since acceleration is a bend in a worldline that is exactly what the geometric approach does.

Are you saying you think the geometric approach does resolve the question without recourse to the phenomenon of acceleration?

Wouldn't you agree it was true that the worldline of the Earth , as plotted in the accl. frame, purely on the basis of spacetime coordinates (time and position measurements) would be a mirror identical shape. With exactly the same path length and equivalent time?

That the reason this is not reciprocal and is invalid is because the bend in the worldline in the Earth based spacetime diagram indicates acceleration. And it is that interpretaion of this bend and the assumptions regarding the phenomenon of acceleration that is the actual rationale for invalidating the the observations and calculations from the other perspective.

 

[Austin0]

(((1)))Gravitational time dilation isn't "equivalent" to velocity time dilation, it's the same exact phenomenon. All of the gravitational time dilation equations are derived from the lorentz transformations.

I was referring to G-dilation as applied to an accelerating frame.

So was I.

Whats the relevance of the distance of the Earth's clock?

=Al68;2796237 ((2)).Because gravitational time dilation is a function of that distance.

Once again we seem to be talking about two different things.

((2)) Here you seem to be referring to actual gravitational time dilation due to the Earth's mass. An inverse function of the difstance I assume?

I am talking about the psuedo-gravitational time dilation occurring soley within the accelerating frame with zero relevance to Earth's clocks.

((1)) [ABove] What do mean by this??

Gravitational dilation as applied in a static mass does not involve velocity . Conceptually it involves acceleration which everybody has concluded is "real" and is distinctly different from inertial velocity.

Dilation may be calculated for an accelerating system on the basis of ICMF's but that does not make Gravitational time dilation the same phenomenon as velocity time dilation.

Not unless you are going to say that acceleration is the same phenomenon as velocity.

Which I would have no problem with actually.

 

[stevmg]

Again -

Can one achieve a non-instantaneous turn around "slow enough" that it would not kill Stella?

stevmg