Again on twin paradox!!

kknull

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

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.

kknull

well,
in special relativity he cannot return home....
but if you want to use special relativity when the twin is travelling 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

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 musician


btw, 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

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).

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".

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

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

http://www.physicsforums.com/showpos...06&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

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

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

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).

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'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

...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

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:

http://www.physicsforums.com/showpos...7&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

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_Ev...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 realise 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..._of_Relativity