Solving Twin Paradox: Earth's Motion in Astronaut's Frame

[kknull]

Again on twin paradox!

Hi!
I'm trying to definitely solve the twin paradox (after 5 years of efforts :) )

In every physics textbook, it is studied the motion of the twin in an inertial frame (so the proper time is that measured in the twin frame), so we can express time delta t in the Earth frame using:

delta tau = int[ sqrt(1/gamma(v(t)) dt ]

or something like the more general formula
http://arxiv.org/pdf/physics/0411233v1

now I want to, and that's my question, study the Earth motion in the astronaut's frame...
so in this case the proper time is that of earth...

in particular, what happens if the astronaut simply accelerate himself, makes a constant speed trip, turns himself and returns home?

 

[revnaknuma]

for special relativity, when he returns home nothing changes.
for general relativity, when he returns home he will be younger than the people who were as old as him before.

 

[HallsofIvy]

for special relativity, when he returns home nothing changes.l

That doesn't follow. The astronaut has accelerated the people staying on Earth have not so even from the point of view of special relativity, the situation is not symmetric.

for general relativity, when he returns home he will be younger than the people who were as old as him before.

 

[kknull]

well,
in special relativity he cannot return home...
but if you want to use special relativity when the twin is traveling at constant speed, you can use lorentz transformations and say that

delta tau = 1/gamma (v) * delta t. <= delta t
where tau is the (proper) time in the Earth's frame, which can be a problem(?) cause at the end Earth's proper time should be > then astronaut's time...

btw,
I need some math explanation in general relativity... I cannot find anything interesting...

 

[Fredrik]

The twin "paradox" is fully resolved by a calculation of the proper times of the two world lines. (That's how you find out what SR says their final ages will be).

To understand why the reasoning that leads to the apparent contradiction is flawed, you need to study the flawed argument. That doesn't seem to be what you want to do.

You're talking about "the astronaut's frame", and by that you seem to be referring to a specific coordinate system that's associated with the astronaut's motion. It can be a somewhat interesting exercise to work out the details (of what events on Earth this coordinate system says is simultaneous with different events on the astronaut's world line), but I don't think it will add much to your understanding of this problem. It will certainly not help you "solve" it. It might however be good for you to make an effort to understand why that particular coordinate system is chosen (yes, chosen) to represent the "accelerating point of view".

Since there must be hundreds of twin paradox threads already, I think you should try to ask very specific questions. All the general stuff has been covered again and again in the other threads.

Are you Swedish by any chance? (Just curious, because of your name).

Edit: I wrote this post before I saw post #4 above. That's exactly the kind of stuff that has been explained over and over and over and over... Could you please have a look at a few of the other threads?

 

[kknull]

The twin "paradox" is fully resolved by a calculation of the proper times of the two world lines. (That's how you find out what SR says their final ages will be).

To understand why the reasoning that leads to the apparent contradiction is flawed, you need to study the flawed argument. That doesn't seem to be what you want to do.

You're talking about "the astronaut's frame", and by that you seem to be referring to a specific coordinate system that's associated with the astronaut's motion. It can be a somewhat interesting exercise to work out the details (of what events on Earth this coordinate system says is simultaneous with different events on the astronaut's world line), but I don't think it will add much to your understanding of this problem. It will certainly not help you "solve" it. It might however be good for you to make an effort to try to understand why that particular coordinate system is chosen (yes, chosen) to represent the "accelerating point of view".

Since there must be hundreds of twin paradox threads already, I think you should try to ask very specific questions. All the general stuff has been covered again and again in the other threads.

Are you Swedish by any chance? (Just curious, because of your name).

OT: well, althought I can understand swedish a bit, I'm not swedish...
I chose this nick before I knew what it means in swedish/danish/norwegian :)
kknull is a musicianbtw, my specific question is how can I express analitically the Earth motion in the astronaut's frame.. I imagine that it can involve a gravitational field (the inertial force of the acceleration can be erased assuming that there's gravitational field in the opposite direction...)

what is always done is to express astronaut's motion in Earth frame...

and yes, it will solve all my problems...
in particular, I have this doubt:
the effect of the accelaration is finite, whereas the effect of the special relativity's time dilation can be undefinitely big ( the astronaut's can accelarate for few seconds and then travel for 100000000 years... so what is the math explanation of what the Earth space and time looks like from the astronaut's point of view?)
edit: for post #5, ok, I'll take a look... do you have some specific link?

 

[Fredrik]

btw, my specific question is how can I express analitically the Earth motion in the astronaut's frame..

How do you define "the astronaut's frame"? (There's more than one way to choose a coordinate system or a frame field to associate with his motion).

in particular, I have this doubt:
the effect of the accelaration is finite, whereas the effect of the special relativity's time dilation can be undefinitely big ( the astronaut's can accelarate for few seconds and then travel for 100000000 years... so what is the math explanation of what the Earth space and time looks like from the astronaut's point of view?)

The "astronaut's point of view" doesn't have an obviously correct definition. We have to choose something like a coordinate system or a frame field and just agree to call this thing "the astronaut's point of view" for the rest of the discussion. If you look at my first post in the thread linked to below (the one with the spacetime diagram) you will see the explanation that's appropriate when we have chosen to represent the astronaut's point of view, not by a single coordinate system, but by his comoving inertial coordinate systems before and after the acceleration.

Edit: That post also contains a link to an article that gives you the explanation that's appropriate when we have chosen to represent the accelerating point of view by "radar time coordinates".

edit: for post #5, ok, I'll take a look... do you have some specific link?

This is one I contributed to that contains good answers from other people as well.

I will address one of your mistakes in #4 right away. Special relativity is defined by the choice to use Minkowski spacetime as a mathematical representation of space and time, and by a few axioms that tell us how the mathematics correspond to results of experiments. The most important one can be stated like this: "A clock measures the proper time of the curve in spacetime that represents its motion". (This axiom tells us that the final ages can be calculated by doing the proper time integral).

Since there's nothing in that definition that requires the curves that represent motion to be straight lines, SR can definitely handle accelerated motion. GR is defined by a different choice of spacetime, a spacetime with properties that are influenced by the properties of the matter in it. The twin "paradox" is just a SR problem. GR doesn't really have anything to do with it.

 

[JesseM]

well,
in special relativity he cannot return home...

Sure he can, special relativity can deal with acceleration in flat spacetime (the simplest way is just to analyze the accelerated object from the perspective of an inertial frame, but even if you use a non-inertial frame it's still considered part of special relativity as long as spacetime has no intrinsic curvature)

 

[Mike_Fontenot]

I'm trying to definitely solve the twin paradox (after 5 years of efforts :) )
[...]
in particular, what happens if the astronaut simply accelerate himself, makes a constant speed trip, turns himself and returns home?

https://www.physicsforums.com/showpost.php?p=2934906&postcount=7 .

Mike Fontenot

 

[kknull]

I don't think I have understand..
so the problem is that I cannot use a constant inertial frame in which the astronaut is at rest?
can I use a frame in every time interval dt which is inertial and comoving with the astronaut and then study the proper time in Earth frame?

 

[kknull]

https://www.physicsforums.com/showpost.php?p=2934906&postcount=7 .

Mike Fontenot

I think you're assuming always that the astronaut is travelling, so you're calculating traveller proper time (so the motion of the traveller) in function of Earth time...

but I want to calculate Earth proper time in function of astronaut's time...

 

[JesseM]

I don't think I have understand..
so the problem is that I cannot use a constant inertial frame in which the astronaut is at rest?
can I use a frame in every time interval dt which is inertial and comoving with the astronaut and then study the proper time in Earth frame?

If you switch between multiple frames you have to worry about the relativity of simultaneity. Suppose I leave my twin on Earth when we are both aged 20, then I move away inertially at 0.6c for 20 years of my time, then at age 40 I instantaneously accelerate so I am now moving back towards Earth at 0.6c, and I continue inertially for another 20 years, and arrive back at Earth at age 60. It's true that you can analyze the outward journey in inertial frame A where I am at rest during that segment, then analyze the return journey in inertial frame B where I am at rest during that segment. In frame A, the Earth twin is only aging at 0.8 my rate, so he'll age only 16 years in the 20 years in this frame between my leaving Earth and turning around, which means frame A says the moment of my turning around when I am aged 40 is simultaneous with my twin being only age 36. However, frame B disagrees about what event on Earth is simultaneous with the event of my turning around when I am aged 40 (that's the 'relativity of simultaneity'); in frame B, the event of my turning around is simultaneous with the Earth twin being age 54, not 36. Then in frame B it's again true that the Earth twin ages at 0.8 the rate that I age, so in the 20 years in this frame between my turning around and returning to Earth the Earth twin only ages 16 years, meaning he'll be 54+16=70 when I return. So he was aging slower during my return voyage in this frame, but he already had a "head start" at the moment of the turnaround, so that's why he's 70 when I return but I'm only 60.

The relativity of simultaneity is a very important concept in SR, and people's failure to understand it is the source of nearly all confusions about basic conceptual matters in SR, so make sure you review this notion and understand it before trying to tackle the twin paradox!

 

[Fredrik]

can I use a frame in every time interval dt which is inertial and comoving with the astronaut and then study the proper time in Earth frame?

If you break up the world line into short segments, and consider the inertial coordinate system that's comoving with the astronaut at say the middle point of each segment, then the coordinate time increase from the start of the segment to the end of it, is approximately equal to the proper time of that segment. The approximation becomes exact in the limit of segment length →0. (This procedure is equivalent to replacing the integral that defines proper time...use the link in my first post in this thread...with a Riemann sum).

I think you're assuming always that the astronaut is travelling, so you're calculating traveller proper time (so the motion of the traveller) in function of Earth time...

but I want to calculate Earth proper time in function of astronaut's time...

Then you have to define a coordinate system that's defined in a large enough region to include both Earth's world line and the astronaut's world line, and assigns position coordinate x=0 to each event on the astronaut's world line. It also has to assign a time coordinate t(p) to each event p on the astronaut's world line, and this time coordinate has to equal the proper time along the curve from the departure event to p. You might want to have a look at the Dolby & Gull paper. I think their way is the most natural way to do it.

 

[Mike_Fontenot]

I think you're assuming always that the astronaut is travelling, so you're calculating traveller proper time (so the motion of the traveller) in function of Earth time...

I'm telling you how, at each instant of the traveler's life, to determine the current age of the unaccelerated person on Earth (according to the traveler).

Mike Fontenot

 

[Fredrik]

I'm telling you how, at each instant of the traveler's life, to determine the current age of the unaccelerated person on Earth (according to the traveler).

...assuming that we have chosen to represent his point of view by a pair of comoving inertial frames.

There are other ways to define the traveler's point of view. The "radar" coordinates used by Dolby and Gull are at least as natural a choice as the comoving inertial frames.

 

[Mike_Fontenot]

...assuming that we have chosen to represent his point of view by a pair of comoving inertial frames.

There are other ways to define the traveler's point of view. The "radar" coordinates used by Dolby and Gull are at least as natural a choice as the comoving inertial frames.

I don't agree. (Fredrik already knows that I don't agree ... this post is for any readers who haven't seen our past debates on this issue).

For me, there are two "show-stoppers" with Dolby & Gull's simultaneity:

1) D&G is non-causal. If the traveler has never accelerated, before or during his outbound leg, D&G says he CANNOT calculate the current age of the home twin at any instant during his outbound leg, because there is no way for him (or anyone else) to know at that instant if he will actually choose to accelerate in the future.

2) Like ANY of the alternative reference frames (other than mine) for the traveler (frames in which the traveler is perpetually stationary), D&G contradicts the traveler's own first-principle conclusions about the current age of the home twin ... conclusions that he arrives at from his own elementary measurements and elementary calculations. I've described the nature of those measurements and calculations in a previous post:

https://www.physicsforums.com/showpost.php?p=3106767&postcount=38 ,

and the calculations are shown in detail in my paper:

"Accelerated Observers in Special Relativity", PHYSICS ESSAYS, December 1999, p629.

Mike Fontenot

 

[harrylin]

Hi!
I'm trying to definitely solve the twin paradox (after 5 years of efforts :) )

In every physics textbook, it is studied the motion of the twin in an inertial frame (so the proper time is that measured in the twin frame), so we can express time delta t in the Earth frame using:

delta tau = int[ sqrt(1/gamma(v(t)) dt ]

or something like the more general formula
http://arxiv.org/pdf/physics/0411233v1

now I want to, and that's my question, study the Earth motion in the astronaut's frame...
so in this case the proper time is that of earth...

in particular, what happens if the astronaut simply accelerate himself, makes a constant speed trip, turns himself and returns home?

That was already explained before it became a paradox, first by Einstein (clocks) and then by Langevin (astronauts).
- http://www.fourmilab.ch/etexts/einstein/specrel/www/ (section 4 near the end)
- http://en.wikisource.org/wiki/The_Evolution_of_Space_and_Time (from p.48, in particular from p.50)

PS: I had not seen the other posts. You may still see this to realize how old the answers are!

And if you really want to see an answer with a gravitational field approach (messy and truly paradoxical if you ask me!), then you may adventure into Einstein's GRT solution here:
- http://en.wikisource.org/wiki/Dialog_about_objections_against_the_theory_of_relativity

 

[JesseM]

1) D&G is non-causal.

How can a coordinate system be "causal" or "non-causal"? Normally causality is understood in terms of the relation between physical events (like the fact that events can only have a causal effect on events that lie in their future light cone), it has nothing to do with what coordinate labels you attach to those events. If you want to come up with some alternative definition of "causal" involving coordinate systems you need to actually give the definition explicitly. And surely the notion of some coordinate systems being "non-causal" would include the fact that time on Earth can actually run backwards in your coordinate system if the traveler accelerates away from Earth?

2) Like ANY of the alternative reference frames (other than mine) for the traveler (frames in which the traveler is perpetually stationary), D&G contradicts the traveler's own first-principle conclusions about the current age of the home twin ... conclusions that he arrives at from his own elementary measurements and elementary calculations. I've described the nature of those measurements and calculations in a previous post:

https://www.physicsforums.com/showpost.php?p=3106767&postcount=38

This comment doesn't define what you mean by "elementary measurements and elementary calculations". I certainly agree that an observer can be said to be moving inertially on any short segment of his worldline where he isn't accelerating, but that doesn't explain why you think it's "elementary" to combine the results from different inertial reference frames to get conclusions which aren't true in anyone inertial frame, like the conclusion that the Earth twin is aging faster as the traveling twin turns around. What you are really doing is constructing a non-inertial frame whose definition of simultaneity at each point on the traveler's worldline matches the definition of their instantaneous inertial rest frame at that point. That's a perfectly good choice of non-inertial frames, but it's not clear what it means to say that this non-inertial frame somehow is based on "elementary measurements and elementary calculations" while other non-inertial frames aren't.

and the calculations are shown in detail in my paper:

"Accelerated Observers in Special Relativity", PHYSICS ESSAYS, December 1999, p629.

The calculations aren't what people disagree with you about, rather they think you are confusing the issue with your vague ill-defined pronouncements that your definition of simultaneity is somehow more physical or "elementary" than others.

 

[kknull]

ok,
I have read some articles, including D&G. They're very interesting.
In particular, in GR the gravity potential depends on the distance of the twins, so the astronaut can travel for many, many years, but then the effect of the gravity will be proportional. That was the answer.

Btw, is D&G fully explainable in SR? I think we have to assume that the space-time metric is invariant in an accelerated reparametrization, don't we? in this case we still need the PE. Is it correct?

 

[kknull]

ok,
I have read some articles, including D&G. They're very interesting.
In particular, in GR the gravity potential depends on the distance of the twins, so the astronaut can travel for many, many years, but then the effect of the gravity will be proportional. That was the answer.

Btw, is D&G fully explainable in SR? I think we have to assume that the space-time metric is invariant in an accelerated reparametrization, don't we? in this case we still need the PE. Is it correct?

 

[Dale]

1) D&G is non-causal. If the traveler has never accelerated, before or during his outbound leg, D&G says he CANNOT calculate the current age of the home twin at any instant during his outbound leg, because there is no way for him (or anyone else) to know at that instant if he will actually choose to accelerate in the future.

So what? Labelling D&D "non-causal" is just a personal definition indicating your personal distaste and does not impact the physics at all.

2) Like ANY of the alternative reference frames (other than mine) for the traveler (frames in which the traveler is perpetually stationary), D&G contradicts the traveler's own first-principle conclusions about the current age of the home twin ... conclusions that he arrives at from his own elementary measurements and elementary calculations.

This is simply wrong, all coordinate systems yield the same predictions for all measurements. You have never even attempted to substantiate this false claim.

 

[Mike_Fontenot]

In particular, in GR the gravity potential depends on the distance of the twins, so the astronaut can travel for many, many years, but then the effect of the gravity will be proportional.

I recommend that you concentrate on fully understanding the standard SR traveling-twin problem, before any consideration of GR. I.e., limit your attention to what happens for two people in relative motion (with and without velocity changes), when there are no large masses involved ... SR is all you need in that case ... bringing GR into it needlessly complicates the issues.

Mike Fontenot

 

[PAllen]

I don't agree. (Fredrik already knows that I don't agree ... this post is for any readers who haven't seen our past debates on this issue).

For me, there are two "show-stoppers" with Dolby & Gull's simultaneity:

1) D&G is non-causal. If the traveler has never accelerated, before or during his outbound leg, D&G says he CANNOT calculate the current age of the home twin at any instant during his outbound leg, because there is no way for him (or anyone else) to know at that instant if he will actually choose to accelerate in the future. Mike Fontenot

My view of this critique is that D&G says the current age of the home twin is undefined without further assumptions. Once you receive light from the home twin, you then know which age of yours corresponded to it (by D&G). Until a given home twin event is causally connected to you, you can only make statements like: If such happens in the future, then such is the simulataneity relation that applies.

Instead of non-causal, this seems like an appropriate conservative postions: refusal to make statements about the unknowable without coupling them to hypotheses about what happens between now and when you can know about them.

[edit: For example, the home twin can board a rocket, be blown up, Earth could be deflected by a planetoid, etc. Thus pretending you can make statements about the home twin's un-observed age constitutes a set of assumptions about the future from the last point you know about. *Any* statements about the age of the distant twin *now* are coupled to assumptions about what isn't knowable. D&G is no worse or better than many other simultaneity conventions, but actually has the *virtue* of making explicit the assumptions you must make to talk about the unknowable.]